Fermat's Library | Quantum metrology with full and fast quantum control annotated/explained version.

Quantum metrology with full and fast quantum control

Pavel Sekatski

1

, Michalis Skotiniotis

1 2

, Jan Kołodyński

3

, and Wolfgang Dür

1

Institut für Theoretische Physik, Universität Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria

2

Física Teòrica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellatera (Barcelona)

Spain

3

ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain

September 6, 2017

We establish general limits on how precise

a parameter, e.g., frequency or the strength

of a magnetic ﬁeld, can be estimated with the

aid of full and fast quantum control. We con-

sider uncorrelated noisy evolutions of N qubits

and show that fast control allows to fully re-

store the Heisenberg scaling (∼ 1/N

2

) for all

rank-one Pauli noise except dephasing. For

all other types of noise the asymptotic quan-

tum enhancement is unavoidably limited to

a constant-factor improvement over the stan-

dard quantum limit (∼1/N) even when allow-

ing for the full power of fast control. The lat-

ter holds both in the single-shot and inﬁnitely-

many repetitions scenarios. However, even

in this case allowing for fast quantum control

helps to improve the asymptotic constant fac-

tor. Furthermore, for frequency estimation

with ﬁnite resource we show how a parallel

scheme utilizing any ﬁxed number of entangled

qubits but no fast quantum control can be out-

performed by a simple, easily implementable,

sequential scheme which only requires entan-

glement between one sensing and one auxiliary

qubit.

1 Introduction

Precision measurements play a fundamental role in

physics and beyond, as they constitute the main in-

gredient for many state-of-the-art applications and ex-

periments [1]. When investigating the limits of known

or speculative theories experimentally one quickly en-

ters into regimes where quantities and parameters

need to be measured with unprecedented precision.

In this context, it is of utmost importance to know

the ultimate limits nature sets on how precise any

given quantity can be determined and how to achieve

this.

These questions are at the focus of quantum metrol-

ogy [2]. Quantum mechanics is a probabilistic the-

ory and the intrinsically stochastic nature of mea-

surements ultimately limits the achievable precision.

When considering classical probes (or particles) inde-

pendently sensing a physical parameter, such as phase

or frequency, the maximum attainable precision (as

quantiﬁed by the Mean Squared Error—MSE) follows

the standard scaling, 1/N , where N is the number of

probes [3]. In turn it was shown that quantum entan-

glement allows one to achieve the so-called Heisen-

berg scaling (HS) in precision, 1/N

2

, a quadratic im-

provement as compared to classical approaches [4, 5].

These precision limits apply to both single-shot pro-

tocols [6–13] as well as protocols utilizing many rep-

etitions [3, 5, 14]. Still, it remains unclear to what

extent such an improvement can be harnessed in prac-

tice under non-idealized conditions, i.e., when taking

unavoidable noise and imperfections into account [15–

17].

Due to the diﬃculty of obtaining exact precision

limits in the presence of noise, several asymptotic

lower bounds have been established for particular

noise models [18–25]. These lower bounds are of-

ten not only cumbersome to evaluate but also hard

to optimize, relying on educated guesses, numerical

methods employing semi-deﬁnite programming or a

combination of the two. Nevertheless, these bounds

show that for typical uncorrelated noise processes the

possible gain due to the usage of quantum resources

is limited to a constant-factor improvement over the

standard scaling, as opposed to a diﬀerent scaling.

On the other hand, it was shown that for some types

of noise—namely noise perpendicular to the Hamilto-

nian that encodes the parameter of interest—the re-

striction to standard scaling can be circumvented, and

HS can be fully restored, by allowing for additional re-

sources, such as perfectly protected auxiliary particles

used to perform quantum error correction [26–32].

In this work we develop a general framework that

provides us with analytic results for all types of noise

processes described by a time-homogeneous master

equation. In order to establish general limits, we ac-

count for the possibility of ancillary resources that do

not take part in the sensing process together with full

and fast quantum control (FFQC) of the system and

the ancillae

1

. Such FFQC allows one to eﬀectively

modify the Lindblad superoperator that describes the

noise process, a possibility which has hitherto not

1

We remark that we do not allow for additional control over

the environment as has been recently considered in [33, 34].

Accepted in Quantum 2017-09-04, click title to verify 1

arXiv:1603.08944v5 [quant-ph] 5 Sep 2017

been considered

2

. Our approach allows us to identify

all types of qubit noise that can be fully corrected, and

thus restore HS in precision. In addition, for all other

types of uncorrelated noise we are able to provide fully

analytic bounds applicable to the entire hierarchy of

metrology schemes. In contrast to the available meth-

ods [19, 21], our bounds can be generally determined

without need to explicitly solve the system dynamics.

Our main results can be summarized as follows:

(i) FFQC allows to restore HS by completely elimi-

nating any Pauli rank-one noise (that is not par-

allel to the Hamiltonian) at the cost of slowing

down the unitary evolution by a constant factor;

(ii) All other noise processes unavoidably limit the

quantum gain to a constant factor improvement

over standard scaling despite full quantum con-

trol. We obtain analytic bounds for the achiev-

able improvement factor.

(iii) For standard scaling-limited noise processes

FFQC may yet allow for signiﬁcant improvement

of precision in case of limited resources. We pro-

vide explicit examples demonstrating the advan-

tage of FFQC-assisted schemes over ones without

intermediate control for both continuous and dis-

crete processes.

Let us already stress that our results do not per-

tain only to the frequentist approach to parameter

estimation, which assumes an inﬁnite number of pro-

tocol repetitions (i.e., suﬃciently large statistics), but

also apply to the Bayesian approach within which one

considers a ﬁnite statistical data or, in the extreme

case, even a single experimental run. Moreover, the

most general protocol involving FFQC lends itself nat-

urally to adaptive strategies in which one is allowed

to modify the protocol (measurements, control oper-

ations etc.) "on-the-ﬂy"—basing on the record of the

measurements already collected.

The paper is organized as follows. In Sec. 2 we recall

some recent results in quantum metrology, and com-

pare them through a hierarchy of quantum metrol-

ogy schemes accounting for various levels of control.

We show that the FFQC scheme, that tops the hi-

erarchy, is the most powerful one allowed by quan-

tum mechanics. Sec. 3 introduces the model of noisy

quantum processes described by a time-homogeneous

master equation, and reviews the concept of quantum

Fisher information (QFI). In Sec. 4, we review upper

bounds on the QFI for metrology schemes, account-

ing for intermediate control, and generalize them to

incorporate FFQC (Sec. 4.2.1) demonstrating (i) and

(ii). Our extension of the bounds to the limit of short

evolution times is crucial to show (ii), and is the key

to obtain analytic bounds (as it removes the necessity

2

To the best of our knowledge, quantum control has been

used to modify the overall system-environment dynamics but

not the dissipative evolution itself [29, 35].

Preparation

...

Control #1

...

(a)

(c)

(d)

(b)

(c)

(d)

Preparation

...

ancillae

(b)

(a)

ancillae

Control #2

Control #k

Control #1

Control #2

Control #m

Figure 1: Hierarchy of quantum metrology protocols for sens-

ing a frequency-like parameter ω that is encoded on each

probe during its noisy evolution E

ω,t

. Within the standard

schemes of type (a), N probes independently sense the pa-

rameter for a time t. In (b), additionally an unlimited number

of ancillae is allowed to perform a more general measurement

strategy that may include a single error-correcting step at the

end of the protocol. In (c), the sensing process is interspersed

at time intervals, δt, with quantum control operations act-

ing on both the sensing probes and ancillae. Finally in (d),

control operations are further allowed to be of inﬁnitesimally

small duration, i.e, δt → dt, so that it is enough to con-

sider a single probe that sequentially senses the parameter

for the elongated total time t

0

= Nt. As full and fast quan-

tum control (FFQC), applied frequently on the global state,

may constitute swap operations any protocol of type (c) can

be simulated. Hence, the schemes form the following hi-

erarchy: (a)⊆(b)⊆(c)⊆(d), when ordered in terms of their

ultimate power.

to solve system dynamics). In Sec. 5 we discuss the

performance of FFQC-assisted metrology both in the

single shot regime as well as the regime of asymptoti-

cally many repetitions. Sec. 6 considers protocols with

limited resources, where we show that FFQC-assisted

schemes outperform parallel schemes without inter-

mediate control (see (iii)). This is demonstrated for

noisy frequency estimation in Sec. 6.1, and for noisy

phase estimation in Sec. 6.2. We summarize and con-

clude our results in Sec. 7. A reader who is familiar

with quantum metrology, or who is primarily inter-

ested in the results, can directly proceed to sections

4.1 and 4.2.4 where the ﬁrst two of our main results

((i) and (ii)) are presented.

Accepted in Quantum 2017-09-04, click title to verify 2

2 Quantum metrology protocols

In a standard quantum metrology protocol, a system

consisting of N probes (photons, atoms etc.) is care-

fully engineered, so that each probe may be used to

independently sense a parameter of interest ω. As de-

picted in Fig. 1(a), the experimentalist prepares then

a suitably entangled state of the N probes, which un-

dergo the sensing process in parallel for a time t, dur-

ing which the unknown parameter ω is imprinted on

the state of the probes independently. The ﬁnal state

of the system is measured in order to most precisely

retrieve information regarding ω.

If the sensing process is noiseless, the strategy (a)

of Fig. 1 is known to optimally achieve the HS [5].

However, various uncorrelated noise types have been

shown to constrain the precision to follow the stan-

dard scaling in the asymptotic N limit, even if one op-

timally prepares the probes in an entangled but noise

robust state [15, 18, 19, 21].

A more powerful metrological protocol is depicted

in Fig. 1(b). Here, in addition to the N sensing

probes, the experimenter is equipped with ancillary

particles that do not take part in the sensing pro-

cess, and may be used to implement single-step error

correction at the ﬁnal measurement stage [22]. No-

tice that the protocol of Fig. 1(b) is more general

than that of Fig. 1(a); indeed one recovers the lat-

ter by choosing not to entangle the sensing probes

with the ancillae. Consequently any bound on pre-

cision valid for scheme (b) also applies to scenario

(a). Methods have been recently proposed that al-

low to derive bounds on the precision for protocols

(b) from a Kraus representation of a single chan-

nel E

ω,t

[21, 22]. Although these methods allowed

to prove the asymptotic standard scaling for various

noise types, the derived bounds cannot be guaranteed

to be tight (even in the asymptotic N limit). Never-

theless, in case of dephasing and particle-loss noise

types, the corresponding bounds have been shown to

be asymptotically achievable already within scheme

(a) [24, 25, 36]. For amplitude-damping noise, how-

ever, scheme (b) has been shown to give a strictly

better scaling than scheme Fig. 1(a) [37].

An interesting situation is the case of the X-noise: a

noise similar to dephasing but with the generator per-

fectly transversal to the Hamiltonian encoding the pa-

rameter. For a ﬁxed sensing time t the bounds [19, 21]

impose asymptotic standard scaling. However, in fre-

quency estimation one strongly beneﬁts from decreas-

ing the sensing time while increasing the number of

probes N. Indeed, by optimizing the sensing time

of the protocols (a) and (b) for each given N, it was

shown that one may asymptotically beat the standard

scaling and achieve the 1/N

5/3

precision scaling [38].

Moreover, it has been shown that one can even re-

store the HS of precision for such strictly transversal

noise after allowing for fast possibly multi-step error

correction [27–30]. This is a particular case of the

metrological scenario depicted in Fig. 1(c). Here, in

addition to employing ancillary particles, the exper-

imenter is capable of freely interjecting the sensing

process with m control pulses that may act on both

probes and ancillae representing, e.g. error correction

steps [27–30], dynamical-decoupling pulses [39–44] or

any general adaptive feedback scheme [45, 46]. No-

tice that protocol (c) is more general than Fig. 1(b)

and one can obtain the latter from the former sim-

ply by allowing the intermediate operations to be the

identity.

The most general and powerful metrology protocol,

and the main focus of the current work, is the one de-

picted in Fig. 1(d). Here, the experimenter prepares

a suitably entangled state between a single probe and

many ancillae, and is capable of frequently interject-

ing the evolution with FFQC—an arbitrary number,

k, of most general intermediate control operations act-

ing on the overall state of probe-plus-ancillae. More-

over, by choosing k suﬃciently large one can ensure

the sensing time spent by the probe in between suc-

cessive FFQC steps to be inﬁnitesimal. It is this ad-

ditional power of frequently interjecting the sensing

process with fast quantum control that we exploit

throughout the remainder of this work to establish

the ultimate bounds on precision. This intermediate

control allows one in fact to modify the noise process,

a possibility that has not been considered in previous

approaches.

To see that this is indeed the most general strategy,

we note that for any protocol (c), with N parallel

probes and total sensing time t, there exists a protocol

of type (d) with total sensing time t

0

= Nt whose

metrological performance is just as good as that of

(c) or better. We note in passing that protocols (c)

and (d) are not equivalent; not only are the timesteps

in protocol (d) assumed to be inﬁnitesimally small,

but protocol (d) allows us to use the same resources

(namely N and t) in a "sequentialized" fashion.

The above metrological schemes apply equally to

the case of Bayesian parameter estimation, i.e., when

considering only a ﬁnite number of experimental rep-

etitions and a prior probability p(ω) representing our

knowledge about the parameter ω. Examples of

such Bayesian scenarios include reference frame align-

ment [9–12], adaptive Bayesian estimation [7, 13, 47],

and single-shot estimation [48, 49].

2.0.1 A remark on error correction

Let us brieﬂy remark that the most general

error correction step (encoding+syndrome read-

out+correction) only requires the dimension of the

space describing the ancillae to be the same as the

system one. Indeed, given that the system containing

the sensing probes has an overall dimension d that is

smaller than the dimension of the ancillary space, any

bipartite pure state of the two can be written using the

Accepted in Quantum 2017-09-04, click title to verify 3

Schmidt decomposition as

P

d

i=1

λ

i

|ii

S

⊗|ψ

i

A

. Here,

{|ii}

d

i=1

denotes a given ﬁxed basis for the system,

while {|ψ

i

i}

d

i=1

is a set of orthonormal states of ancil-

lae that spans a d-dimensional subspace. Thus, since

any error correction protocol may be implemented by

restricting to states of the above form, it suﬃces to

choose the overall space describing the ancillae to be

of same dimension as the system.

3 Background

In order to assess the performance of any of the proto-

cols introduced in Sec. 2, we review below the crucial

tool of quantum metrology—the QFI. In addition, in

Sec. 3.2 we describe the general time-homogeneous

qubit noise processes we consider throughout this

work.

Notation. Before doing so, let us ﬁrst introduce the

notation we utilize. In what follows, we denote the

derivative with respect to the estimated parameter ω

(whenever applied to operators, vectors or matrices;

diﬀerentiating adequately the entries) as

˙

•:=

d•

dω

; and

shorten the trigonometric functions to c

θ

:=cos(θ) and

s

θ

:= sin(θ). ||A|| := max

ψ

|hψ|A |ψi| stands for the

operator norm, whereas and I for the identity oper-

ator and identity linear map respectively. We use bold

face to denote vectors, so that r := (x, y, z)

T

repre-

sents a 3-dimensional complex vector, while x, y and

z the Cartesian unit vectors. We denote the standard

Pauli 4- and 3-vectors as ς := (σ

0

, σ

1

, σ

2

, σ

3

)

T

and

σ := (σ

1

, σ

2

, σ

3

)

T

respectively (σ

0

:= ). The scalar

and outer products of vectors are then deﬁned in the

usual way irrespectively of the type of vector entries,

i.e., v

†

v

0

:=

P

i

v

†

i

v

0

i

and [v

0

v

†

]

ij

:= v

0

i

v

†

j

. We write

σ

n

:= n

T

σ to denote a Pauli operator in the spa-

tial direction n = (s

θ

c

φ

, s

θ

s

φ

, c

θ

)

T

. We reserve the

special font, M, for matrices whose entries, M

ij

, are

(complex) numbers, so that the multiplication of all

types of vectors can then be deﬁned in the standard

way: Mv =(

P

j

M

1j

v

j

,

P

j

M

2j

v

j

, . . . )

T

. For example,

any linear qubit map, L, can be unambiguously repre-

sented by constructing its corresponding matrix L (in

the Pauli basis) such that L =ς

†

Lς =

P

3

µ,ν=0

σ

µ

L

µν

σ

ν

.

3.1 QFI and its role in quantum metrology

In all metrology schemes discussed in Sec. 2 the in-

formation about the estimated parameter of interest

is encoded in the ﬁnal state of the system consisting

generally of both probes and ancillae. Thus, we de-

note the ﬁnal state of the system in any of the schemes

(a,b,c,d) of Fig. 1 as %

ω

. It is then %

ω

that is mea-

sured, in order to most precisely construct an estimate

of the parameter of interest, ω.

How best to quantify the precision of this estimate

depends crucially on how the information about ω has

been collected. If we are freely allowed to repeat the

protocol suﬃciently many times and collect statistics,

then the best precision of estimation, as quantiﬁed by

the mean squared error (MSE), is determined by the

well-known Cramér-Rao bound (CRB) [50, 51]

δ

2

ω ≥

1

νF(%

ω

)

, (1)

where F(%

ω

) is the QFI of the state and ν is the num-

ber of repetitions. On the other hand, if the informa-

tion about ω is inferred from the outcome of just a

single repetition of the protocol, then the single-shot

MSE has to be averaged over the prior distribution

representing the knowledge we possess about ω prior

to its estimation. Then, the average mean squared

error (AvMSE ) can be lower-bounded by either the

Bayesian Cramér-Rao bound (BCRB) [49] (for well

behaved prior distributions) or the Ziv-Zakai bound

(ZZB) [48].

Whether considering the CRB, BCRB, or

ZZB, the crucial quantity of interest is the

QFI: F(%

ω

). Given the spectral decomposition

%

ω

=

P

i

p

i

(ω) |ψ

i

(ω)ihψ

i

(ω)|, the QFI of the state %

ω

evaluated with respect to the estimated parameter ω

generally reads

F(%

ω

) := 2

X

i,j

p

j

+p

j

6=0

1

p

i

(ω) + p

j

(ω)

|hψ

i

(ω)| ˙%

ω

|ψ

j

(ω)i|

2

.

(2)

On the other hand, the QFI may be equivalently de-

ﬁned by using its relation with the ﬁdelity between

quantum states [52], F (ρ, σ):=tr

p

√

σρ

√

σ:

F(%

ω

) := 8 lim

dω→0

1 − F (%

ω

, %

ω+dω

)

dω

2

. (3)

The QFI satisﬁes the following important proper-

ties. It is additive, so that if the ﬁnal state of the

system is a product state, i.e., %

ω

= ρ

⊗N

ω

, then its

QFI satisﬁes F(ρ

⊗N

ω

)= NF(ρ

ω

) and thus gives rise to

standard scaling [3]. If the ﬁnal state of the probes is

pure, %

ω

=|ψ

ω

ihψ

ω

|, then

F(%

ω

) = 4





˙

ψ

ω

|

˙

ψ

ω



−





˙

ψ

ω

|ψ

ω





2



. (4)

Moreover, if the parameter is also encoded via

a unitary, so that |ψ

ω

i = e

−iωH

|ψi where H is

some Hamiltonian, then F(%

ω

) = 4(hψ|H

2

|ψi −

hψ|H |ψi

2

) =: 4 Var

|ψi

(H). In this case the QFI

is maximized by preparing the probes in the state

|ψi =

1

√

2

(|λ

max

i+|λ

min

i), where



λ

max/min



are the

eigenstates corresponding to the maximum/minimum

eigenvalues of H. Furthermore, if the N probes are

subjected to a unitary evolution generated by a lo-

cal Hamiltonian, e.g., H =

1

2

P

N

n=0

σ

(n)

3

, then such a

state corresponds to the GHZ state and attains the

HS by virtue of the CRB (1) [3]. However, note that

the QFI is important not only for metrology but also

for entanglement detection [53–55]; deﬁnes a natural

Accepted in Quantum 2017-09-04, click title to verify 4

geometric distance between quantum states [56], thus

serving as a tool to derive speed limits on quantum

evolution [56, 57], as well as constitutes a measure of

macroscopicity [58]. The QFI is the primary focus of

this work and we further discuss its role in quantify-

ing the precision of estimation in metrology protocols

in more detail in Sec. 5.

3.2 Time-homogeneous qubit evolution

The interaction dynamics via which a probe senses

the parameter is generally described by a quantum

channel [59] (see Fig. 1), E

ω,t

, that encodes the pa-

rameter ω onto the state of the probe over the inter-

rogation time t: ρ

ω

(t)=E

ω,t

(ρ). The channel consists

of a unitary evolution part, encoding the parameter

of interest, and a non-unitary (noise) part describ-

ing additional system-environment interactions. In

this work we shall consider that the non-unitary part

arises due to uncorrelated noise processes described

by a time-homogeneous master equation of Lindblad

form—often referred to as the semigroup dynamics

[60]—which provides an appropriate description for

most physically relevant noise processes. In particu-

lar, in such a typical setting the environment is as-

sumed to be suﬃciently large such that it disturbs

the system in the same fashion independently of the

system state and the time instance [61]. Thus, the

time-homogeneity of evolution allows us to unambigu-

ously apply the FFQC techniques as any operation

applied on the system does not aﬀect the environ-

ment. Moreover, one may then freely swap the sensing

particles at any time with the ancillary ones, which

experience afterwards the same ﬁxed noisy dynam-

ics. In contrast, noise processes described by time-

inhomogeneous master equations have been recently

considered within the context of quantum metrology

in [62–65], where time-inhomogeneity has been shown

to be beneﬁcial at short time-scales— the so-called

Zeno regime—where the bath can no longer be as-

sumed to be uncorrelated from the system. Let us

stress that such a regime, however, does not allow for

the general control operations considered here to be

unambiguously applied without explicitly modelling

the environment [35, 66].

We describe the evolution of a single qubit probe

by following time-homogeneous master equation

dρ

ω

(t)

dt

= −i

ω

2

[σ

3

, ρ

ω

(t)] + L(ρ

ω

(t)), (5)

where ω is the parameter to be estimated and the

noise is given by the Lindblad super-operator, L. The

Liouvillian is then generally deﬁned as

L(ρ) :=

1

2

3

X

µ,ν=0

L

µν

([σ

µ

ρ, σ

ν

] + [σ

µ

, ρσ

ν

]) , (6)

where L is the matrix representation of the Lindblad

superoperator – an Hermitian positive semi-deﬁnite

matrix whose entries are independent of t and ω.

We note that all the noise terms L

0i

and L

i0

can be

straightforwardly corrected within FFQC as they lead

to an additional Hamiltonian evolution term in the

Eq. (5), sometimes referred to as Lamb-shift. This

ω-independent Hamiltonian term can be cancelled by

continuously applying the inverse unitary rotation. In

what follows we assume that the control operations

already incorporate such unitary and will, thus, be

concerned with the restriction of L to the subspace

spanned by {σ

1

, σ

2

, σ

3

}, which we denote as

¯

L.

We can group all the relevant noise processes into

three important families:

• Rank-one Pauli noise of strength γ

L

1P

n

(ρ) :=

γ

2

(σ

n

ρσ

n

− ρ), (7)

with the particular case of dephasing noise L

1P

z

for

which σ

z

= σ

3

.

• General rank-one noise L

1G

r

deﬁned by the matrix

¯

L

1G

r

:= r r

†

, (8)

where r = (x, y, z)

T

and the strength of the noise

is denoted γ/2 = |r|

2

. The special case Re(r) ×

Im(r) = 0 corresponds to the rank-one Pauli noise

of Eq. (7), whereas whenever |Re(r)|= |Im(r)| and

Re(r)

T

Im(r) = 0 an amplitude damping channel is

recovered, which represents spontaneous emission

along some particular direction.

• Rank-two Pauli noise L

2P

Ω

deﬁned by the matrix

¯

L

2P

Ω

:=

1

2

R

T

Ω

γ

1

γ

2

0

!

R

Ω

, (9)

where Ω ∈ SO(3) and R

Ω

= R

z

(ϕ)R

y

(θ)R

z

(ξ) is

its matrix representation written in the Euler form

with angles (ϕ, θ, ξ). The special case of R

Ω

= cor-

responds to asymmetric X-Y noise, which we con-

veniently parametrize with γ

1

=γ p and γ

2

=γ(1−p)

for 0≤p≤1 to deﬁne

L

X-Y

(ρ) :=

γ

2

(p σ

1

ρ σ

1

+ (1 − p)σ

2

ρ σ

2

− ρ).(10)

In order to explicitly determine the form of the

quantum channel describing the qubit evolution, one

has to integrate Eq. (5) [67]. The resulting dynamics

can then be expressed, e.g, with help of the corre-

sponding dynamical matrix, S (speciﬁed in the Pauli

operator basis), or via a Kraus representation as fol-

lows:

E

ω,t

(ρ) =

3

X

µ,ν=0

S

µν

(ω, t) σ

µ

ρ σ

ν

(11)

=

X

i

K

i

(ω, t) ρ K

i

(ω, t)

†

. (12)

Note that the Kraus representation of Eq. (12) is not

unique as starting from a set of r linearly independent

Accepted in Quantum 2017-09-04, click title to verify 5

Kraus operators, K = (K

1

, . . . , K

r

)

T

, we may simply

construct another valid set K

0

=uK choosing u to be

any (potentially ω-,t- dependent) r×r unitary matrix.

4 Noisy metrology with full and fast

control

In this section we derive the main results of our work.

In Sec. 4.1 we outline the optimal FFQC proto-

col suitable for correcting rank-one Pauli noise that

is not parallel to the Hamiltonian (7). In Sec. 4.2

we introduce the necessary tools required to bound

the QFI for FFQC (Fig. 1(d)), and show that any

other noise unavoidably leads to a linear scaling of the

QFI. We exemplify our ﬁndings by deriving analyti-

cal bounds for the QFI for several physically relevant

noise-types. Finally, in Sec. 4.3 we discuss diﬀerent

aspects in which FFQC allows to outperform scenar-

ios without fast control, using the cases of X-Y noise

(7) and transversal rank-one Pauli noise (40c) as ex-

amples.

4.1 Removing rank-one Pauli noise

Here we construct a general FFQC strategy that al-

lows one to correct for any rank-one Pauli noise L

1P

n

introduced in Eq. (7) with n 6= z. For convenience,

we perform a change of basis so that the noise is gen-

erated by σ

1

(n=x), whereas ω is encoded via a ro-

tated Hamiltonian σ

θ

:= s

θ

σ

3

+ c

θ

σ

1

. We consider

the qubit probe to be aided by an ancillary qubit

and deﬁne the two-qubit code space as a subset of

H

S

⊗ H

A

⊃ H

C

:= span{|00i, |11i}, with correspond-

ing projector Π

C

:= |00ih00| + |11ih11|, and the error

space, H

E

:=span{|01i, |10i}, with Π

E

:= − Π

C

.

Let the probe-plus-ancilla be prepared in a pure

state % = |ψihψ| with |ψi= α |00i + β |11i ∈ H

C

. The

dynamics on the probe-plus-ancilla is then described

by Eq. (7) with only the probe system evolving

d%

dt

= −i

ω

2

[σ

θ

⊗ , %] + L

1P

x

⊗ I

A

(%). (13)

Integrating Eq. (13) over an elementary timestep dt,

we may write the probe-plus-ancilla state, up to ﬁrst

order in dt, as

%(dt) = %(1 −

γ

2

dt) − i

ω

2

[σ

θ

⊗ , %]dt (14)

+

γ

2

(σ

1

⊗ )%(σ

1

⊗ )dt + O(dt

2

).

Projecting %(dt) onto the code and error subspaces

yields the unnormalized states %

C

(dt) = Π

C

%(dt)Π

C

and %

E

(dt)= Π

E

%(dt)Π

E

given by

%

C

(dt) = %(1 −

γ

2

dt) − i s

θ

ω

2

[σ

3

⊗ , %]dt + O(dt

2

),

%

E

(dt) =

γ

2

(σ

1

⊗ )%(σ

1

⊗ ) dt + O(dt

2

) (15)

respectively.

If an error is detected, we simply apply σ

1

on the

sensing probe (so after correction ¯%

E

(dt) := (σ

1

⊗

)%

E

(dt)(σ

1

⊗ )), and otherwise do nothing. Hence,

the state of probe-plus-ancilla after an inﬁnitesimal-

timestep followed by fast error correction is the mix-

ture

¯%(dt) := %

C

(dt) + ¯%

E

(dt)

= % − i s

θ

ω

2

[σ

3

⊗ , %] dt + O(dt

2

), (16)

which, to ﬁrst order in dt, is equivalent to a unitary

evolution under the projected Hamiltonian

ω

2

s

θ

σ

3

. As

the measurement may always be adjusted to compen-

sate for a known rotation of the Hamiltonian encoding

the parameter, the above strategy perfectly corrects

rank-one Pauli noise (7) at the price of slowing the

evolution down by a factor s

θ

=

p

1 − (n

T

z)

2

.

We summarize the above result in the following ob-

servation:

Result 1. A general FFQC strategy allows one to

completely eliminate the impact of any rank-one Pauli

noise L

1P

n

(7) that is not exactly parallel to the

parameter-encoding Hamiltonian. The resulting dy-

namics of the system are then described by a noise-

less evolution with a rotated Hamiltonian and the es-

timated parameter being rescaled to

p

1 − (n

T

z)

2

ω,

which still yields HS in precision.

Note that in the above derivation nothing forbids us

from replacing the scalar parameter ω with any opera-

tor B acting on an additional system described by the

Hilbert space H

B

. Consequently, one can eﬀectively

modify the noisy dynamics of any %

BS

∈ B(H

B

⊗H

S

):

d%

BS

dt

= −i

1

2

[B ⊗ σ

θ

, %

BS

] + I

B

⊗L

1P

x

(%

BS

), (17)

by adding an ancillary qubit and implementing the

FFQC strategy described above. Then, the dynamics

of the error-corrected ¯%

BS

= tr

A

{¯%

BSA

} is governed

by

d¯%

BS

dt

= −i

p

1 − (n

T

z)

2

[B ⊗ σ

3

, ¯%

BS

], (18)

with the rank-one Pauli noise removed. Hence, our

strategy can be directly used for the implementation

of unitary gates U = e

ig B⊗σ

3

in the presence of rank-

one Pauli noise on the qubit.

4.2 Other noise-types: unavoidable linear scal-

ing

In this subsection we prove our main no-go result,

F

L

≤ 4 α

L

t

0

, showing that even with FFQC the QFI

is bound to a linear scaling for all uncorrelated noise-

types other than rank-one Pauli noise, and to min-

imize the constant α

L

appearing in this bound for

several practically relevant noise-types introduced in

Accepted in Quantum 2017-09-04, click title to verify 6

Sec. 3.2. These results are presented in Sec. 4.2.4

and Sec. 4.2.5 respectively. The preceding sections

are devoted to elaborate the method that allows us to

derive these results. In Sec. 4.2.1 and Sec. 4.2.2 we

describe the channel extension (CE) method used to

upper bound the QFI [18, 21, 22, 37] and then review

the bounds it yields for strategies (a), (b) and (c) of

Fig. 1. However, the CE method fails to yield a mean-

ingful bound when considering the limit in which the

sensing process lasts for an inﬁnitesimal time-step. In

Sec. 4.2.3 we resolve this issue by amending the CE

method in order to obtain an upper bound on the QFI

for strategies that employ FFQC.

We stress that we are able to derive analytic bounds

by extending the CE method to the limit dt → 0,

in which the problem actually simpliﬁes: one does

not have to solve the master equation and determine

the full dynamics! This is even more remarkable as

these bounds apply to any metrological scheme (see

the hierarchy in Fig. 1)

4.2.1 Upper bounds on the QFI

We would like to compare the maximum achievable

QFI in the various metrology schemes introduced in

Sec. 2 and, in particular, quantify the improvement (if

any) between protocols (b), (c) and (d). To this end

we use variations of the CE method [18, 21, 22, 37]

that allow one to upper-bound the QFI depending

on the scenario considered. Crucially, they constrain

the QFI using solely the properties of the quantum

channel responsible for encoding the parameter.

CE bound. The QFI of a state representing the

output of a quantum channel encoding the parameter

may always be upper-bounded purely by the channel

properties after performing a channel extension (CE).

Given a probe in a state ρ and a quantum channel E

ω,t

one may always construct the CE bound on the QFI

of the output state E

ω,t

(ρ) [18]:

F(E

ω,t

(ρ)) ≤ max

%

F(E

ω,t

⊗ I

A

(%)) ≤ 4 ||α(t)||,

(19)

where %∈B(H

S

⊗H

A

) represents a joined (extended)

state of the probe and an ancilla, and the operator

α(t) :=

X

i

˙

K

†

i

˙

K

i

=

˙

K

†

˙

K (20)

for any of the Kraus representation Eq. (12) of the

probe channel. Here,

˙

K

i

are the derivatives of the

Kraus operators with respect to ω (in what follows,

we write explicit dependencies of the Kraus operators

on t and ω only if necessary)

Although the ﬁrst inequality in Eq. (19) may not

be tight—entangling the probe and ancilla may in-

crease the achievable QFI—the second is. A valid

Kraus representation of E

ω,t

for which the second in-

equality is saturated is guaranteed to exist as long

as dim(H

A

) ≥dim(H

S

) [18]. In consequence, the CE

bound (19) directly determines the maximal attain-

able QFI for the ancilla-assisted parallel scheme de-

picted in Fig. 1(b) with N =1 [22].

Parallel CE bound. The CE bound (19) may be

directly applied to upper-bound the ultimate attain-

able QFI of any parallel scheme of Fig. 1(b) that em-

ploys N probes and ancillae. In scenario (b), the ﬁnal

state generally reads %

(b)

N

(t)= E

⊗N

ω,t

⊗I

A

(%

(b)

N

). Hence,

applying the CE bound (19) to channel E

⊗N

ω,t

⊗ I

A

that describes the overall evolution and ﬁxing a ten-

sor product Kraus representation such that

E

⊗N

ω,t

(ρ

N

) =

X

i

1

,...,i

N

K

i

1

⊗...⊗K

i

N

ρ

N

K

†

i

1

⊗...⊗K

†

i

N

, (21)

one obtains the parallel CE bound [18, 21, 22]:

F



%

(b)

N

(t)



≤ max

%

(b)

N

F



E

⊗N

ω,t

⊗ I

A

(%

(b)

N

)



≤ 4N||α(t)|| + 4N(N − 1)||β(t)||

2

, (22)

where %

(b)

N

∈ B(H

⊗N

S

⊗ H

A

) describes the initial

probes-plus-ancillae state and

β(t) := i

X

i

˙

K

†

i

K

i

= i

˙

K

†

K (23)

which depends solely on Kraus operators and their

derivatives. In order to obtain Eq. (22) from Eq. (19)

note that due to the tensor product structure the term

˙

K

†

i

˙

K

i

=

d

dω

(K

†

i

1

⊗ ... ⊗ K

†

i

N

)

d

dω

(K

i

1

⊗ ... ⊗ K

i

N

) in

Eq. (19) contains N terms for which in both parenthe-

sis the derivative applies on the Kraus operator acting

on the same probe and N(N − 1) terms where it ap-

plies on the Kraus operators acting on two diﬀerent

probes [18]. Yet, in contrast to Eq. (19), the par-

allel CE bound (22) is then not generally attainable,

even after optimizing Eq. (22) over all Kraus represen-

tations of the single-probe channel. This is because

the Kraus decomposition for the map E

⊗N

ω,t

⊗I

A

that

makes Eq. (19) saturable might not be of a tensor-

product form, as assumed in Eq. (21).

Sequential CE bound. Now, we would like to ap-

ply the CE methods to the two strategies involving

control, i.e., (c) and (d) of Fig. 1. However, as argued

in Sec. 2, any scheme of type (c) employing N probes

with m control steps (each lasting δt) can always be

mimicked by a scheme Fig. 1(d) employing a single

probe but lasting N-times longer, i.e., with t

0

= Nt

and involving more k = N m steps each still of dura-

tion δt. Hence, we may always upper-bound the QFI

of any scheme depicted in Fig. 1(c) as

F



%

(c)

N,m

(t)



≤ max

schemes (d)

k=N m, t

0

=Nt

F



%

(d)

k

(t

0

)



(24)

Accepted in Quantum 2017-09-04, click title to verify 7

where the maximization is over all sequential schemes

of type (d) with step duration δt = N/m. Thus, in

what follows, we focus on applying the CE methods

to the scenario of Fig. 1(d). Note that by decreasing

δt further, or equivalently by raising the number of

control steps k, we may only increase the right hand

side of Eq. (24). Hence, the optimal FFQC scheme

yielding the maximum in Eq. (24) must correspond

to the limit of k → ∞, or equivalently, δt → 0. It

is so, as any FFQC scheme of step duration δt can

be mimicked by a protocol with shorter δt

0

= δt/k

(and any k ≥ 2) and setting some of the intermediate

control operations to be trivial, i.e., the identity.

For sequential strategies employing intermediate

control a CE-based upper bound on the correspond-

ing QFI has recently been derived in Ref. [37]. Let the

output state for the sequential protocol of Fig. 1(d)

be given as

%

(d)

k

(t

0

) =

X

i

(k)

K

i

(k)

(t

0

) %

(d)

K

†

i

(k)

(t

0

) (25)

with %

(d)

∈B(H

S

⊗ H

A

) being the initial state. How-

ever, in contrast to Eq. (21), the overall Kraus oper-

ators

K

i

(k)

(t):= (K

i

k

(δt)⊗

A

)U

k

. . . (K

i

1

(δt)⊗

A

)U

1

(26)

have a composition rather than tensor-product struc-

ture, where U

`

stands for the a control operation act-

ing on both probe and ancillae at the `th step , while

K

i

`

are the Kraus operators of the channel E

ω,δt

act-

ing solely on the probe at the `th step. An application

of the general CE bound (19) to this case was derived

in Ref. [37], leading to the sequential CE bound on

the QFI attainable using any sequential scheme with

FFQC (with timestep δt):

F



%

(d)

k

(t

0

)



≤ 4k||α(δt)||+ (27)

+ 4k(k − 1)||β(δt)||(||α(δt)|| + ||β(δt)|| + 1),

which applies to any protocol (d) of Fig. 1 with time-

steps of duration δt, and hence also to all schemes (c)

of Fig. 1 upon substituting k =N m.

Notice that, just as in the case of the parallel CE

bound (22), there may not exist a Kraus representa-

tion of the single-probe channel such that the sequen-

tial CE bound of Eq. (27) is guaranteed to be tight.

This holds also for the optimal sequential strategy (d)

that maximizes the corresponding QFI (i.e., the one

maximizing the right hand side of Eq. (24)).

4.2.2 Optimization of the CE bounds

All the CE bounds presented this far: stan-

dard (Eq. (19)), parallel (Eq. (22)) and sequential

(Eq. (27)); rely solely on the structure of the single-

probe channel E

ω,t

encoding the parameter and its

particular Kraus representation (12). Thus, in order

to obtain the tightest versions of these bounds one

should minimize them over all equivalent Kraus rep-

resentations, K

0

= uK (with u

†

u = ), describing the

single-probe dynamics. Although such a gauge free-

dom is generally parameter dependent, i.e., u = u(ω),

the operators α (20) and β (23) depend solely on

h := i u

†

˙

u. Hence, it always suﬃces to search only

through Kraus representations satisfying K

0

=K and

˙

K

0

=

˙

K−ihK or, in other words, any CE-based bound

considered may always be optimized by simply per-

forming its minimization over all Hermitian matrices

h [18].

In contrast to the sequential CE bound (27),

the standard (19) and parallel (22) CE bounds are

quadratic (convex) in h, and thus may be minimized

numerically in a systematic manner by means of semi-

deﬁnite programming (SDP) given some K [21, 22].

On the other hand, when considering t (or δt in case

of Eq. (27)) to be ﬁxed and the asymptotic limit of

N (22) (N m (24), or k (27)), it is always optimal, if

possible (contrary to the unitary case), to set β(t)=0

in the CE bounds, so that they asymptotically be-

come linear in N (Nm, or k respectively). In such a

regime, as the constraint β(t) = 0 is linear in h, one

may always apply the SDP methods to minimize the

remaining coeﬃcient 4||α(t)||, which is then emergent

in all the bounds [21, 22].

Equivalently, the same argumentation can be made

starting from the dynamical matrix S of the channel

E

ω,t

, deﬁned in Eq. (11), instead of its Kraus represen-

tation. The two are directly related (see also App. A),

as any valid vector of Kraus operators K can always

be expressed in a complete operator basis, e.g. the

Pauli basis in case of a single-qubit maps, via a ma-

trix M satisfying

K = Mς and M

†

M = S

T

. (28)

As a result, one may rewrite the operators α (20)

and β (23) with help of the matrix M and a general

Hermitian matrix h to be optimized as

α(t) = ς

†

(

˙

M

†

+ iM

†

h) (

˙

M − ihM) ς , (29)

β(t) = ς

†

(

˙

M

†

+ iM

†

h) M ς . (30)

In what follows, we always choose M = M

†

=

√

S

T

and thus unambiguously ﬁx the starting Kraus rep-

resentation K in Eq. (28). We then make use of

Eqs. (29) and (30) in order to analytically optimize

over h the inﬁnitesimal-timestep version of the sequen-

tial CE bound (27) as we now explain.

4.2.3 Inﬁnitesimal-timestep CE bound

In the most powerful FFQC setting of Fig. 1(d), one

allows for the control operations to be arbitrarily fast

and thus assumes the steps of the protocol to last an

inﬁnitesimal duration dt. Hence, one deals then with

a sequential strategy of inﬁnitesimal step duration, for

which the sequential bound of Eq. (27) can be directly

Accepted in Quantum 2017-09-04, click title to verify 8

applied after setting k →

t

0

dt

and taking the limit δt ≡

dt→0 (see App. B). In order for the bound to remain

meaningful, we adjust the derivation of Ref. [37] in

App. B, so that it now reads:

F



%

(d)

t

0

/dt

(t

0

)



≤ 4

t

0

dt

||α(dt)|| (31)

+4



t

0

dt



2

||β(dt)||



1

√

dt

||α(dt)||+||β(dt)||+

√

dt



.

We refer to Eq. (31) as the inﬁnitesimal-timestep CE

bound.

In order to derive the ultimate precision bounds

valid in the presence of noise and FFQC, we compute

the inﬁnitesimal-timestep CE bound (31) after substi-

tuting the expressions for α and β given by Eqs. (29)

and (30). We explicitly minimize Eq. (31) over Kraus

representations as described in Sec. 4.2.2, by expand-

ing the Kraus operators, K(dt), and the Hermitian

matrix, h(dt), up to small orders of

√

dt. As we are

interested in the regime of asymptotic resources, i.e.,

the limit t

0

dt (or equivalently N tδt in case of the

scheme (c)), we may assume without loss of general-

ity that it is always optimal to make the second term

in Eq. (31) vanish by setting β(dt) = 0. We outline

the methods we use below, with detailed calculations

deferred to App. C.

Consider the dynamical matrix S, deﬁned in

Eq. (11), of the inﬁnitesimal time channel E

ω,dt

. Ex-

panding it in dt around t = 0 gives S(dt) := S

(0)

+

S

(1)

dt+O(dt

2

). The evolution of the probe after an

inﬁnitesimally short time is then given by

E

ω,dt

(ρ) =

3

X

µ,ν=0

S

µν

(dt) σ

µ

ρ σ

ν

. (32)

=

3

X

µ,ν=0

(S

(0)

µν

+ S

(1)

µν

dt) σ

µ

ρ σ

ν

+ O(dt

2

).

The above expansion is unambiguously speciﬁed up

to O(dt

2

) by the master equation in Eq. (5)

E

ω,dt

(ρ) = ρ+



−i

ω

2

[σ

3

, ρ] + L(ρ)



dt+ O(dt

2

), (33)

so we may directly relate the short-time expansion

of S to the Liouvillian of Eq. (6) without explicitly

integrating Eq. (5):

S

(0)

= diag(1, 0, 0, 0), (34)

S

(1)

=







−

P

3

i=1

¯

L

ii

Im

¯

L

23

Im

¯

L

31

Im

¯

L

12

+ i

ω

2

Im

¯

L

23

Im

¯

L

31

¯

L

Im

¯

L

12

− i

ω

2







,

(35)

where

¯

L is just the restriction of the Liouvillian L (6)

to the subspace spanned by the Pauli operators. We

remark that in the limit dt → 0 these are the only

meaningful terms, as higher orders in the expansion

do not aﬀect the ﬁnal state (which is precisely the

reason why the master equation formalism is valid).

From Eqs. (34) and (35) we directly obtain the the

expansion of M in

√

dt, using the convenient choice

M= M

†

=

√

S

T

.

Next, we expand the inﬁnitesimal-timestep CE

bound (31) in orders of

√

dt by deﬁning the ex-

pansions for operators α(dt) :=

P

`≥0

α

(`)

(dt)

`/2

and

β(dt) (similarly). The deﬁnitions Eqs. (29) and (30)

of α(dt) and β(dt) allow one to relate their expansions

to the ones of the matrix M (ﬁxed by the Liouvillian

via Eqs. (34) and (35)) and the Hermitian matrix h

(which is still free) order by order. In the last step one

optimizes the matrix h in order to obtain the tight-

est inﬁnitesimal-timestep CE bound (31) in the limit

dt→0.

We include the full details in App. B, yet let us

summarize here that a non-trivial bound in Eq. (31)

can only be obtained if the expansion terms satisfy

α

(0)

= α

(1)

= β

(0)

= β

(1)

= 0, these condition are al-

ways satisﬁed if the low-order terms in the expansion

of the matrix h are constrained. If it is possible to

choose h such that β

(2)

= β

(3)

= 0 also hold then the

inﬁnitesimal-timestep CE bound (31) becomes

F



%

(d)

t

0

/dt

(t

0

)



≤ 4||α

(2)

||t

0

+ (t

0

)

2

O(

√

dt). (36)

bounding the QFI to a linear scaling in t

0

in the limit

dt →0 with k = t

0

/dt. Note that β

(3)

can always be

set to zero without aﬀecting lower order terms, while

the existence of a solution for β

(2)

= 0 depends on the

noise, in particular it is impossible for rank-one Pauli

noise as implied by the result of Sec. 4.1. If there

exists a matrix h such that α

(0)

= α

(1)

= β

(0)

= β

(1)

=

β

(2)

=0 we obtain

F



%

(d)

t

0

/dt

(t

0

)



≤

dt→0

4 α

L

t

0

with (37)

α

L

:= min

h such that α

(i)

=β

(j)

=0

||α

(2)

||, (38)

for i ≤ 1 and j ≤ 3.

Note that as the sequential scheme with FFQC is

the most powerful one (see Sec. 2), Eq. (37) holds for

all protocols depicted in Fig. 1 after accordingly set-

ting t

0

= Nt. In particular, Eq. (37) also applies in

case of control with ﬁnite time-step δt (scheme (c))

and in the absence of quantum control when δt = t

(schemes (a) and (b)). Finally, let us remark that

although in the case of parallel strategies with ancil-

lae of Fig. 1(b) it is the parallel CE bound (22) that

must yield tighter limits on precision for ﬁnite t, one

may show (following exact argumentation as in the

previous paragraphs) that the linearly scaling parallel

CE bound must always converge to its inﬁnitesimal-

timestep equivalent of Eq. (37) when ﬁxing t

0

= Nt

and considering the t →0 limit, in which the parallel

protocol becomes inﬁnitesimally short but involves an

inﬁnite number of probes.

Accepted in Quantum 2017-09-04, click title to verify 9

4.2.4 Universal asymptotic linear bound

Thanks to the inﬁnitesimal-timestep CE bound (31)

yielding Eq. (37), we may formulate a general obser-

vation about the scaling of the QFI applicable to any

of the schemes depicted in Fig. 1:

Result 2. For all noise processes described by Eq. (6)

except rank-one Pauli noise-types that are not parallel

to the parameter-encoding Hamiltonian, the QFI in

any scheme of Fig. 1 is upper-bounded by

F

L

≤ 4 α

L

t

0

, (39)

where α

L

is a constant that solely depends on the par-

ticular form of the Liouvillian L deﬁned in Eq. (6),

while t

0

is the eﬀective protocol time as deﬁned via the

most powerful scheme (d) with FFQC in Fig. 1.

We explicitly prove Res. 2 in App. C, but outline

the derivation here. In App. C.1, for any qubit Liou-

villian L of Eq. (6), except the rank-one Pauli noise of

(7) with σ

n

6= σ

3

, we give the form of the hermitian

matrix h for which α

(0)

= α

(1)

= β

(0)

= β

(1)

= β

(2)

=

β

(3)

=0. Hence, there exists some ﬁnite α

L

for which

(37) holds. We ﬁrst show that dephasing noise, as

well as any rank-one noise which is not of Pauli type,

imposes a linear scaling of the QFI due to Eq. (37).

Next, we consider noise processes described by Liou-

villians whose rank is strictly greater than one. Any

such noise process corresponds to a matrix represen-

tation of the Lindblad superoperator (6) that may be

written as a direct sum of matrices of lower rank. We

show that the QFI of noise processes of rank greater

than one is always smaller than the QFI of any of

its orthogonal components. Thus, the only rank-two

noise processes that, in principle, could still allow for

quadratic scaling of the QFI are the ones correspond-

ing to a direct sum of two orthogonal rank-one Pauli

noises. For this case, however, we explicitly show that

the QFI scales linearly with t

0

. Moreover, this also set-

tles the case of the rank-three noise types as any such

noise can be expressed as a direct sum that contains

either a rank-one non-Pauli or a rank-two Pauli noise,

or both. Finally, in App. C.2 we derive explicit forms

of α

L

for the exemplary noise-types discussed below.

4.2.5 Exemplary noise-types

Although we have shown the validity of Eq. (39) for

most types of noise processes, it still remains to com-

pute α

L

for a general Liouvillian (see Eq. (6)) by ade-

quately minimizing Eq. (38). In App. C.2 we perform

such minimization in a completely analytic manner.

In particular, we minimize ||α

(2)

|| in Eq. (37) over

Hermitian matrices h (under the constraint of β

(2)

=0

and α

(`)

= β

(`)

= 0 for ` ≤1) for all the exemplary Li-

ouvillians stated in Sec. 3.2. The corresponding CE

bounds for dephasing, general rank-one, and rank-two

Pauli noise types respectively read

F

1P

z

≤

t

0

2γ

(40a)

F

1G

r

≤

t

0

4

max{|x + iy|, |x − iy|}

2

(|Re(r)|

2

|Im(r)|

2

− (Re(r)

T

Im(r))

2

)

(40b)

F

2P

Ω

≤

t

0

2γ

1

γ

2



c

2

θ

(γ

1

+ γ

2

) + s

2

θ

(γ

1

s

2

ϕ

+ γ

2

c

2

ϕ

)



(40c)

=⇒ F

X-Y

≤

t

0

2γ p(1 − p)

(40d)

We note that in the case of general rank-one noise,

yielding F

1G

r

in Eq. (40b), we analytically ﬁnd the

optimal h for the cases where one of the components

of r (x, y, or z) vanishes (see App. C.2). This allows

us to use the structure of the optimal h as an ansatz

for the case of general rank-one noise L

1G

. However,

although the minimization of ||α

(2)

|| subject to the

constraint β

(2)

= 0 can be done for any ﬁxed Liouvil-

lian, it generally contains a large number of param-

eters to be optimized. Hence, we derive a valid, but

not provably tightest, analytical bound for the case of

general noise.

4.3 The gain allowed by fast control

We now discuss the impact of fast control, in particu-

lar the gap in the achievable QFI between the scheme

(b) and the schemes with FFQC, i.e., (c) and (d) (with

δt →0 throughout this section) in Fig. 1. To do the

comparison we focus on the example of the X-Y noise

L

X-Y

in Eq. (10), and the rank-one transversal Pauli

noise L

1P

x

in Eq. (7) with σ

n

= σ

1

, which is also the

limiting case of X-Y noise for p = 1. For X-Y noise

the master equation (5) can be analytically solved,

hence the parallel CE bound Eq. (22) allows one to

upper-bound the QFI of the parallel scheme F

(b)

(t)

(see App. D).

4.3.1 Transversal rank-one Pauli noise

We begin with the rank-one transversal Pauli noise

L

1P

x

= L

X-Y

|

p=1

. From Res. 1 and from previous re-

sults [27–30], we know that this noise can be com-

pletely removed by FFQC without harming the evolu-

tion. Consequently, the strategies with FFQC attain

the QFI given by F

(d)

(t

0

) = (t

0

)

2

and F

(c)

(t) = (N t)

2

respectively. Remarkably, in order to attain such QFI

within (d) one only requires one qubit and one ancilla

in parallel: this is enough to implement the FFQC

strategy of Sec. 4.1 and get rid of the noise, and it

is known that in the noiseless case a sequential single

qubit strategy is optimal [5]. For the parallel strat-

egy (b) the CE bound reads F

(b)

≤4N α

(b)

ω,γ,p=1

(t) (see

App. D). We plot 4 α

(b)

ω,γ,p

for p = ω = γ = 1 in Fig. 2

Accepted in Quantum 2017-09-04, click title to verify 10

(thin solid line). This clearly shows the advantage of-

fered by FFQC: for any ﬁxed t in strategy (b) the at-

tainable F

(b)

is bound to a linear scaling in N. More-

over, for any γ and ω as t increases the bound even-

tually starts to decrease (actually it has been shown

that F

(b)

(t)/t is maximized for t ∝ 1/N

1

3

[38]). Inter-

estingly, the bound α

ω,γ,0

diverges when t approaches

zero. This is to be expected, as in this limit a super-

linear scaling in N is known to be possible already

within scenario (a) [38, 68].

4.3.2 X-Y noise

Now let us turn to X-Y noise L

X-Y

(with 0 < p <

1). For the parallel strategy (b), the same bound

F

(b)

≤ 4Nα

(b)

ω,γ,p

(t) applies and the function α

(b)

ω,γ,p

(t)

is now well-behaved (it starts at zero for t = 0). We

plot α

(b)

ω,γ,p

(t) for ω = γ = 1 and p = 0.1 in Fig. 2

(thick solid line). The situation for FFQC schemes

is a bit more subtle, and allows to nicely illustrate

diﬀerent aspects in which fast control is helpful. First

there is the ultimate bound of Eq. (40c) which holds

for any FFQC strategy and reads F

(d)

X-Y

≤

t

0

2γp(1−p)

(F

(c)

X-Y

≤

Nt

2γp(1−p)

for scheme (c)) but might not be

attainable (dashed line in Fig. 2).

Now let us consider a particular FFQC strategy,

which is described in details in Sec. D.1. Using the

error correction code of Sec. 4.1 one can detect if an

error happens at a given timestep. However, as there

is no way to tell if the error corresponds to a σ

1

or

σ

2

the noise cannot be fully corrected. Still, one can

chose to correct the most probable error term, say p ≤

1/2 and it is σ

2

. Hence, if the guess was right the error

is corrected σ

2

σ

2

= , while the other case leads to a

z-error σ

2

σ

1

= −iσ

3

. This strategy modiﬁes the X-Y

noise process to an announced dephasing noise (one

knows how many correction steps were performed),

for which the same bound Eq. (40d) holds as all our

control operations are unitaries.

Next, consider a suboptimal variant of the above

scheme (we will show that this is strictly subopti-

mal in Sec. 6) where we forget the error register, i.e.,

the knowledge about the number of correction steps.

In this case the resulting eﬀective noise process is

the usual dephasing (L

1P

z

of Eq. (7)) with strength

γ

z

= p γ, and the bounds F

(d)

≤

t

0

2γp

and F

(c)

≤

tN

2γp

of Eq. (40a) apply (dotted line in Fig. 2). Notice that

these bounds are attainable; it is known that the CE

bound for dephasing may be asymptotically achieved

already within strategy (a) when considering the limit

of short dynamics t →0 and large number of probes

N →∞ [24, 25, 36]. Crucially, when considering the

more powerful strategies (c) and (d), such a regime

can always be mimicked by employing an unbounded

number of ancillary qubits. These are then continu-

ously swapped with the sensing probes and measured

so that the required protocol of type (a) is indeed re-

0 20 40 60 80

500

1000

1500

2 4

60

100

200

300

t

Figure 2: X-Y noise: Scaling of the QFI as function of t for

various protocols and ω = 1, γ = 0.1 and p = 0.1. The solid

line corresponds to the CE bound on F

(b)

(t)/N for strat-

egy (b) without FFQC (the original X-Y noise). The dashed

straight is the ultimate upper-bound

t

2γp(1−p)

Eq. (40c). The

dotted straight line corresponds to the attainable bound for

the particular FFQC strategy of Sec. 4.3.2 where one cor-

rects the dominant noise but discards the error record, it

corresponds to

t

2γp

from Eq. (40a). The dot-dashed line is

the attainable bound for the same strategy but with ﬁxed

number of ancillae equal to N and no intermediate measure-

ments. X-noise: Finally, the thin line is the CE bound on

F

(b)

(t)/N for the X-noise with p = γ = ω = 1, note that it

diverges as t → 0 since strategy (b) allows to super-classical

scaling in N for vanishing t. The inset is a zoom on the

region close to the origin.

covered.

A natural question to ask then is what happens if

we keep the number of ancillary qubits constant, and

equal to N in (c) or (d). In this case, the FFQC strat-

egy described above allows one to mimic the scheme

(a) with dephasing noise L

1P

z

of strength γ

z

= p γ

and a ﬁxed number of probes N. For such an eﬀec-

tive noise-type, the parallel CE bound is known to

yield F

(a)

(t) ≤ N

t

2

η

2

(1−η

2

)

with η = e

−pγt

[22], which is

attainable when N is large (dot-dashed line in Fig. 2).

Hence, it is also attainable by strategy (c) with the

number of ancillary qubits equal the number of probes

N, or by strategy (d) with N + 1 qubits in total (only

one qubit is needed as an ancilla to run the error cor-

rection protocol, while the others can be used for con-

secutive preparation of the required initial entangled

states).

4.3.3 Advantages of FFQC

In summary, we see that FFCQ oﬀers several advan-

tages. First, it allows to modify the noise process par-

ticularly rank-one transversal Pauli noise which can

be completely removed, while X-Y noise can be trans-

formed to a weaker dephasing noise (dot-dashed line

in Fig. 2). In addition to this, if the number of avail-

able ancillary qubits is large FFQC oﬀers the possibil-

ity to prolong the short-time dynamics by “paralleliz-

Accepted in Quantum 2017-09-04, click title to verify 11

ing the evolution” (dotted line in Fig. 2)—mimicking

a parallel evolution with short running time but large

N and the modiﬁed noise. Finally, it also gives the

possibility to keep the register of errors that happened

during the evolution (dashed line in Fig. 2). To get

a feeling of how the latter improves the QFI, think

of the frequentist scenario where for the construction

of the ﬁnal estimator one values more the runs where

less or no errors happened (we explore this in more

depth in Sec. 6). In any case, FFQC undoubtedly out-

performs the parallel scheme (b) (solid line in Fig. 2)

by a large amount.

We showed how the introduction of FFQC leads

to an improved scaling of the QFI. Yet an interest-

ing question is if there exists a gap between the two

strategies (c) and (d). If the frequency with which one

applies the controls in (c) is limited, i.e. the duration

of each step δt is ﬁxed, it is easy to see that there

exists a gap. The bound derived in the App. D also

applies to this case and yields F

(c)

≤ 4Nm α

(b)

ω,γ,p

(δt),

and we just demonstrated that it can be outperformed

by FFQC. However, it is not clear if there is a gap

between (c) and (d) when δt →0 and t = t

0

/N. Fur-

thermore, it is not clear these strategies outperform

scheme (b) for t → 0 with N =

t

0

t

, as all the CE

bounds coincide in this limit. On the other hand, the

fact that the CE bounds coincide does not disprove

the existence of a gap, as none of the bounds is nec-

essarily tight. Finally, we note that protocol (d) has

following important advantage: one can read the er-

ror syndrome at any time-step before deciding on the

optimal strategy for the future, whereas for protocols

(b) and (c) some decisions have to be made before-

hand. We will come back to this particular point,

and show for a simple phase estimation example how

this diﬀerence allows to boost the attainable QFI in

Sec. 6.2.

5 Implications on attainable precision

in metrology schemes

Hitherto our analysis focused on the scaling of the

QFI evaluated on the ﬁnal system state of the metrol-

ogy protocols depicted in Fig. 1 in presence of general

time-homogeneous Liouvillian noise. We now discuss

the implications of FFQC on the precision with which

one can estimate the parameter ω. As already men-

tioned in Sec. 3.1, how to best quantify the estimation

precision depends crucially on how information about

ω is obtained. In Sec. 5.1 we consider the implications

of FFQC on precision for the case of asymptotically

many repetitions, whereas Sec. 5.2 deals with single-

shot estimation. Regardless of how one chooses to

quantify the precision, whenever a noiseless estima-

tion scenario exhibits a quantum improvement in pre-

cision scaling then this improvement may always be

maintained in the presence of rank-one Pauli noise by

employing the FFQC protocol described in Sec. 4.1.

We note again that the ability of our scheme to ame-

liorate for any rank-one Pauli noise ﬁnds applications

beyond metrology, i.e., in the design of high ﬁdelity

gates, or indeed any other protocol where such noise

terms may appear.

The achievable precision in estimating ω in any

metrological protocol is dictated by the resources at

hand. In what follows we shall consider two diﬀer-

ent ways of quantifying the resources of any metro-

logical protocol: time-particles and number of probes.

In the time-particles approach the total resource per

experimental run is deﬁned as the product between

the number of probes N and the protocol duration

t. Thus, for all the types of protocols depicted Fig. 1

the time-particle resource is equal to t

0

= Nt. In

the number of probes approach one is primarily inter-

ested in how estimation precision scales with N only.

In particular, this is how resources are deﬁned in fre-

quency estimation [15, 19, 22, 38], where the time for

each experimental run, t, is bounded from above so

that one can consistently consider the limit of many

experimental repetitions by letting the overall exper-

iment last suﬃciently long. In all cases we show how

our results yield noise dependent upper bounds on

the attainable precision and restrict it to a classical

scaling in terms of the resource.

5.1 Precision in the presence of free repetitions

We now consider precision bounds for estimating pa-

rameter ω using FFQC for the case where the protocol

is repeated asymptotically many times so that CRB

is applicable. We ﬁrst consider the time-particles pic-

ture before moving onto the scenario in which the

number of probes N is the resource.

5.1.1 Time-particles

Given ν repetitions of the protocol and using the

CRB, we bound the ultimate MSE attainable by the

most general FFQC strategy in the presence of rank-

one Pauli noise as

δω

2

ν ≥

1

(1 − (n

T

z)

2

)

1

t

0

2

. (41)

On the other hand, stemming from Eq. (39) the MSE

for all other noise processes may be lower-bounded as

follows

δω

2

ν ≥

1

4α

L

1

t

0

. (42)

Yet, in contrast to Eq. (41), Eq. (42) is not guaranteed

to be achievable in the limit ν → ∞.

5.1.2 Frequency estimation

We now turn to the setting of frequency estimation in

which the resource of interest is the number of probes

Accepted in Quantum 2017-09-04, click title to verify 12

N. Notice that the FFQC protocol in this case is the

one depicted in Fig. 1(c). Yet, its QFI can always be

upper-bounded by that of scheme (d) after setting t

0

=

Nt. In order to ensure a large number of repetitions

we ﬁx the time of a single run t and demand that the

total time T := νt  t. Hence, we can again use the

CRBs of Eqs. (41) and (42) and obtain the precision

bounds:

δω

2

T ≥

1

(1 − (n

T

z)

2

)

1

N

2

t

, (43)

δω

2

T ≥

1

4α

L

1

N

. (44)

Indeed, Eq. (43) proves that HS is achievable for rank-

one Pauli noises. On the other hand, for all other

noise processes, thanks to Eq. (39) being linear in t

0

,

the bound of Eq. (44) is independent of the single-

run duration t. This suggests that FFQC allows us to

indeﬁnitely maintain the ultimate precision normally

exhibited only at very short time scales (see Sec. 4.3).

5.2 Single-shot precision bounds

We now consider precision bounds applicable in the

single-shot scenario, i.e., when ν = 1. As a result

we are forced to consider the AvMSE as the ﬁgure of

merit, where the average is with respect to the prob-

ability distribution p

0

(ω) describing our prior knowl-

edge about the parameter. The AvMSE can then be

lower-bounded with the help of BCRB and ZZB (see

Sec. 3.1). However, note that the FFQC-assisted se-

quential strategies of Fig. 1(d) with suﬃciently large t

0

also incorporate all many repetition protocols. This

is because one can always set the intermediate con-

trol gates to implement measurements and state re-

preparations. Hence, we expect all the single-shot

bounds to reproduce the ones derived in the regime

of many repetitions when considering t

0

→ ∞.

5.2.1 Pauli rank-one noise

As by repeating we may only improve the precision,

whenever the precision is forced to follow the stan-

dard scaling in the many-repetitions regime, it must

be standard scaling-bounded in case of the single-shot

estimation. Thus, it is most important to ask whether

the HS can still be attained in presence of rank-one

Pauli noise within strategies (c) and (d) with only a

single run. In particular any protocol that achieves

the maximum attainable precision in the absence of

noise can maintain this precision in the presence of

rank-one Pauli noise by utilizing FFQC. For example,

by incorporating FFQC in the protocol in [47] one

can maintain its optimal performance for any rank-

one Pauli noise process; the number of measurements

N in [47] correspond to the control-steps k in our pro-

tocol (d). In App. F, we provide an explicit strategy

which for a uniform prior—a prior distribution that

is constant over a ﬁnite interval, and zero elsewhere—

leads to the AvMSE satisfying

hδω

2

i

FFQC

≤

π

4

κ

4(1 − (n

T

z)

2

)

1

t

02

(45)

for (d) and (c) (after substituting t

0

= N t), where

κ = 6.74. This shows that, at least for this prior,

quadratic scaling of precision in the total resource t

0

2

(or (Nt)

2

) is attainable by a single shot strategy. In

the context of frequency estimation of Sec. 5.1, this

demonstrates that if one can decide whether to divide

the total time T in many runs of a ﬁxed duration

t or perform a single shot strategy with FFQC, one

should choose the latter as it achieves precision that

scales quadratically hδω

2

i ∼ 1/(N T )

2

in N and T ,

and not only in N , as in Eq. (43).

5.2.2 Bayesian Cramér-Rao bound

The BCRB is deﬁned as [49]

hδω

2

i ≥

1

hF

L

i + F



p

0

(ω)



, (46)

where F



p

0

(ω)



= h(∂

ω

log p

0

(ω))

2

i is the classical

Fisher information of the prior distribution and h. . . i

denotes averaging with respect to the prior distribu-

tion p

0

(ω). Hence, using Eq. (39), we obtain the fol-

lowing lower bound on the AvMSE :

hδω

2

i ≥

1

4α

L

t

0

+ F



p

0

(ω)



. (47)

Notice that if F



p

0

(ω)



is ﬁnite, by substituting t

0

→

νt

0

above we adequately recover the many-repetition

bound of Eq. (42).

5.2.3 Ziv-Zakai bound

A more general bound that also allows for irregular

priors is the ZZB introduced in [48]. In App. D we

generalize the bound of [48] so that it can be applied

also to mixed states. As a result we can lower bound

the AvMSE for large enough t

0

by

hδω

2

i ≥

1

12α

L

t

0

(48)

for any prior that fulﬁlls some very mild regularity

conditions (see App. D). Again, by substituting t

0

→

νt

0

we adequately recover the many-repetition bound

of Eq. (42).

6 Importance of quantum control for

metrology with limited resources

In this section we further analyse the beneﬁts of con-

tinuous quantum control for metrology in presence of

Accepted in Quantum 2017-09-04, click title to verify 13

noise that can not be completely removed. In addi-

tion, we adopt a practical perspective where the re-

sources available for the implementation of the scheme

are severely limited. This allows us to keep the analy-

sis simple, but also makes the proposed protocols eas-

ily implementable in real experiments. In Sec. 6.1 to

cope with an unbalanced X-Y noise we propose a sim-

ple FFQC scheme that only requires one sensing and

one ancillary qubit, but can still outperform a paral-

lel scheme with a large number of entangled probes

and ancillae (but no continuous quantum control). In

Sec. 6.2 we show that also in the case of phase esti-

mation the possibility to perform intermediate con-

trol (applied between the two successive applications

of the channel on the probe qubits as shown in Fig. 6)

leads to an improvement in the attainable QFI.

6.1 Frequency estimation with X-Y noise

We study the setting of frequency estimation in which

the probes sensing the parameter are also aﬀected by

the X-Y noise speciﬁed in Eq. (10). In accordance to

Eq. (40d, 44) the attainable precision for this noise

is asymptotically constrained to a constant improve-

ment over standard scaling, even when considering

the most powerful FFQC scheme of Fig. 1(d) (with

the exception of the limiting cases p = 0 or p = 1,

where one of the components vanishes and one recov-

ers the correctable perpendicular dephasing noise [27–

30, 38]). Also in these cases strategies with FFQC can

outperform those without yielding a large improve-

ment factor as we showed in Sec. 4.3. In this sec-

tion we strengthen this point by demonstrating how

the most general parallel scheme of Fig. 1(b) with N

qubits and N ancillae is outperformed by a simple and

experimentally tangible strategy, which only requires

entanglement between one sensing and one ancillary

qubit but employs fast control.

More precisely, we compare the most general paral-

lel strategy Fig. 3(b) with simple sequential strategies

Fig. 3(s1) and Fig. 3(sN), where one is only allowed to

entangle probes and ancillae two by two rather then

manipulate a global entangled state as required by

(b). The two strategies (s1) and (sN) are actually the

same, but correspond to two diﬀerent ways of treat-

ing the total resources. In (s1) the experimentalist

only manipulates one qubit and one ancilla and does

the experiment for a total duration T . This protocol

has the same duration as (b) but uses N times less

“calls of the master equation”, i.e., one eﬀectively ap-

plies the single-qubit evolution for a time that is N

times shorter as compared to (b). The scheme (sN)

is equivalent to (s1) upon replacing T with NT , such

that it uses the same amount of time-particles as (b).

Equivalently (sN) corresponds to running N proto-

cols (s1) simultaneously. In this respect it has the

same number of qubits and the same time duration

as (b). However, in contrast to (b), the scheme (sN)

is granted with fast control on the one hand, but only

requires two-qubit entanglement on the other. We

demonstrate that, even with such restrictions, for any

ﬁxed N both strategies (sN) and (s1) outperform any

parallel schemes (b) given that the asymmetry in the

X-Y noise is high enough.

Our results show that the use of control and error

correction techniques allow one to attain resolutions

with a two-qubit (probe-plus-ancillae) setup which

outperform ones reached when considering systems

containing large-scale entanglement. We believe that

our results may support current state-of-art quantum

metrology experiments with nitrogen-vacancy (NV)

centres, in which the error correction protocols have

already been implemented with great success [69, 70].

In particular, as the dominant noise is such systems

has been argued to be nearly transversal [28, 29] and,

hence, highly asymmetric [68], our work proves that

such systems are indeed capable of attaining resolu-

tions unreachable by the entanglement-based schemes

employing comparable resources.

6.1.1 Parallel strategy with N qubits and N ancillae

A general parallel frequency estimation scenario is de-

picted in Fig. 1(b) and Fig. 3(b). For a regular prior

knowledge and suﬃciently large T (see Sec. 5.2.2), or

with the explicit assumption of suﬃciently many rep-

etitions (T /t  1), the ultimate attainable precision

is determined by the CRB:

T δ

2

ω ≥ min

t

F



%

(b)

N

(t)



. (49)

The right hand side in Eq. (49) is the inverse of the

maximal QFI rate for a given scheme of type (b) em-

ploying N sensing qubits. It is the optimized for the

single-run duration that we dub t

(b)

opt

. The optimal

time in general depends on the system size (the probe

number N) and the form of the noise (the Liouvil-

lian in Eq. (6))

3

. Following the methods described

in Sec. 4.2.1 (see also App. D), we can upper-bound

the maximal QFI rate by employing the parallel CE

bound (22):

F



%

(b)

N

(t

(b)

opt

)



t

(b)

opt

≤ N f

(b)↑

N

, where

f

(b)↑

N

:= 4 max

t

min

h(t)

||α(t)|| + (N − 1)||β(t)||

2

t

(50)

is the so-obtained upper bound on the maximal QFI

rate per probe. We adequately label then as t

(b)↑

opt

the

optimal t maximising Eq. (50). As a result, we may

generally lower-bound the precision dictated by the

3

Note that in the absence of noise it is optimal to set t = T

as large as possible, which makes the CRB (49) not applicable.

Accepted in Quantum 2017-09-04, click title to verify 14

...

N

2N

T

...

T

(b)

...

T

...

(s1)

... ... ...

... ...

(sN)

NT

... ... ...

...

Figure 3: (s1): A control-assisted sequential protocol that only involves one probe and one ancillary qubit. (b): The most

general parallel protocol of Fig. 1(b) that requires to manipulate entangled states of N probe and N ancillary qubits. (sN): A

control-assisted sequential protocol that employs N independent probe-plus-ancilla entangled pairs running in parallel.

CRB (49) as follows:

(b): T δ

2

ω ≥

1

Nf

(b)↑

N

, (51)

at the price of the saturability that cannot be guar-

anteed any more, even in the T/t

(b)↑

opt

1 limit.

In order to perform the minimization in (50), we re-

sort to the SDP-based methods of Ref. [22] reviewed in

Sec. 4.2.2. To this end we solve the master Eq. (10) for

the X-Y noise, and compute the Kraus representation

of the resulting channel, E

X-Y

ω,t

, given in App. D. For

any ﬁxed N and t this allows us to perform the mini-

mization over the Hamiltonian h(t) numerically, while

for N = 1

4

and N →∞ we obtain the corresponding

expressions analytically for all times. Finally, to ob-

tain f

(b)↑

N

we numerically search for the optimal t

(b)↑

opt

in Eq. (50). The detailed analysis on the solution

of the master equation and derivation of the bounds

may be found in App. D. Note that the balanced X-Y

noise (p = 1/2) is phase-covariant, i.e., it commutes

with the Hamiltonian, so that for this particular case

the bound (50) can be obtained analytically [65].

6.1.2 Simple sequential strategy

For the sequential strategy, all the probe plus ancilla

pairs behave independently, and since the QFI is addi-

tive in this case in order to establish the global perfor-

mance it is suﬃcient to consider a single pair up to the

ﬁrst measurement in Fig. 3(s). We denote the state of

a probe-plus-ancilla at any time by %(t) ∈B(C

2

⊗C

2

).

The maximal QFI rate per probe then reads

f

(s)

:=max

t

F(%(t))

t

=

F(%(t

(s)

opt

))

t

(s)

opt

(52)

with t

(s)

opt

being now determined solely by the noise.

The sequential FFQC strategy was already

sketched in Sec. 4.3.2 and is formally described in

4

For which the CE bound (50) is guaranteed to be tight, see

Eq. (19).

Sec. D.1. It consists of continuously checking if the

probe+ancilla state is in the code H

C

or the error

subspace H

E

. At each time step the probability of de-

tecting an error is

γdt

2

, and if an error is detected the

most probable error term, say σ

1

, is canceled by ap-

plying the same unitary on the probe σ

1

σ

1

= . This

maps the state back into the code space but slightly

degrades it with a residual dephasing coming from the

other noise term σ

1

σ

2

= iσ

3

.

As one keeps track of the number of errors m that

happened during such an FFQC assisted evolution,

but disregards the exact times the errors occurred, the

ﬁnal state at time t possesses a direct-sum structure

¯%(t)=

∞

M

m=0

p(m; t)¯%

m

(t). (53)

The probability of errors follows a Poissonian distri-

bution p(m; t) = e

−γt/2

(γt/2)

m

/m!, while the state

¯%

m

(t) conditional to the detection of m errors is given

in (112).

The corresponding QFI may be straightforwardly

evaluated:

F(¯%(t)) =

X

m

p(m; t)F



¯%

m

(t)



= t

2

e

−γt 2p(1−p)

,

(54)

and yields the following maximal QFI rate per probe

(52):

f

(s)

=

1

2γe

1

p(1 − p)

, (55)

which is attained after optimally setting t

(s)

opt

=

1/(2γp(1 − p)).

In the regime of long experimental duration

T/t

(s)

opt

1 that we are interested in, the CRBs (49)

for the two strategies Fig. 4(s1) and Fig. 4(sN) respec-

tively read:

(s1): T δ

2

ω ≥

1

f

(s)

(56)

(sN): T δ

2

ω ≥

1

Nf

(s)

, (57)

Accepted in Quantum 2017-09-04, click title to verify 15

Figure 4: Maximal QFI rate per probe f

(s)

attainable by the

sequential strategies (dashed lines) of Fig. 3(s1) and (sN),

as compared to the upper bound, f

(b)↑

N

(solid curves), valid

for all parallel protocols of Fig. 3(b) in presence of X-Y noise

(ω =1, γ =0.05). For p =0.25 (black) the curves indicate the

sequential strategy (sN) to be superior for N < 5, whereas

for p=0.05 (red) up to N < 316 (while (s1) is superior up to

N < 4—dotted red curve). The inset depicts dependence on

the probe number of t

(b)↑

opt

that maximizes Eq. (50). Despite

not being smooth at ﬁnite N , t

(b)↑

opt

asymptotically follows

1/N

1/3

scaling (dashed line).

and are guaranteed to be attainable.

Before we proceed to the comparison of strategies

(s) and (b), let us brieﬂy comment on the role of

the error register, and how it allows to boost the

attainable performance. As we argued in Sec. 4.3.2

discarding the error register is equivalent to eﬀec-

tively modifying the X-Y noise to a dephasing noise

of strength p γ. Concretely, if the error register is

discarded, the state at time t is a mere mixture

¯%(t) =

P

∞

m=0

p(m; t)¯%

m

(t) rather than a direct sum

as in Eq. (53). The corresponding maximal QFI rate

per probe is then given by f

(s)

=1/(2γep), which is re-

duced by a factor 1/(1−p) as compared to the strategy

including the error register and described by Eq. (55).

The intuition behind this gap is rather simple. When

inferring the value of ω after a certain number of runs

it is helpful to know how noisy each run was. This

information, contained in the error register, allows to

properly ponder the data obtained in each run in or-

der to optimally construct the global estimator.

6.1.3 Comparison of the strategies

In Fig. 4, we explicitly present the maximal QFI rate

per probe attained by the sequential strategy f

(s)

of

Eq. (55), and compare it to the bound limiting the

performance of any parallel scheme f

(b)↑

N

of Eq. (50),

which we plot as a function of the number of probes

N. This corresponds to a comparison of strategies

(sN) and (b) described in Fig. 3. In particular, after

ﬁxing ω = 1, γ = 0.05, we show that when the noise

generated by σ

2

is three times more dominant than

the one generated by σ

1

, i.e., p = 1/4, the sequential

Figure 5: Threshold numbers of probes, N

(s1)

th

and N

(sN)

th

,

below which the two sequential strategies (s1) and (sN) are

superior to all parallel protocols (b) in Fig. 3. as a function of

the X-Y noise asymmetry p for ω = 1 and γ = {0.1, 0.2, 0.3}

(top to bottom). Red lines are plotted assuming sequential

protocol performance to be evaluated per probe (sN), while

blue lines for the sequential protocol employing only a single

probe-plus-ancilla pair (s1). In the former case, N

(sN)

th

≥ 2

with equality always occurring strictly at p = 1/2, while in

the single-pair scenario there exists a range of 0.13 /p ≤ 0.5

for which trivially N

(s1)

th

=1. In both cases, N

th

diverges with

noise asymmetry, as p→0, and eventually follows the p

−3/2

,

p

−3/5

scalings respectively (dashed lines).

strategy is guaranteed to be superior for N less than

5. However, with the increase of noise asymmetry

the superiority of the sequential strategy drastically

improves, so that for p = 0.05 only if the probe num-

ber N is greater than 316 the parallel schemes (b)

can potentially beat the sequential one (sN). In this

case, in order to also compare (b) to the strategy (s1),

which employs only a single (rather than N) probe-

plus-ancilla qubit pair for the same duration T , we

also plot f

(s)

/N. Strikingly, the parallel strategies (b)

are still outperformed by (s1) as long as N <4.

To compare the strategies more directly we explic-

itly present in Fig. 5 the threshold probe numbers

below which the sequential strategies are guaranteed

to be superior over all parallel schemes. Concretely,

to compare (sN) with (b) we plot N

(sN)

th

such that

f

(s)

> f

(b)↑

N

for all N < N

(sN)

th

(the top curves), and

to compare (s1) with (b) we plot N

(s1)

th

such that

f

(s)

/N > f

(b)↑

N

for all N < N

(s1)

th

(the bottom curves).

Note that N

(sN)

th

≥ 2 for any value of p, as in the

worst case of balanced noise (p = 1/2) one may ex-

plicitly show that f

(b)↑

N=2

=f

(s)

, see App. D. Regarding

the other threshold, although N

(s1)

th

= 1 in the range

0.13 / p ≤ 0.5, meaning that the sequential strategy

(s1) only outperforms parallel strategies with a single

probe-plus-ancilla pair; N

(s1)

th

still diverges to inﬁnity

when p →0 and the noise approaches the perpendic-

ular dephasing. We observe that the thresholds scale

as N

(sN)

th

∝ p

−3/2

and N

(s1)

th

∝ p

−3/5

. This proves

that both sequential strategies (s1) and (sN) can out-

Accepted in Quantum 2017-09-04, click title to verify 16

(i)

(ii)

Figure 6: Two diﬀerent strategies for phase estimation in-

volving two applications of the noisy channel E

θ

presented in

Sec. 6.2. In strategy (i) the two probe and ancillary qubits

are prepared in a suitably entangled state and the channels

are applied to the sensing systems in parallel before they are

ﬁnally measured. This strategy intrinsically includes error-

correction but only at the ﬁnal step (see strategy (iii) in [37]).

(ii) The sequential strategy from Sec. 6.2, after the applica-

tion of the ﬁrst channel one reads out the error syndrome

and decides on the optimal strategy for the next step (see

strategy (iv) in [37]).

perform the most general parallel one of type (b) for

any N, given suﬃcient noise asymmetry.

6.2 Phase estimation with bit- and bit-phase-

ﬂip noise

We now return to the canonical phase estimation sce-

nario [5], which may yet be interpreted as the fre-

quency estimation considered before but with the evo-

lution time t being ﬁxed, such that the parameter of

interest is θ = ωt. In this case the sensing interac-

tion of a single probe of duration t constitutes the

elementary block, drawn as a square box in Fig. 6.

Each box corresponds to a CPTP map E

θ

incorpo-

rating the noise [37]. In this context one is free to

use ancillary particles to beneﬁt from error correction

techniques, but also to combine the boxes in diﬀer-

ent ways. The two schemes depicted in Fig. 6(i) and

Fig. 6(ii) are the elementary phase estimation equiva-

lents of the schemes depicted in Fig. 1(b) and Fig. 1(d)

respectively.

We provide a simple example demonstrating that

a sequential strategy outperforms the parallel one.

Speciﬁcally, consider the situation where each line in

Fig. 6 corresponds to a qubit and the map E

θ

is a

combination of the unitary evolution given by U

θ

=

e

−i

θ

2

σ

3

and of the noise which consists of a random ap-

plication of either a bit ﬂip (σ

1

) or bit-phase ﬂip (σ

2

)

with equal probability

1−p

2

. Accordingly, E

θ

has a

Kraus representation of the form {K

0

=

√

p U

θ

, K

1

=

√

1 − p |1ih0|, K

2

=

√

1 − p |0ih1|}. Note that this

noise channel corresponds to a Pauli X − Y noise

channel which, however, diﬀers from the solution of

the master equation for the balanced X − Y noise

described in Sec. 6.1.

In the parallel strategy of Fig. 6(i) one starts with

a suitably entangled state of two sensing qubits and

two ancillae. The channel is applied on both qubits

in parallel, and the ﬁnal state is measured (this in-

cludes error-correction at the ﬁnal stage see strat-

egy (iii) in [37]). Without loss of generality the in-

put two-qubit plus two-ancilla state is given by |ψi =

P

1

i,j=0

a

ij

|i, ji

Q

|i, ji

A

with a

ij

∈ R. This encoding

is optimal as it allows us to distinguish between all

the diﬀerent branches of the noisy evolution. More-

over, within every branch the state remains pure, as

the channel corresponds to either leaving the sensing

qubit untouched—with probability p—or projecting

it onto the {|0i, |1i} basis (followed by an irrelevant

ﬂip of the qubit) which can be detected by our error

correction code. As all the branches can be detected,

and within every branch the state remains pure, the

QFI of the ﬁnal state is given by the mean variance

of H =

1

2

(σ

3

⊗ + ⊗σ

3

) over all the branches. The

mean variance can be straightforwardly maximized by

a brute force numerical optimization over the three di-

mensional manifold spanned by the coeﬃcients a

ij

. It

turns out that the optimal states are symmetric un-

der permutations of the two qubits (a

01

= a

10

) and,

moreover, satisfy a

00

= a

11

. Using these conditions at

the outset one can analytically determine the optimal

state to be

|ψi

opt

=

q

x

opt

2

(|00i

Q

|00i

S

+ |11i

Q

|11i

S

)

+

q

1−x

opt

2

(|01i

Q

|01i

S

+ |10i

Q

|10i

S

), (58)

with x

opt

= min(1,

2−p

4(1−p)

). The corresponding maxi-

mal QFI is

F

(i)

=

(

−

(p−2)

2

p

2(p−1)

p ≤

2

3

4p

2

otherwise.

(59)

Now consider the following sequential strategy with

intermediate error correction involving two sensing

qubits and a single ancilla, see Figure 6(ii). The ﬁrst

sensing qubit and the ancilla are prepared in the state

1

√

2



|00i

Q

1

A

+ |11i

Q

1

A



. After the application of the

ﬁrst box on the ﬁrst sensing qubit but before the sec-

ond one we perform our error detection scheme. If

an error is detected the state contains no information

about the parameter, hence one discard Q

1

and pre-

pare the second qubit and the ancilla in the same state

1

√

2



|00i

Q

2

A

+ |11i

Q

2

A



. If no error is detected after

the ﬁrst application of the channel the second sensing

qubit is introduced and entangled with the ﬁrst qubit

such that the overall state is given by

r

1

2



e

−i

θ

2

|0i

Q1

(

√

y |00i

Q2,A

+

p

1 − y |11i

Q2,A

)

+e

i

θ

2

|1i

Q1

(

√

y |11i

Q2,A

+

p

1 − y |00i

Q2,A

)



.

(60)

One easily checks that the state that optimizes the

ﬁnal QFI (after the application of the second box) is

the one with y = y

opt

= min(1,

1

2(1−p)

). Finally the

overall QFI of this scheme is given by

F

(ii)

=

(

p(3p + 1) p ≥

1

2

p

(

p

2

−2p+2

)

1−p

otherwise.

(61)

Accepted in Quantum 2017-09-04, click title to verify 17

0.2

0.4

0.6

0.8

1.0

0.05

0.10

0.15

0.20

p

Figure 7: Comparison of the QFI’s for strategies (i) and (ii).

The curve corresponds to the diﬀerence of QFI between the

strategies (ii) in Eq. (61) and (i) Eq. (59).

In Figure 7, we plot the diﬀerence between QFI’s of

the strategies (ii) in Eq. (61) and (i) in Eq. (59). One

clearly sees that the strategy with intermediate con-

trol Fig. 6(ii) performs better than the parallel strat-

egy Fig. 6(i). Moreover there is a gap for all values of

p except the trivial cases: the noiseless case p = 1 and

the case where no information about θ is contained in

the channel p = 0.

This simple example shows that intermediate

control (FFQC) allows for an improvement in the

achievable accuracy. Notice that such an advantage

can be maintained for any ﬁnite number of blocks

as one can choose to introduce new sensing qubits,

appropriately entangled with the already existing

ones, only after one has obtained information about

the errors from the intermediate error correcting

steps. In the parallel strategy however, all the qubits

have to be initialized in an entangled state, which

is highly fragile to subsequent errors. However the

question whether this improvement vanishes in the

asymptotic case—as conjectured in [37]—or persists,

as one might suspect from the ﬁnite N results shown

here, remains open.

7 Summary and Outlook

We have considered the general limits of quantum

metrology where one is, in principle, equipped with

a full-scale quantum computer to assist in the sensing

process. We have shown that one can use techniques

from quantum error correction to detect or correct for

certain kinds of errors while maintaining the sensing

capabilities of the system. In particular, we discover

that the use of FFQC allows to restore the Heisen-

berg scaling of precision for all all rank-one Pauli noise

processes, except the Pauli noise that is identical to

the Hamiltonian, at the cost of a slowing down the

evolution by a constant factor. For all other noise-

types, we have shown that the QFI is limited to a

linear scaling with the number of resources. As we

demonstrated, this result forbids the Heisenberg scal-

ing of the estimation precision and provides noise de-

pendent upper bounds for all metrological scenarios,

local or Bayesian. Remarkably, considering the ulti-

mate metrological scheme (FFQC) actually lead to a

simpliﬁcation of the problem, as in the limit of short

evolution steps dt → 0 the CE method does not re-

quire to solve the master equation, this allowed us to

obtain fully analytic bounds on the achievable preci-

sion for relevant noise processes.

However, even in the cases where FFQC does not

allow for full restoration of the Heisenberg scaling,

one can achieve a signiﬁcant improvement over par-

allel strategies that operate on limited resources, and

do not utilize FFQC. We have demonstrated this for

the example of asymmetric X-Y noise where the im-

provement over parallel schemes employing the same

number of resources but do not use FFQC can be

signiﬁcant. Moreover, we have put forward simple

sequential protocols that make use of only a single

sensing and auxiliary system, but that nevertheless

outperform parallel entanglement-based strategies for

limited number of sensing systems. From a practical

perspective, the existence of simple practical schemes

that operate with only a single sensing and auxiliary

system, are of high relevance, and may open the way

for practical noisy metrology with a signiﬁcant quan-

tum advantage. This demonstrates that fast quan-

tum control and error correction are powerful tools in

quantum metrology and may enable practical sensing

with a signiﬁcant quantum enhancement even in the

presence of noise.

An important question one may ask is how our re-

sults generalize to higher dimensional systems. Note

that in higher dimensions the parameter encoding

Hamiltonian may have support only on a subspace

of the Hilbert space of the probe system, which may

allow for more involved error correcting codes.

Note added. After making this work available on-

line, the error correction protocol presented in Sec. 4.1

capable of removing any rank-one Pauli noise has been

successfully implemented in an NV-centre experiment

to compensate for transversal dephasing noise [71].

On the other hand, a dissipation-based scheme has

been proposed to correct for such noise in ultra-cold

ion sensing experiments [72]. Lastly, the general re-

sults presented in Sec. 4.1 and Sec. 4.2.4 have been

very recently generalized [73, 74] beyond the case of

qubit probes, considered in this paper.

Acknowledgements

This work has been supported by Austrian Science

Fund (FWF: P24273-N16, P28000-N27), Swiss Na-

tional Science Foundation Grant (P2GEP2_151964),

Spanish Ministry National plans FOQUS and

Accepted in Quantum 2017-09-04, click title to verify 18

MINECO (Severo Ochoa Grant No. SEV-2015-

0522), Spanish MINECO FIS2013-40627-P, Gener-

alitat de Catalunya CIRIT 2014 SGR 966 Fun-

dació Privada Cellex, Generalitat de Catalunya Grant

(No. SGR875), as well as received funding from the

European Union’s Horizon 2020 research and innova-

tion programme under the Marie Skłodowska-Curie

Grant: Q-METAPP (No. 655161).

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Accepted in Quantum 2017-09-04, click title to verify 22

A Relation between the Kraus representations and the dynamical matrix

Any Kraus operator, K

i

=

P

k

M

ik

σ

k

, can be expressed in the operator basis given by the four Pauli operators

in the qubit case, which we write for the vector of Kraus operators as

K = Mς . (62)

The action of the channel on the density matrix is expressed in the Kraus representation as

E(ρ) =

X

i

K

i

ρK

†

i

=

X

i

(

X

k

M

ik

σ

k

)ρ(

X

`

M

∗

i`

σ

`

) (63)

=

X

k`

(

X

i

M

†

`i

M

ik

)σ

k

ρσ

`

=

X

k`

(M

†

M)

`k

σ

k

ρσ

`

.

Imposing the equality with the dynamical matrix representation (11) one arrives at

M

†

M = S

T

. (64)

From this expression it follows that all Kraus representation satisfying M

0

= uM (with a unitary u) lead to the

same dynamical matrix and therefore correspond to the same channel. Remark that in general M may not be

a square but a rectangular n × 4 matrix, where n is the number of Kraus operators in a given representation.

However, any M admits a singular value decomposition of the form

M = u D

n×4

v

†

, (65)

with u– a unitary n × n matrix , v– a unitary 4 × 4 matrix and D

n×4

– a rectangular n × 4 diagonal matrix.

Now M

0

= D

n×4

v

†

is also a valid Kraus representation, and so is the 4 × 4 matrix M

00

= D

4×4

v

†

(where to get

to D

4×4

from D

n×4

we either remove all the n − 4 zero lines in the case n > 4, or add 4 − n zero lines in the

case n < 4). Finally, we chose the Hermitian 4 × 4 matrix

M

000

= v D

4×4

v

†

(66)

to be our canonical Kraus representation for the channel.

B Inﬁnitesimal-timestep CE bound

Here, we prove the inﬁnitesimal-timestep CE bound of Eq. (31) that may be applied also to schemes that

incorporate FFQC, i.e., the ones of type (d) in Fig. 1. We essentially follow the derivation contained in the

Supplementary Material of Ref. [37] except the last step, at which we slightly generalize the bound presented

therein.

In particular, one should follow the reasoning up to Eq. (S15) of Ref. [37], at which a more general upper

bound (for any real

√

x) may be derived via

||

X

k

˙

K

†

k

iAK

k

+ h.c.|| = ||i

X

k

(

√

x

˙

K

†

k

)A(

1

√

x

K

k

) − (

1

√

x

K

†

k

)A(

√

x

˙

K

k

)|| (67)

= ||i

X

k

(

√

x

˙

K

k

+ i

1

√

x

K

k



†

A(

√

x

˙

K

k

+ i

1

√

x

K

k

) − λ

˙

K

†

k

A

˙

K

k

−

1

x

K

†

k

AK

k

|| (68)

Using the triangle inequality and the one stated in Eq. (S13) of Ref. [37], we arrive at a generalized version of

Eq. (S15) therein:

||

X

k

˙

K

†

k

iAK

k

+ h.c.|| ≤ 2||A||



x||

X

k

˙

K

†

k

˙

K

k

|| + ||

X

k

˙

K

†

k

K

k

|| +

1

x



, (69)

which in turn results in the generalization of Eq. (S19):

F ≤ 4 k||α|| + 4 k(k − 1)||β||



x||α|| + ||β|| +

1

x



. (70)

By taking the limit of inﬁnitesimal timesteps k = t

0

/dt and writing explicitly the time-dependences, we obtain

the bound:

F ≤ 4

t

0

dt

||α(dt)|| + 4

(t

0

)

2

dt

2

||β(dt)||



x||α(dt)|| + ||β(dt)|| +

1

x



, (71)

Accepted in Quantum 2017-09-04, click title to verify 23

whose tightest form we would like determine in the dt →0 limit. Hence, as otherwise Eq. (71) must diverge, we

can restrict to Kraus representations for which α(dt) = α

(2)

dt+O(dt

3/2

) and β(dt) = β

(2)

dt+β

(3)

dt

3/2

+O(dt

2

).

As we are willing to show a linear scaling of the QFI in t

0

due to vanishing of the second term in Eq. (71), we

set x = 1/

√

dt and compute the Taylor expansion:

F ≤ 4 t

0



||α

(2)

|| + O(dt)



+ 4 t

02



β

(2)

dt

+

β

(3)

√

dt

+ O(1)



h

||α

(2)

|| + 1



√

dt + O(dt)

i

. (72)

Thus, if we are able to ﬁnd a Kraus representation for which both β

(2)

= 0 and β

(3)

= 0, the above expression

does not diverge in the dt→0 limit but rather provides the desired upper bound—Eq. (37) in the main text:

F ≤ 4 t

0

||α

(2)

|| with β = O(dt

2

), (73)

which importantly imposes the SQL-like scaling due to its linearity in t

0

.

C Asymptotic FFQC-valid CE bounds for qubit Liouvillians

C.1 Existence of asymptotic bound for any Liouvillian but the non-parallel rank-one Pauli noise

In what follows, we perform analytic minimization of the inﬁnitesimal-timestep CE bound introduced in

Sec. 4.2.3 that lead to the Eq. (37). For ease of notation we denote the matrix S

(1)

in Eq. (35) by

S

(1)

=



s

(1)

s

(1)†

s

(1)

¯

S

(1)



. (74)

Recall that our goal is to minimize over all Kraus decompositions K = Mς of the channel E

ω,dt

up to ﬁrst

order in dt. Expanding the matrix M up to ﬁrst order in dt, M = M

(0)

+

√

dt M

(1)

+ dt M

(2)

, and using Eq. (28)

the matrices M

(`)

satisfy

M

(0)

M

(0)

= S

(0)

M

(0)

M

(1)

+ M

(1)

M

(0)

= 0

M

(1)

M

(1)

+ M

(0)

M

(2)

+ M

(2)

M

(0)

= S

(1)

. (75)

These constraints enforce the following structure on the matrices M

(`)

M

(0)

= diag(1, 0, 0, 0),

M

(1)

=



0 0

T

0

¯

M

(1)

=

√

¯

L



, M

(2)

=



1

2

s

(1)

s

(1)†

s

(1)

¯

M

(2)



, (76)

where

¯

M

(2)

does not contribute to the ﬁrst order expansion of the channel and can thus be chosen at will.

By performing a power series expansion of the matrices α and β in terms of dt

α = α

(0)

+

√

dt α

(1)

+ dt α

(2)

+ O(dt

3/2

)

β = β

(0)

+

√

dt β

(1)

+ dtβ

(2)

+ dt

3/2

β

(3)

+ O(dt

2

), (77)

and using Eq. (29) and Eq. (30) with h = h

(0)

+ h

(1)

√

dt + h

(2)

dt the minimization over all equivalent Kraus

operators is equivalent to searching over Hermitian matrices h

(`)

that minimize the bound of Eq. (37).

To yield a non-trivial bound a Kraus decomposition has to satisfy α

(0)

= α

(1)

= β

(0)

= β

(1)

= 0. Expanding

Eq. (29) and Eq. (30) in powers of dt and minimizing order by order imposes the following structure for h

(0)

, h

(1)

h

(0)

=



0 0

0 H

(0)



, h

(1)

=



0 h

(1)†

h

(1)

H

(1)



, (78)

and the coeﬃcients α

(2)

and β

(2)

are given by

α

(2)

= ς

†



h

(1)

†

h

(1)

h

(1)†

H

(0)

¯

M

(1)

¯

M

(1)

H

(0)

h

(1)

¯

M

(1)

H

(0)

H

(0)

¯

M

(1)



ς (79)

β

(2)

= i



−1

2

σ

3

+ ς

†



h

(2)

00

h

(1)†

¯

M

(1)

¯

M

(1)

h

(1)

¯

M

(1)

H

(0)

¯

M

(1)



ς



. (80)

Accepted in Quantum 2017-09-04, click title to verify 24

Moreover, the coeﬃcient β

(3)

in Eq. (77) reads

β

(3)

= iς

†



1

2

h

(3)

00

+ s

(1)†

h

(1)

h

(1)

†

¯

M

(2)

+ (s

(1)†

H

(0)

+ h

(2)†

)

¯

M

(1)

0

1

2

¯

M

(1)

H

(1)

¯

M

(1)

+

¯

M

(2)

H

(0)

¯

M

(1)



+ h.c.



ς + iς

†



0 (0 0

1

2

)

†

¯

M

(1)

0 0



ς ,

(81)

where h.c. stands for the Hermitian conjugate. As none of the parameters h

(3)

00

, H

(1)

,

¯

M

(2)

, and h

(2)

appearing

in Eq. (81) enter into the expressions for α

(2)

and β

(2)

the norm of β

(3)

can be set to zero independently of all

lower orders of coeﬃcients for α and β. Indeed, the following choice of the free parameters

h

(3)

00

= − (s

(1)†

h

(1)

+ h

(1)

†

· s

(1)

)

H

(1)

=2 H

(0)

¯

M

(2)

= −

¯

M

(1)

Re(h

(2)

) =Re



h

(1)

− H

(0)

s

(1)



−

1

4





0

1





(82)

sets β

(3)

to zero.

C.1.1 Rank-one noise

We start with a general rank-one Liouvillian

¯

L

1G

r

deﬁned in Eq. (8)

¯

L

1G

r

= r r

†

=

γ

2

v v

†

(83)

with r = (x, y, z)

T

,

γ

2

= |x|

2

+ |y|

2

+ |z|

2

and the vector v has unit length such that v v

†

is a rank-1 projector.

Equation (76) implies

¯

M

(1)

=

p

γ

2

v v

†

. Expressing h

(1)

= (h

r

+ ih

i

)v + h

⊥

v

⊥

in a basis containing v allows to

re-write Eq. (80) as

β

(2)

= i(−

σ

3

2

+ ς

†



h

(2)

00

(h

r

− ih

i

)r

†

h.c. H

(0)

11

r r

†



ς ), (84)

where H

(0)

11

= v

†

H

(0)

v. We decompose the vector r = r

r

+ i r

i

, with r

r

= Re(r) and r

i

= Im(r). A little bit of

algebra allows one to rewrite Eq. (84) as

β

(2)

= iσ

0

(h

(2)

00

+

γ

2

H

(0)

11

)

+ i



−

σ

3

2

+ 2 σ

†

(h

r

− h

i

r

i

+ H

(0)

11

r

× r

i

)



, (85)

where we used the property [r

†

i

σ, r

†

r

σ] = 2i(r

i

× r

r

)

†

σ. We wish to set the operator β

(2)

to zero. The σ

0

term can be trivially put to zero by choosing h

(00)

2

= −

γ

2

H

(0)

11

. The remaining three terms can be put to zero

if the following conditions are fulﬁlled. Either r

r

and r

i

are linearly independent—in which case the vectors

r

, r

i

, r

r

× r

i

span the whole vector space; or r

r

and r

i

are parallel to each other and to the vector z—which

corresponds to the dephasing noise.

Hence, β

(2)

can be set to zero, so that Eq. (37) holds for any rank-one noise except for Pauli noise which is

not parallel to σ

3

. For this later case we have outlined a FFQC strategy in Sec. 4 that eﬀectively removes such

noise at the cost of slowing down the evolution by a constant factor.

C.1.2 Liouvillians of higher rank and CE bound

Let us now consider a general Lindbladian, it can always be written in the diagonal form

¯

L = r

1

r

†

1

M

r

2

r

†

2

M

r

3

r

†

3

(86)

with r

†

i

r

j

= 0 for i 6= j (as suggested by the direct sum). Again by v

j

=

r

j

|r

j

|

we denote the normalized vecors.

Now we show that if at least one Liouvillian

¯

L

j

= r

j

r

†

j

(e.g., j = 1) implies SQL scaling, then also

¯

L leads to

SQL scaling. To see this simply pick h

(1)

= (h

r

+ ih

i

)v

1

and H

(0)

= H

(0)

11

v

1

v

†

1

, Eq. (80) implies

β

(2)

= i



−

σ

3

2

+ ς †



h

(2)

00

(h

r

− ih

i

)r

†

1

h.c. H

(0)

11

r

1

r

†

1



ς



. (87)

Accepted in Quantum 2017-09-04, click title to verify 25

This expression is exactly equal to Eq. (84), consequently it can be set to zero if the Liouvillian

¯

L

1

leads to

standard scaling. Accordingly any higher rank Liouvillian that contains a non-Pauli rank-1 noise in its diagonal

decomposition leads to standard scaling.

C.1.3 Rank-two noise

Now consider a general Lindbadian of rank-two. Because of what we just showed in the previous section, the

only case which can potentially lead to Heisenberg scaling is the rank-two Pauli noise in Eq. (40c)

¯

L

2P

=

1

2

R

T

Ω





γ

1

γ

2

0





R

Ω

= r

1

r

†

1

M

r

2

r

†

2

,

(88)

with orthogonal real vecrtors r

1

=

p

γ

1

2

v

1

and r

2

=

p

γ

2

v

2

. The choice h

(2)

00

= 0, h

(1)

= h

1

v

1

+ h

2

v

2

(with

h

1

, h

2

∈ R), and

H

(0)

=





0 −ic 0

ic 0 0

0 0 0





(89)

expressed in the basis {v

1

, v

2

, v

3

= v

1

× v

2

} yields in Eq. (80)

β

(2)

= i



−

σ

3

2

+ 2 σ

†

(h

1

r

1

+ h

2

r

2

+ c r

1

× r

2

)



. (90)

As the three vectors {r

1

, r

2

, r

1

× r

2

} form a basis, the expression above can always be set to zero. Hence,

Eq. (37) holds for any rank two noise process as well.

C.1.4 Rank-three noise

Finally let us consider the case of rank-three noise. In this case the matrix

¯

M

(1)

is invertible such that β

(2)

in

Eq. (80) can be trivially set to zero by choosing h

(1)

=

¯

M

(1)−1

(0, 0, 1/4)

T

, h

(2)

00

= 0 and

¯

H

0

= 0.

Thus, we have shown that even in the most general situation where the experimentalist has full quantum

control any Liouvillian with the exception of the rank-one Pauli noise (not parallel to the generator of the

evolution) yields a QFI that is upper bounded by, F ≤ 4t||α

(2)

(h)||, i.e., the standard scaling.

C.2 Optimization of the bound for the exemplary noise-types

We derive below for exemplary noise types the general bounds on QFI presented Eq. (40) that encapsulate also

the FFQC-assisted schemes and show their optimality.

C.2.1 Rank-one noise

Let’s go back to the general rank-one Liouvillian in Eq. (83)

¯

L

1G

r

= r r

†

=

γ

2

v

1

v

†

1

. (91)

We call r

R

= Re(r) = (x

R

, y

R

, z

R

)

T

and r

I

= Im(r) = (x

I

, y

I

, z

I

)

T

, and choose the two other unit vectors v

2

and

v

3

such that the triplet {v

1

, v

2

, v

3

} forms an orthonormal basis. Expressing the component of the Hamiltonian

matrix in this basis, we may restrict to writing:

h

(1)

=





h

R

1

+ ih

I

1

h

R

2

+ ih

I

2

h

R

3

+ ih

I

3





H

(0)

=





H

(0)

11

H

(0)

12

H

(0)

13

H

(0)∗

12

H

(0)∗

13





,

(92)

as the missing terms do not enter in α

(2)

in Eq. (79), and thus do not inﬂuence the ﬁnal bound. Setting β

(2)

= 0

in Eq. (85) uniquely speciﬁes the parameters h

R

1

, h

I

1

and H

(0)

11











h

R

1

=

(x

2

I

+y

2

I

)z

R

−(x

I

x

R

+y

I

y

R

)z

I

4(|r

R

|

2

|r

I

|

2

−(r

†

R

r

R

)

2

)

h

I

1

=

(x

I

x

R

+y

I

y

R

)z

R

−(x

2

R

+y

2

R

)z

I

4(|r

R

|

2

|r

I

|

2

−(r

†

R

r

R

)

2

)

H

(0)

11

=

x

R

y

I

−x

I

y

R

4(|r

R

|

2

|r

I

|

2

−(r

†

R

r

R

)

2

)

.

(93)

Accepted in Quantum 2017-09-04, click title to verify 26

The rest of the parameters are free and have to be chosen such that ||α

(2)

|| is minimal. To this end we rewrite

Eq. (79)

α

(2)

= (h

(1)†

h

(1)

+

γ

2

v

†

1

H

(0)

H

(0)

v

1

) σ

0

+ (h

(1)†

H

(0)

v

1

) r

†

σ + (v

†

1

H

(0)

h

(1)

) σ

†

r

+ (2v

†

1

H

(0)

H

(0)

v

1

) σ

†

(r

R

× r

I

). (94)

A direct way to obtain an upper-bound is to set all the free parameters to zero, this yields

α|

(2)

0

= σ

0

(|h

R

1

|

2

+ |h

I

2

|

2

+ |r|

2

(H

(0)

11

)

2

) (95)

+H

(0)

11

σ

†



(h

R

1

+ ih

I

1

)r + (h

R

1

− ih

I

1

)r

∗

+ 2H

(0)

11

r

R

× r

I



and

||α|

(2)

0

|| =

|x|

2

+ |y|

2

+ 2|x

I

y

R

− x

R

y

I

|

16(|r

R

|

2

|r

I

|

2

− (r

R

· r

I

)

2

)

=

max{|x + iy|, |x − iy|}

2

16(|r

R

|

2

|r

I

|

2

− (r

R

· r

I

)

2

)

. (96)

This holds as an upper bound for any r, but might not be optimal in general. Now consider the deviation

from this bound in case where the free parameters are not set to zero, and denote h

1

= h

R

1

+ ih

I

1

, h =

(h

R

2

+ ih

I

2

, h

R

3

+ ih

I

3

)

T

and H = (H

(0)

12

, H

(0)

13

)

T

. One easily obtains

||α

(2)

|| − ||α|

(2)

0

|| = |h|

2

+ |H|

2



|r|

2

+ 2|r

R

× r

I

|



+



(H

†

h + H

(0)

11

h

1

) r + (h

†

H + H

(0)

11

h

∗

1

) r

∗



−



(H

(0)

11

h

1

) r + (H

(0)

11

h

∗

1

) r

∗



. (97)

In the case where one of the coeﬃcients of r is zero the only negative term in the expression above vanishes

making ||α|

(2)

0

|| the optimal bound. As follows from Eq. (93), if x = 0 (up to a rotation around z) this happens

because H

(0)

11

= 0, while the case z = 0 implies h

1

= 0. In general, the bound in Eq. (96) could be potentially

improved.

C.2.2 Pauli noise in the z-direction

This case corresponds to the much studied case of dephasing noise [15, 19, 21–23]. As the noise commutes with

the evolution our FFQC can not detect it and therefore our FFQC strategy cannot improve the metrological

performance. Nevertheless we treat it here for the sake of completeness.

The Liouvillian is given by

¯

L

1P

z

=





0

γ

2





= r r

†

, (98)

with r = (0, 0,

p

γ/2). It is easy to see from Eq. (85) that β

(2)

= 0 implies 4

p

γ/2 h

r

= 1. Whereas the norm

of α

(2)

in Eq. (79) is given by

||α

(2)

|| = |h

(1)

|

2

+ r

†

H

(0)

H

(0)

r

+|h

(1)†

H

(0)

r + r H

(0)

h

(1)

|. (99)

Which is minimized by

||α

(2)

|| = h

2

r

=

1

8γ

(100)

for h

(1)

= h

r

|r|

and H

(0)

= 0.

Accepted in Quantum 2017-09-04, click title to verify 27

C.2.3 Rank-two Pauli noise

Finally, let us consider the case of rank two Liouvillian noise. For convenience we perform a basis rotation so

that the Liouvillian is given by X-Y Pauli noise

¯

L

2P

=

1

2





γ

1

γ

2

0





, (101)

whereas the evolution is given by

ω

2

(s

θ

c

ϕ

, s

θ

s

ϕ

, c

θ

)

T

σ. In the rotated basis, the relevant elements in the series

expansion of the Hermitian operator h read:

h

(1)

=





h

r

1

+ ih

i

1

h

r

2

+ ih

i

2

h

r

3

+ ih

i

3





, (102)

H

(0)

=





H

(0)

11

R

(0)

12

− iI

(0)

12

H

(0)

13

R

(0)

12

+ iI

(0)

12

H

(0)

22

H

(0)

23

H

(0)∗

13

H

(0)∗

23

H

(0)

33





. (103)

The requirement that β

(2)

= 0 imposes the following conditions











h

r

1

= −

s

θ

c

ϕ

4

√

γ

1

/2

h

r

2

= −

s

θ

s

ϕ

4

√

γ

2

/2

I

(0)

12

= −

c

θ

2

√

γ

1

γ

2

.

(104)

For α

(2)

one gets a cumbersome expression, whose exact form is not important. It is suﬃcient to notice that

it can be written as

α

(2)

=

σ

0

2



2h

(1)†

h

(1)

+ γ

1

(|H

(0)

11

|

2

+ R

(0)2

12

+ I

(0)2

12

+ |H

(0)

13

|

2

)

+γ

2

(|H

(0)

22

|

2

+ R

(0)2

12

+ I

(0)2

12

+ |H

(0)

23

|

2

)



+ f(h

(1)

, H

(0)

) σ

†

ˆ

n,

(105)

where f(h

(1)

,

¯

H

(0)

) = 0 when all the parameters except h

r

1

, h

r

2

and I

(0)

12

are set to zero. Since this is also the

choice that minimizes the pre-factor of σ

0

in the expression above, it is the optimal choice for the norm of the

whole operator. This implies the optimal bound

||α

(2)

|| = (h

r

1

)

2

+ (h

r

2

)

2

+

γ

1

+ γ

2

I

(0)2

12

=

c

2

θ

(γ

1

+ γ

2

) + s

2

θ

(γ

1

s

2

ϕ

+ γ

2

c

2

ϕ

)

8γ

1

γ

2

. (106)

D Analysis of the X-Y noise

The full master equation incorporating the X-Y noise introduced in Eq. (10) reads

dρ

ω

(t)

dt

= −i

ω

2

[σ

3

, ρ

ω

(t)] +

γ

2

[p σ

1

ρ

ω

(t)σ

1

+ (1 − p)σ

2

ρ

ω

(t)σ

2

− ρ

ω

(t)] (107)

and can be integrated analytically, e.g., by methods described in [67], to explicitly derive the probe dynamics.

D.1 Simple FFQC strategy

Let us carefully explain the sequential FFQC strategy used in Sec. 4.3.2 and Sec. 6.1.2, which is inspired

by the strategy of Sec. 4.1 for perpendicular dephasing [28]. Recall that neglecting the higher orders in dt

the probe-plus-ancilla state after an evolution of inﬁnitesimal duration can follow two possible branches, see

Eq. (14):

%(t + dt) = %

C

(t + dt) + %

E

(t + dt). (108)

Accepted in Quantum 2017-09-04, click title to verify 28

Either no error happened %

C

(t+ dt) = (1−

γ

2

dt)%(t)−i

ω

2

dt[σ

3

⊗ , %(t)] and the parameter was imprinted on the

state or with probability

γ

2

dt an error occurred %

E

(t+dt) =

γ

2

dt



p (σ

1

⊗ )%(t)(σ

1

⊗ )+(1−p)(σ

2

⊗ )%(t)(σ

2

⊗ )



.

If in addition the state at time t belongs to the code space % ∈ B(H

C

) with H

C

= span(|00i, |11i) and

H

E

= span(|01i, |10i), the two branches of the inﬁnitesimal evolution can be distinguished in a non-demolition

manner by projecting onto the code the error spaces via a parity measurement (since %

C

(t + dt) ∈ B(H

C

) and

%

E

(t + dt) ∈ B(H

E

)). If the state happens to be in the correct branch we simply leave it alone, wheres if an

error is detected our strategy consists of mapping the error branch back into the code space by correcting for

the most probable error term that we assume to be σ

2

⊗ (0 < p ≤1/2) without loss of generality. Thus, after

the correction step the error branch reads ¯%

E

(t + dt) = (σ

2

⊗ )%

E

(t + dt)(σ

2

⊗ ) with

¯%

E

(t + dt) =

γ

2

dt



(1 − p)%(t) + p(σ

3

⊗ )%(t)(σ

3

⊗ )



+ O(dt

2

). (109)

Note that here we assume that the control steps are performed fast enough, so that Eq. (109) holds up to ﬁrst

order in dt

5

. Importantly, after the correction the ﬁnal state belongs to the code space again ¯%(t + dt) ∈ B(H

C

),

and we can continue running the strategy. Another important point is that we keep track of the number of errors

that have been detected before a time t in a classical error register. This can be made explicit by introducing an

error register mode R initialized in the state |0i

R

for each probe qubit, and a Hermitian operator that increases

its count by one R

+

|mi

R

= |m + 1i

R

. For example the register can be modelled by a bosonic mode a with

R

+

=

1

a

†

a

†

. Such a FFQC strategy leads to an eﬀective evolution of the probe+register system (the ancilla

does not play a role at this stage) given by the master equation

d

dt

%

ω

(t) = −i

ω

2

[σ

3

⊗

R

, %

ω

(t)]+

γ

2



−%

ω

(t) +

γ

2

( ⊗ R

+

)



(1 − p)%

ω

(t) + p(σ

3

⊗

R

)%

ω

(t)(σ

3

⊗

R

)



( ⊗ R

+

)



.

(110)

For a single probe initialized in a state |+i =

1

√

2



1



(or

|00i+|11i

√

2

if we remember the ancillary qubit) the ﬁnal

state resulting from such an eﬀective evolution after time t reads

%

ω

(t) =

∞

X

m=0

p(m; t) ¯%

m

(t) ⊗ |mihm|

R

, (111)

where the total number of errors m detected during time t follows the Poissonian distribution p(m; t) =

e

−γt/2

(γt/2)

m

/m!, and the state of the probe conditional on the occurrence of m errors is

¯%

m

(t) =

1

2

1 (1 − 2p)

m

e

iωt

(1 − 2p)

m

e

−iωt

1

!

. (112)

D.2 Description of the probe channel

In particular, one may derive the Choi-Jamiolkowski (CJ) matrix [75] representing the probe map E

X-Y

ω,t

, which

is deﬁned then as P(ω, t) :=E

X-Y

ω,t

⊗ I[I] with |Ii=|00i + |11i and reads

P(ω, t) = 2







C

γ

2

0 0 C

Ω

− i˜ωS

Ω

0 S

γ

2

−˜γS

Ω

0

0 −˜γS

Ω

S

γ

2

0

C

Ω

+ i˜ωS

Ω

0 0 C

γ

2







, (113)

where

C

x

:=

1

2

e

−

γt

2

cosh(xt), S

x

:=

1

2

e

−

γt

2

sinh(xt),

µ :=

γ(1 − 2p)

2

, Ω :=

p

µ

2

− ω

2

,

˜γ :=

µ

Ω

, ˜ω :=

ω

Ω

. (114)

For t >0 the CJ-matrix (113) is of rank four, so the probe channel, E

X-Y

ω,t

, is full-rank always possessing four Kraus

operators in Eq. (12). By performing the spectral decomposition of the CJ-matrix, P(ω, t) =

P

4

i=1

λ

i

|e

i

ihe

i

|,

5

Note that the control steps need not be ultra fast, just suﬃciently fast so that the ﬁrst order approximation holds. The duration

of these control-pulses depends on the strength of the decoherence process.

Accepted in Quantum 2017-09-04, click title to verify 29

we obtain the CJ-canonical Kraus representation of channel E

X-Y

ω,t

after identifying

√

λ

i

|e

i

i= K

i

⊗ |Ii for all

i, where

K

1

=

q

S

γ

2

− ˜γS

Ω

σ

1

, K

2

= −i

q

S

γ

2

+ ˜γS

Ω

σ

2

, (115)

K

3

=

r

C

γ

2

−

q

e

−γt

4

+ (˜γS

Ω

)

2





−

p

e

−γt

4

+(˜γS

Ω

)

2

C

Ω

+i˜ωS

Ω

0

0 1





,

K

4

=

r

C

γ

2

+

q

e

−γt

4

+ (˜γS

Ω

)

2





p

e

−γt

4

+(˜γS

Ω

)

2

C

Ω

+i˜ωS

Ω

0

0 1





.

On the other hand, we may rewrite the action of the probe channel, E

X-Y

ω,t

, in the Pauli basis and thus derive

its corresponding dynamical matrix S deﬁned in Eq. (11):

S(ω, t) =







C

γ

2

+ C

Ω

0 0 i˜ωS

Ω

0 S

γ

2

− ˜γS

Ω

0 0

0 0 S

γ

2

+ ˜γS

Ω

0

−i˜ωS

Ω

0 0 C

γ

2

− C

Ω







. (116)

In the special case of balanced X-Y noise, for which p = 1/2, all the above expressions dramatically simplify

(µ = ˜γ = 0, Ω = iω, ˜ω = −i), as the Liouvillian part of Eq. (107) yields a noisy channel that commutes with

parameter encoding, or in other words, the channel E

X-Y

ω,t

becomes then phase-coviariant with respect to rotations

generated by σ

3

[65].

D.3 Parallel CE bound

We follow the methods of Ref. [22] summarised in Sec. 4.2.2, in order to compute the parallel CE bound (22)

for the X-Y noise and the scenario depicted Fig. 3(b):

F

(b)↑

X-Y

(N, t) := 4 min

h(t)



N||α(t)|| + N(N − 1)||β(t)||

2



, (117)

which then allows to determine an upper bound on the maximal QFI rate per probe, deﬁned in Eq. (51):

f

(b)↑

N

:= max

t

F

(b)↑

X-Y

(N, t)

N t

. (118)

For given γ, ω, N and t, we substitute into Eq. (117) the canonical Kraus operators of Eq. (115), K(t) =

(K

1

, . . . , K

4

)

T

, and their shifted derivatives,

˙

K(t) − ih(t)K(t), so that the minimisation can be eﬃciently per-

formed by means of semi-deﬁnite programming (SDP) [22]. In order to compute f

(b)↑

N

, we repeat such procedure

for various t, while numerically optimising Eq. (118) over t that then yields t

(b)↑

opt

.

However, in the special single-qubit (N = 1) and asymptotic cases (N → ∞), we are able to determine the

analytic expressions for F

(b)

↑

(t) in Eq. (117) after correctly choosing an analytic ansatz form (motivated by

the numerical SDPs) for the Hermitian matrix h(t) and explicitly performing the minimisation in Eq. (117).

Although such expressions cannot be claimed to be derived in a fully analytic manner, their correctness can be

veriﬁed numerically to arbitrary precision thanks to the SDP formulation.

Nevertheless, due to their cumbersome form we present below only their simpliﬁed versions for the special

case of balanced X-Y noise. On the other hand, in case of the asymptotic (N → ∞) regime, we provide a

compact derivation of slightly loosened CE bound (117), which possesses much simpler form but nonetheless

approximates Eq. (117) to negligible precision, what we have numerically veriﬁed.

D.4 Parallel CE bound for p= 1/2

In the case of balanced X-Y noise we explicitly compute the CE bound (117) thanks to the recently derived

analytic generalisation of the SDP-formulation of Eq. (117) for phase-covariant, unital qubit maps [65]:

F

(b)↑

X-Y

(N, t)



p=

1

2

=

N

2

t

2

1 + 2N e

γt

S

γ

2

, (119)

Accepted in Quantum 2017-09-04, click title to verify 30

which is obtained after choosing as the channel Kraus representation the canonical one of Eq. (115), and

optimally setting in Eq. (117):

h

opt

(t)|

p=

1

2

=

t

2κ







0 1 − N 0 0

1 − N 0 0 0

0 0 −κ κ − 1

0 0 κ − 1 −κ







(120)

with κ=1 + 2N e

γt

S

γ

2

.

Thanks to analytic form of Eq. (119), we explicitly compute the corresponding upper bound on maximal QFI

rate (118):

f

(b)↑

N



p=

1

2

=

2

γ

N



1 + W



2−N

eN



2 −



1 − e

1+W

(

2−N

eN

)



N

(121)

which occurs at t

(b)↑

opt

(N) =

1+W

(

2−N

eN

)

γ

with W (x) representing the Lambert function.

Note that for N = 2, f

(b)↑

N=2



p=

1

2

=2/(γe), so that the bound (121) coincides with the QFI rate attained by our

proposed sequential strategy of Fig. 3(a2), i.e., f

(s)

in Eq. (55). As for any smaller p (more noise asymmetry),

the sequential strategy can perform only better in comparison to the parallel schemes (b) of Fig. 1—see also

Fig. 5—this proves that for any p there always exist a non-trivial range of N > 1 for which the sequential

protocol of Fig. 3(a2) outperforms all the parallel schemes.

D.5 Asymptotic parallel CE bound

In the asymptotic regime of N →∞ and ﬁxed t, it is always optimal to set β(t) = 0 in Eq. (117) if possible, so

that the parallel CE bound (117) then reads

F

(b)↑

X-Y

(N →∞, t) = 4N min

h(t) s.t. β(t)=0

||α(t)||. (122)

We determine an upper-bound on the r.h.s. of Eq. (122) making use of the M-matrix formulation introduced

in Sec. 4.2.2 and the alternative expressions for α and β of Eqs. (29) and (30).

Basing on the dynamical matrix S stated Eq. (116), we derive the M matrix following Eq. (28) and choosing

M= M

†

=

√

S

T

:

M(ω, t) =







ε + ζ 0 0 −iχ

0 ∆ 0 0

0 0 Γ 0

iχ 0 0 ε − ζ







, (123)

with

ε

2

=

Θ

+

2

, ζ

2

=

C

Ω

2

2Θ

−

, χ

2

=

˜ωS

Ω

2

2Θ

−

, (124)

∆

2

= S

γ

2

− ˜γS

Ω

, Γ = S

γ

2

+ ˜γS

Ω

, (125)

and Θ

±

=C

γ

2

±

q

C

γ

2

− C

Ω

2

− ˜ωS

Ω

2

.

We inspect Eq. (30) and, in order to satisfy the asymptotic condition β(t) = 0, we set following an educated

guess:

h =

˙

ζχ − ζ ˙χ

∆ Γ







0 0 0 0

0

σ

2

0

0 0 0 0







, (126)

which then after evaluating α

(b)

ω,γ,p

(t):= ||α(t)|| according to Eq. (29) yields an upper bound on Eq. (122):

F

(b)↑

X-Y

(N →∞, t) ≤ 4 N α

(b)

ω,γ,p

(t), (127)

Accepted in Quantum 2017-09-04, click title to verify 31

where

α

(b)

ω,γ,p

(t) = 2( ˙ε

2

+

˙

ζ

2

+ ˙χ

2

) +

+

˙

∆

2

+

˙

Γ

2

+

(∆

2

+ Γ

2

)(ζ ˙χ −

˙

ζχ)

2

∆

2

Γ

2

(128)

is of complex but importantly of fully analytic form.

E Ziv-Zakai precision bound and the QFI

Consider the parameter ω with prior distribution p

0

(ω). The state at time t is denoted as ρ

ω,t

. The Ziv-Zakai

bound on the AvMSE of the parameter reads [48]

hδω

2

i ≥

Z

∞

0

dττ

Z

∞

−∞

dω min[p

0

(ω), p

0

(ω + τ )]Pr

e

(ω, ω + τ ),

where Pr

e

(ω, ω+τ) is minimum error probability of the binary hypothesis testing between ω and ω+τ . Quantum

mechanically it is given by the trace distance [48]

Pr

e

(ω, ω + τ ) ≥

1

2

(1 − D[ρ

ω,t

, ρ

ω+τ,t

])

:=

1

2

(1 −

1

2

||ρ

ω,t

− ρ

ω+τ,t

||

1

). (129)

We use the triangle inequality to decompose the trace distance in elementary steps

D[ρ

T

ω

, ρ

T

ω+τ

] ≤ min



1,

Z

ω+τ

ω

D[ρ

τ

0

,t

, ρ

τ

0

+dτ

0

,t

]dτ

0



, (130)

and the local expansion of the trace distance D[ρ

τ

0

,t

, ρ

τ

0

+dτ

0

,t

] ≤ dτ

0

√

F(ρ

τ

0

,t

)

2

we rewrite the error probability

as

Pr

e

(ω, ω + τ ) ≥

1

2

(1 − min



1,

Z

ω+τ

ω

dτ

0

p

F(ρ

τ

0

,t

)

2



(131)

Now we are in position to plug in the bound for the QFI, as we showed for the SQL Liouvillians the QFI of the

state at time t satisﬁes F

L

≤ 4α

L

t, such that the error probability fulﬁlls

Pr

e

(ω, ω + τ ) ≥

1

2

max(0, 1 − τ

√

α

L

t). (132)

And the Ziv-Zakai bound reads

hδω

2

i ≥

1

2

Z

1

√

α

L

t

0

dττ (1 − τ

√

α

L

t)

Z

∞

−∞

dω min[p

0

(ω), p

0

(ω + τ )]

Which in the case t  1 leads to

hδω

2

i ≥

1

12α

L

t

, (133)

for any prior which respects the regularity condition lim

τ→0

R

∞

−∞

dω min[p

0

(ω), p

0

(ω + τ )] → 1.

F Strategy for noiseless frequency estimation

In the context of phase alignment it is known [7] that the state minimizing the cost-function sin

2

(

ϕ

2

) is given by

|Ψi =

r

2

M + 2

N

X

n=0

sin(

π(n + 1)

N + 2

) |ni, (134)

where |ni is the symmetric eigenstate of the total spin J

z

=

1

2

P

N

k=1

σ

(k)

3

corresponding to the eigenvalue

2n−N

2

.

We call the rotated state |Ψ

ϕ

i = e

iϕJ

z

|Ψi. For the measurement given by the covariant POVM {|θihθ|} with

|θi = (N + 1)

−1/2

P

e

iθn

|ni and the state |Ψ

ϕ

i we denote the probability to observe an outcome θ by

p(θ − ϕ) = p(θ|ϕ) := |hθ|Ψ

ϕ

i|

2

. (135)

Accepted in Quantum 2017-09-04, click title to verify 32

It has been shown in [7] that the probability p(δ) in Eq. (135) yields a cost

C =

Z

π

−π

sin

2

(

δ

2

)p(δ)dδ =

π

2

4N

2

(136)

in the regime of large N . Note that the cost-function sin

2

(

δ

2

) is appropriate in the context of phase estimation

where the estimated parameter is deﬁned a circle ϕ ∈ [−π, π). But in the context of frequency estimation the

parameter of interest is deﬁned on the whole line ω ∈ (−∞, ∞) (similar for the resulting phase ϕ = ωt), and the

appropriate cost-function is the variance (MSE). We show below how it can be lower bounded from the value

of C.

Using the fact that δ

2

≤ sin

2

(

δ

2

)π

2

on the interval [−π, π] we get a bound on the second moment.

V =

Z

π

−π

δ

2

p(δ)dδ ≤ π

2

C =

π

4

4N

2

. (137)

On the other hand, one can also upper-bound the tail probabilities from the cost C

T (r) =

Z

π

rπ

p(δ)dδ ≤

C

2 sin

2

(rπ/2)

(138)

The average mean square error over all possible outcomes of the covariant POVM is given by

hδϕ

2

i =

Z

π

−π

dθ p(θ)

Z

dϕ(ϕ − ˜ϕ

θ

)

2

p(ϕ|θ) (139)

=

Z

π

−π

dθ

Z

dϕ(ϕ − ˜ϕ

θ

)

2

p(θ − ϕ)p

0

(ϕ), (140)

where p

0

(ϕ) is the prior knowledge and ˜ϕ

θ

is the optimal estimator of the phase given the measurement outcome

θ. Choosing an explicit estimator ˜ϕ

θ

= θ (which might not be optimal for the MSE) and assume

p

0

(ϕ) =

(

1

2πr

|ϕ| ≤ rπ

0 |ϕ| > rπ.

(141)

This implies

hδϕ

2

i =

Z

π

−π

dθ

Z

dϕ(ϕ − θ)

2

p(θ − ϕ)p

0

(ϕ) (142)

=

1

2πr

Z

π

−π

dθ

Z

rπ

−rπ

dϕ(ϕ − θ)

2

p(θ − ϕ). (143)

Now we can change the integration variables to Σ = ϕ + θ and ∆ = ϕ − θ. The domain of integration then is

a rotated rectangle, but since the integrand is positive we can enlarge the domain to the minimal square that

contains it without lowering the integral. This implies

hδϕ

2

i ≤

1

2πr

R

(1+r)π

−(1+r)π

R

(1+r)π

−(1+r)π

dΣd∆

2

∆

2

p(∆)

=

r+1

2r

R

(1+r)π

−(1+r)π

∆

2

p(∆)d∆

=

r+1

2r



V + 2

R

π(1+r)

π

∆

2

p(∆)d∆



. (144)

Finally, because p(∆) is periodic, the second term in brackets in the last expression is upper-bounded by

Z

π(1+r)

π

∆

2

p(∆)d∆ ≤ π

2

(1 + r)

2

Z

π

(1−r)π

p(∆) d∆ = π

2

(1 + r)

2

T (1 − r) (145)

≤ C

π

2

(1 + r)

2

2 sin

2

((1 − r)π/2)

, (146)

which yields

hδϕ

2

i ≤ C

(r + 1)π

2

2r



1 +

(1 + r)

2

sin

2

((1 − r)π/2)



. (147)

Accepted in Quantum 2017-09-04, click title to verify 33

Now, minimizing the above bound over r and substituting for the cost C according to Eq. (136), we obtain

hδϕ

2

i ≤ C κ π

2

≤

π

4

κ

4

1

N

2

, (148)

which is attained for r

opt

≈ 0.33 that gives κ ≈ 6.74.

It remains to show how the above bound applies within the frequency estimation scenario. Let us assume the

prior distribution of the frequency parameter to also have an interval form:

p

0

(ω) =

(

1

2ω

0

if |ω| ≤ ω

0

0 if |ω| > ω

0

.

(149)

Remark that if the interval is not centred at zero, it can be always shifted be means of FFQC. Given a total

resource t

0

= N t, we can construct a circuit that is equivalent to running N =

t

0

ω

0

πr

opt

parallel qubits prepared

in the state (134) for a time t =

r

opt

π

ω

0

. This choice maps the frequency estimation task on the estimation of

the phase ϕ = ωt = ω

r

opt

π

ω

0

for the prior deﬁned in Eq. (141) with r = r

opt

, which we have solved above.

Consequently, the MSE for the frequency reads

hδω

2

i =

hδϕ

2

i

t

2

≤

π

4

κ

4

1

t

2

N

2

=

π

4

κ

4

1

(t

0

)

2

. (150)

However, in contract to the case of phase estimation, note that if the evolution is slowed down—as it is for the

rank-one Pauli noise—the time t (and, hence, t

0

) has to be set longer, what aﬀects the frequency MSE above.

Accepted in Quantum 2017-09-04, click title to verify 34

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