A resource efficient approach for quantum and classical
simulations of gauge theories in particle physics
Jan F. Haase
1,2
, Luca Dellantonio
1,2
, Alessio Celi
3,4
, Danny Paulson
1,2
, Angus Kan
1,2
,
Karl Jansen
5
, and Christine A. Muschik
1,2,6
1
Department of Physics & Astronomy, University of Waterloo, Waterloo, ON, Canada, N2L 3G1
2
Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada, N2L 3G1
3
Departament de F
´
ısica, Universitat Aut
`
onoma de Barcelona, E-08193 Bellaterra, Spain
4
Center for Quantum Physics, Faculty of Mathematics, Computer Science and Physics, University of Innsbruck, Innsbruck A-6020, Austria
5
NIC, DESY, Platanenallee 6, D-15738 Zeuthen, Germany
6
Perimeter Institute for Theoretical Physics, Waterloo, ON, Canada, N2L 2Y5
Gauge theories establish the standard model
of particle physics, and lattice gauge the-
ory (LGT) calculations employing Markov
Chain Monte Carlo (MCMC) methods have
been pivotal in our understanding of fun-
damental interactions. The present limita-
tions of MCMC techniques may be overcome
by Hamiltonian-based simulations on classi-
cal or quantum devices, which further pro-
vide the potential to address questions that
lay beyond the capabilities of the current
approaches. However, for continuous gauge
groups, Hamiltonian-based formulations in-
volve infinite-dimensional gauge degrees of
freedom that can solely be handled by trunca-
tion. Current truncation schemes require dra-
matically increasing computational resources
at small values of the bare couplings, where
magnetic field effects become important. Such
limitation precludes one from ‘taking the con-
tinuous limit’ while working with finite re-
sources. To overcome this limitation, we
provide a resource-efficient protocol to sim-
ulate LGTs with continuous gauge groups
in the Hamiltonian formulation. Our new
method allows for calculations at arbitrary
values of the bare coupling and lattice spac-
ing. The approach consists of the combina-
tion of a Hilbert space truncation with a regu-
larization of the gauge group, which permits
an efficient description of the magnetically-
dominated regime. We focus here on Abelian
gauge theories and use 2 + 1 dimensional quan-
tum electrodynamics as a benchmark exam-
ple to demonstrate this efficient framework to
achieve the continuum limit in LGTs. This
possibility is a key requirement to make quan-
Jan F. Haase: jan.frhaase@gmail.com, contributed equally
Luca Dellantonio: luca.dellantonio@uwaterloo.ca, contributed
equally
Alessio Celi: alessio.celi@uab.cat, contributed equally
titative predictions at the field theory level
and offers the long-term perspective to utilise
quantum simulations to compute physically
meaningful quantities in regimes that are pre-
cluded to quantum Monte Carlo.
Contents
1 Introduction 2
2 Minimal encoding of LGTs with continuous
gauge groups 4
2.1 QED in two dimensions . . . . . . . . 4
2.2 QED Hamiltonian for physical states . 5
3 Transformation into the magnetic representa-
tion 8
4 Performance and application of the new ap-
proach 11
4.1 Phenomenological analysis . . . . . . . 11
4.2 Fidelity and convergence of the two
representations . . . . . . . . . . . . . 12
4.3 Estimation of hi . . . . . . . . . . . . 14
5 Generalisations: Dynamical matter and arbi-
trary torus 15
5.1 Including dynamical charges . . . . . . 15
5.2 Hamiltonian for an arbitrary torus and
charges . . . . . . . . . . . . . . . . . 16
6 Conclusions and outlook 17
7 Acknowledgements 18
References 18
A Dimensions of the (2+1) dimensional QED
Hamiltonian 20
B Hamiltonian in the link formalism and link-to-
rotator translation rules 21
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arXiv:2006.14160v3 [quant-ph] 21 Jan 2021
B.1 Effective Hamiltonian for a minimal
lattice in the link formulation . . . . . 21
B.2 An intuitive picture of the rotator and
link operator relation . . . . . . . . . . 22
B.3 Gauge field creation in the rotator and
string picture . . . . . . . . . . . . . . 22
C Diagonalisation of the magnetic gauge field
Hamiltonian 23
D Asymptotic behaviour of the ground state ex-
pectation value of 24
E Truncation effects in the strong coupling
regime 24
F Numerical determination of L
opt
25
1 Introduction
Gauge theories are the basis of high energy physics
and the foundation of the standard model (SM).
They describe the elementary interactions between
particles, which are mediated by the electroweak and
strong forces [1–3], making the SM one of the most
successful theories with tremendous predictive power
[4]. Still, there are numerous phenomena which can-
not be explained by the SM. Examples include the
nature of dark matter, the hierarchy of forces and
quark masses, the matter antimatter asymmetry and
the amount of CP violation [5]. Answering these ques-
tions and accessing physics beyond the SM, though,
often requires the study of non-perturbative effects.
A very successful approach to address non-
pertubative phenomena is lattice gauge theory (LGT)
[6–8], as proposed by Kenneth Wilson in 1974 [9]. In
LGT, Feynman’s path integral formulation of quan-
tum field theories (QFTs) is employed on an Eu-
clidean space-time grid. Such a discretised form of
the path integral allows for numerical simulations uti-
lizing Markov Chain Monte Carlo (MCMC) methods.
The prime target of LGT is quantum chromodynam-
ics (QCD), i.e. the theory of strong interactions be-
tween quarks and gluons. In this field, LGT has been
extremely successful, allowing for example the com-
putation of the the low-lying baryon spectrum [10],
the structure of hadrons, fundamental parameters of
the theory and many more [11–14].
However, many of the aforementioned open ques-
tions in modern physics cannot be addressed within
the standard approach, due to the sign-problem [15–
17] that renders MCMC methods ineffective. A pos-
sible solution is to employ a Hamiltonian formula-
tion of the underlying model. Classical Hamiltonian-
based simulations using tensor network states (TNS),
including fermionic projected entangled-pair states,
have been successful [18–28], but are so far restricted
to mostly one spatial dimension (for link model 2D
calculations with DMRG and tree tensor network see
e.g [29, 30]). Consequently, there is a necessity for new
approaches to both access higher dimensions and ad-
dress problems where standard MCMC methods fail.
It is presently not known whether efficient classical
methods can be developed to overcome this problem.
Hamiltonian-based simulations on quantum hard-
ware provide an alternative route, since there is no
such fundamental obstacle to simulating QFTs in
higher dimensions [32–35]. Therefore, this approach
holds the potential to address questions that can-
not be answered with current and even future clas-
sical computers. The rapidly evolving experimen-
tal capabilities of quantum technologies [36, 37] have
led to proof-of-concept demonstrations of simulators
tackling one-dimensional theories [38–44]. Extending
these results to higher dimensions is a lively area of
research [32, 45–54], since it represents a crucial step
for this field, and realisations on ‘Noisy Intermediate-
Scale Quantum’ devices [55, 56], i.e. current quantum
hardware, require novel approaches to make this leap.
To meet this challenge, we provide a resource-
efficient approach that facilitates the quantum simula-
tion of higher dimensional LGTs that would otherwise
be out of reach for current and near-term quantum
hardware, which is exemplified Table 1. In addition,
purely classical simulations based on the Hamiltonian
formalism also benefit from our resource-optimised
approach. Hence, we bring both quantum and clas-
sical calculations closer to developing computational
strategies that do not rely on Monte Carlo methods,
and thus circumvent their fundamental limitations.
Our new approach addresses the important problem
of reaching the continuum limit (in which the lattice
spacing approaches zero) with finite computational re-
sources. Since QFTs are continuous in their time and
space variables, the need to take a controlled con-
tinuum limit is inherent to any lattice approach and
necessary to extract physically relevant results from a
lattice simulation.
Taking QCD as a concrete example, we require
an accurate description for particles interacting at
both short and long distances. Lattice QCD and
other LGTs offer the unique tool to investigate both
regimes. At long distances, e.g. the bound state spec-
trum can be computed. At short distances, and after
taking the continuum limit, it is possible to connect
the perturbative results derived with QFTs with non-
perturbative simulations, thus assessing the range in
which perturbation theory is valid. However, taking
the continuum limit is in general computationally ex-
pensive. MCMC methods, for instance, have the in-
trinsic problem of autocorrelations, that become more
and more severe when decreasing the lattice spacing.
This drawback in turn leads to a significant increase in
the computational cost, and fixes the smallest value of
the lattice spacing that can be reached. On one hand,
Hamiltonian approaches circumvent this problem. On
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Table 1: Computational cost for different approaches. We estimate the number of states required to reach a 1% accuracy
in the expectation value of the two-dimensional plaquette in QED (see Sec. 4.3) when compared to the value we obtain with
our method considering a maximum of 9261 basis elements. The three columns refer, from left to right, to the standard
approach described in Sec. 2.1, our approach (see Sec. 3) using a fixed group Z
N
, and finally our optimised strategy, in which
the order of the group N is scaled with the bare coupling g (see Sec. 4). The shown savings in computational resources bring
quantum simulations with current technology within reach. Note that 125 states correspond to seven qubits. We present a
robust implementation strategy for ion-based quantum computers in [31].
the other, however, Hamiltonian-based formulations
face the challenge that for continuous (Abelian and
non-Abelian) gauge groups, local gauge degrees of
freedom are defined in an infinite dimensional Hilbert
space. As a consequence, any simulation – classical or
quantum – requires a truncation of the gauge fields,
which is inherently at conflict with the required con-
tinuum limit.
In this work, we present a practical solution to over-
come this crucial bottleneck and to allow for resource-
efficient Hamiltonian simulations of LGTs. Although
our approach is general and applicable to LGTs of
any dimension, we consider two-dimensional quantum
electrodynamics (QED) as a benchmark example.
Truncation of the gauge fields is typically performed
in the ‘electric basis’, i.e. the basis in which the elec-
tric Hamiltonian and Gauss’ law are diagonal. As
such, truncation preserves the gauge symmetry, and
the resulting models are known as gauge magnets or
link models [57–59], which are of direct relevance in
condensed matter physics [60–63]. As recently shown
in Ref. [54], spin-1/2 truncations are within the reach
of current quantum simulators. From the perspective
of fundamental particle interactions, electric trunca-
tions can result in an accurate description of the sys-
tem in the strong coupling regime. However, by de-
creasing the value of the coupling or equivalently the
lattice spacing, the magnetic contributions to the en-
ergy become increasingly important and the number
of states that have to be included in the electric basis
grows dramatically (a similar increase can be realised
by adding an auxiliary spatial dimension to the lat-
tice [64]). An alternative approach to describe the
gauge degrees of freedom is to approximate continu-
ous gauge groups with discrete ones, for instance, to
approximate U(1) with Z
2L+1
(L ∈ N) [65–67]. Such
approaches also face similar limitations as the ones de-
scribed above, as L has to be progressively increased
with decreasing coupling.
A natural solution to simulate the weak coupling
regime consists of exploiting the self-duality [68] of the
electric and magnetic terms by Fourier transforming
the Hamiltonian and working in the ‘magnetic basis’,
i.e. the basis in which magnetic interactions are diag-
onal, as suggested in [69]. However, the fact that the
magnetic degrees of freedom are continuous variables
with a gapless spectrum, poses intricate challenges for
a resource-efficient truncation scheme, that have yet
(to the best of our knowledge) to be addressed. In
this work, we provide a practical solution by combin-
ing state truncation with a gauge group discretisa-
tion that is dynamically adjusted to the value of the
coupling. This approach allows for controlled simula-
tions at all values of the bare coupling, smoothly con-
necting the weak, strong and intermediate coupling
regimes. As a proof-of-principle of this new approach
and its ability to faithfully simulate non-perturbative
phenomena, we target the renormalised coupling in
QED in 2 + 1 dimensions.
To observe non-perturbative phenomena such as
confinement, the simulated physical length scale needs
to be larger than the scale at which confinement sets
in. As a result, large lattice sizes are required and
the number of lattice points grows rapidly when ap-
proaching the continuum limit of the theory. This
results in computational requirements that cannot be
satisfied using current classical and quantum comput-
ers. Still, as previously done in the pioneering work
by Creutz [70], we can study the bare coupling depen-
dence of the local plaquette operator. This quantity
allows us to benchmark our formalism and to show
that a smooth connection between the weak and the
strong coupling regimes can be established. In addi-
tion, our method allows for estimating the precision
with which a given truncation approximates the un-
truncated results.
The paper is organised as follows. In Sec. 2, we re-
view lattice QED in 2 + 1 dimensions as an example
of Abelian and non-Abelian gauge theories with con-
tinuous groups and magnetic interactions. We con-
sider lattices with periodic boundary conditions and
reformulate the lattice Hamiltonian in terms of gauge-
invariant degrees of freedom. By eliminating redun-
dant variables, we obtain an effective Hamiltonian de-
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scription that allows for simulations at a low compu-
tational cost. In Sec. 3, we introduce a new magnetic
representation of lattice QED that is equipped with a
regularisation in terms of a Z
2L+1
group and an effi-
cient truncation scheme. In Sec. 4, we study the per-
formance of our method and benchmark its precision
by calculating the expectation value of the plaquette
operator on a periodic plaquette in the static charge
limit. We show that both the truncation cut-off pa-
rameter, i.e. the maximum number of gauge basis
states included in the simulation, and L, the dimen-
sion of the Z
2L+1
group, can be used as adjustable
variational parameters. Both are used to optimise the
simulation and estimate its accuracy. In the following
Sec. 5, we present the generalisation to an arbitrary,
two-dimensional periodic lattice with dynamical mat-
ter. Finally, we outline the prospects of this method
for classical and quantum simulations is in Sec. 6
2 Minimal encoding of LGTs with con-
tinuous gauge groups
In this chapter, we provide a Hamiltonian formula-
tion for LGTs with continuous gauge groups that al-
lows for resource-efficient classical and quantum simu-
lations. First, we review the standard Kogut-Susskind
Hamiltonian subject to Gauss’ law (the local con-
straints ensuring gauge invariance) in Sec. 2.1, con-
sidering QED on a square lattice as a paradigmatic
example. In Sec. 2.2, we proceed to provide a minimal
formulation of the QED lattice Hamiltonian, in which
redundant degrees of freedom have been removed.
2.1 QED in two dimensions
We review here the bottom-up construction of the lat-
tice Hamiltonian as originally presented in [71]. For
the sake of simplicity, we consider QED in 2 + 1 di-
mensions which displays key features of phenomeno-
logically relevant theories like QCD, including chiral
symmetry breaking and a renormalisation of the cou-
pling constant [72], features that are absent in one
spatial dimension.
The Hamiltonian of Abelian and non-Abelian gauge
theories in two (or more) dimensions is constructed in
terms of electric and magnetic fields, and their cou-
pling to charges. In continuous Abelian U(1) gauge
theories like QED (and similarly for non-Abelian
gauge theories like QCD), electric and magnetic fields
are defined through the vector potential A
µ
, with
E
µ
= ∂
t
A
µ
and B = ∂
x
A
y
− ∂
y
A
x
(in the unitary
gauge A
0
= 0). Here t, x, y are the time and space
coordinate in two dimensions, and µ = x, y.
Gauge invariance, i.e. invariance of the Hamilto-
nian under local phase (symmetry) transformations
of the charges, follows directly from the invariance
of E
µ
and B under A
µ
→ A
µ
+ ∂
µ
θ(x, y), where
θ(x, y) is an arbitrary scalar function. Due to the uni-
tary gauge, only spatial, time-independent transfor-
mations are considered. The electric field is sourced
by the charges through Gauss’ law,
P
µ
∂
µ
E
µ
= 4πρ,
where ρ is the charge density.
In LGTs [9], the charges occupy the sites n =
(n
x
, n
y
) of the lattice while the electromagnetic fields
are defined on the links. The links are denoted by
their starting site n and their direction e
µ
(µ = x, y),
as shown in Fig. 1. The electric interactions are de-
fined in terms of the electric field operator
ˆ
E
n,e
µ
,
which is Hermitian, possesses a discrete spectrum and
acts on the link connecting the sites with coordinates
n and n + e
µ
. For each link, one further defines a
Wilson operator
ˆ
U
n,e
µ
, as the lowering operator for
the electric field, [
ˆ
E
n,e
µ
,
ˆ
U
n
0
,e
ν
] = −δ
n,n
0
δ
µ,ν
ˆ
U
n,e
µ
.
The Wilson operator measures the phase proportional
to the bare coupling g acquired by a unit charge
moved along the link (n, e
µ
) of length a, i.e.
ˆ
U
n,e
µ
∼
exp{iag
ˆ
A
µ
(n)}. The magnetic interactions are given
by (oriented) products of Wilson operators on the
links around the plaquettes of the lattice. These
operators are used to construct the Kogut-Susskind
Hamiltonian as
ˆ
H =
ˆ
H
gauge
+
ˆ
H
matter
. Let us discuss
first the pure gauge part that describes the limit of
static charges
ˆ
H
gauge
=
ˆ
H
E
+
ˆ
H
B
,
ˆ
H
E
=
g
2
2
X
n
ˆ
E
2
n,e
x
+
ˆ
E
2
n,e
y
,
ˆ
H
B
= −
1
2g
2
a
2
X
n
ˆ
P
n
+
ˆ
P
†
n
. (1)
Here, the sums run over both components of the sites
n = (n
x
, n
y
) and
ˆ
P
n
=
ˆ
U
n,e
x
ˆ
U
n+e
x
,e
y
ˆ
U
†
n+e
y
,e
x
ˆ
U
†
n,e
y
(2)
is the plaquette operator. It is easy to check that
Eq. (1) reduces to the pure gauge U(1) Hamiltonian
in the continuum,
ˆ
H ∝
R
dxE(x)
2
+ B(x)
2
, when
the lattice spacing a is sent to zero (see App. A).
The Hamiltonian in Eq. (1) is gauge-invariant as it
commutes with the lattice version of Gauss’ law
"
X
µ=x,y
ˆ
E
n,e
µ
−
ˆ
E
n−e
µ
,e
µ
− ˆq
n
−
ˆ
Q
n
#
|Φi = 0 ∀n
⇐⇒ |Φi ∈ {physical states}, (3)
that determines what states are physical for a given
distribution of charges. Here, ˆq
n
is the operator mea-
suring the charge on the site n and |Φi represents the
state of the whole lattice, including both links and
sites. Furthermore, the operators
ˆ
Q
n
denote possible
static charges which we set to zero in the following.
The eigenstates of the electric field operators
ˆ
E
n,e
µ
|E
n,e
µ
i = E
n,e
µ
|E
n,e
µ
i, E
n,e
µ
∈ Z (4)
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form a basis for the link degrees of freedom. In par-
ticular, the physical states can be easily identified in
this basis via Eq. (3).
Let us now consider moving charges. To ensure
gauge invariance, their motion is required to respect
Gauss’ law, i.e. a charge q moving between two sites
has to change the electric field along the path by −q.
In other words, the lowering operator
ˆ
U has to be
applied q times to the links on the path to preserve
gauge-invariance. Since
ˆ
U
q
= exp{iqag
ˆ
A}, the so-
called minimal coupling condition [7] is recovered in
the continuum limit a → 0, which is equivalent to
replacing derivatives of matter fields by the covariant
derivatives, i.e. shifting the particles’ momentum by
a gauge field contribution ˆp
µ
7→ ˆp
µ
− qg
ˆ
A
µ
.
In QED, charges are represented by Dirac fermions.
In the staggered representation [71], they are repre-
sented on a square lattice as ordinary fermions at half
filling, with staggered chemical potential that plays
the role of the mass term. Their Hamiltonian is
ˆ
H
matter
=
ˆ
H
M
+
ˆ
H
K
, where
ˆ
H
M
and
ˆ
H
K
are the
mass and kinetic contributions, respectively
ˆ
H
M
= m
X
n
(−1)
n
x
+n
y
ˆ
Ψ
†
n
ˆ
Ψ
n
, (5)
ˆ
H
K
= κ
X
n
X
µ=x,y
h
ˆ
Ψ
†
n
ˆ
U
†
n,e
µ
q
ˆ
Ψ
n+e
µ
+ H.c.
i
.(6)
Here, m and q are the particles’ effective mass and
(integer) charge, κ the kinetic strength and
ˆ
Ψ
(†)
n
the
fermionic lowering (raising) operator for site n. Since
ˆ
H
M
identifies the Dirac vacuum with the state with
all odd sites occupied, creating (destroying) a particle
at even (odd) site is equivalent to creating a (−)q-
charged “fermion” (“antifermion”) in the Dirac vac-
uum. Thus the tunneling processes in the kinetic term
correspond to the creation or annihilation of particle-
antiparticle pairs and the corresponding change in the
electric field string connecting them. The charge op-
erator ˆq
n
is given by
ˆq
n
= q
ˆ
Ψ
†
n
ˆ
Ψ
n
−
2
[1 − (−1)
n
x
+n
y
]
, (7)
where q is an integer number which we set to one
in the following. Note that we rescaled the fermion
field by a factor
√
α as discussed in App. A, which
establishes the relations
m =
M
α
and κ =
1
2aα
, (8)
with M being the bare mass of the particles.
We conclude this section with a few comments on
the structure of the pure gauge part of the Kogut-
Susskind Hamiltonian in Eq. (1). There, the elec-
tric and magnetic terms show an apparent asymmetry
that obscures the electromagnetic duality in QED in
the continuum and in the Wilson lattice formulation
[9].
The symmetry between electric and magnetic fields
in QED and in Wilson’s action-formulation is due
to the fact that time and space are treated on the
same footing. Wilson’s lattice action theory is for-
mulated on a space-time grid with lattice spacing a
µ
,
µ = t, x, y, z. In this case, an isotropic continuum
limit is taken in which the lattice spacings in both the
temporal and the spatial directions approach zero. In
the Hamiltonian formulation, time is a continuous pa-
rameter. Accordingly, the above procedure is broken
into two steps. Firstly, the continuum limit with re-
spect to time a
t
→ 0 is taken, which results in the
Hamiltonian lattice formulation used here. In a sec-
ond step, the continuum limit has to be taken with
respect to space a
x,y,z
→ 0 to obtain physical results.
In the Hamiltonian formulation, the electric field
operators
ˆ
E form an algebra and are non-compact, as
their integer spectrum takes values from minus infin-
ity to infinity. By contrast, the Wilson operators
ˆ
U
and hence the plaquette operators
ˆ
P , form a group.
The moduli of their expectation value is one, as is
the case for the Wilson action. More specifically, in
Wilson’s action formulation, it follows from operators
ˆ
U = exp{iga
µ
ˆ
A
µ
} that
ˆ
A
µ
is compact as it is defined
from −π/(ga
µ
) to π/(ga
µ
), where a
µ
is the lattice
spacing in the µ-direction, which includes both space
and time as µ = t, x, y, z. The Kogut-Susskind Hamil-
tonian can be obtained from the Wilson action by tak-
ing the continuum limit in the time direction a
t
→ 0
[70]. Thus, the asymmetry between the electric and
magnetic terms in the Hamiltonian formulation disap-
pears when the continuum limit is taken in the spatial
direction.
While a fully non-compact formulation of Hamilto-
nian LGT is possible [73] (for the different outcomes
of the two approaches see e.g. [74, 75]), we do not
discuss this approach here as it is not advantageous
for quantum simulations. As we show in Sec. 3, it
is instead convenient to write the electric term in a
compact form.
2.2 QED Hamiltonian for physical states
As outlined in the previous section, gauge-invariance
constrains the dynamics to the physical states only,
i.e. those satisfying Gauss’ law in Eq. (3). Practi-
cally, unphysical states have to be suppressed, e.g.
via energy penalties [76]. In any case, quantum states
that are not physical represent an exponential over-
head for classical and quantum computation (also af-
ter a proper truncation, see Sec. 4). Furthermore, in
noisy near term quantum devices or simulation proto-
cols where the Hamiltonian has to be split up, e.g. to
simulate time-evolutions employing a product formu-
las such as the Trotter expansion [77], implementing
or imposing Gauss’ law during the simulation may be
complicated, or even impossible.
It is thus convenient to eliminate the redundant de-
grees of freedom by solving the constraint at each lat-
tice site. In one dimension, such a procedure allows
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Figure 1: Two-dimensional lattice gauge theory with periodic boundary conditions. A single cell of the periodic 2D lattice
in (a) is made of four links, oriented towards the positive x and y directions. Each lattice site is indicated by a unique vector n,
which marks the lower left corner of each single plaquette. The associated operator
ˆ
P
n
accounts for the electric field quanta
circulating along the sketched path. The periodic lattice spans the surface of a torus, shown in the middle, whose minimal
instance is assembled by four sites and the corresponding electric fields [thick lines, same color coding as in (a)]. Unwrapping
this minimal torus yields the geometry shown in (b). We identify the strings
ˆ
R
x
and
ˆ
R
y
and the four rotators
ˆ
R
j
, j = 1, 2, 3, 4.
The eigenstates of the strings and three of the rotators (we arbitrarily remove
ˆ
R
4
, dashed loop) form a basis for the physical
states of the pure gauge theory. To describe the physical states for a generic charge configuration we add three charge strings
(dotted green arrows) that correspond to a conventional physical state for the given charge configuration.
one to completely eliminate the gauge field, yielding
an effective Hamiltonian containing only matter terms
(but long-range interactions) [78, 79]. A similar ap-
proach is applicable in higher dimensions, with the
difference that the gauge field has also physical de-
grees of freedom. Here, we show how to formulate
an effective Hamiltonian that directly incorporates
the constraints of Eq. (3) by employing a convenient
parametrization of the physical states that yields an
intuitive description of the system.
For the sake of clarity, we consider the minimal in-
stance of a periodic two-dimensional gauge theory: a
square lattice formed by four lattice points. The gen-
eralisation to an arbitrary lattice on a torus is derived
in Sec. 5. Due to the periodic boundary conditions,
this minimal system can equivalently be represented
as a torus with four faces, or as four distinct plaque-
ttes consisting of eight links [see Fig. 1(b)]. Due to
charge conservation
P
n
ˆq
n
= 0, only three out of the
four constraints given by Gauss’ law [Eq. (3)] are in-
dependent. Consequently, three of the eight links in
the lattice are redundant, and the electric Hamilto-
nian in Eq. (1) can be solely expressed in terms of the
remaining five (see App. B.1 for details).
Describing the system in terms of these five links,
however, entails serious drawbacks. The effective
Hamiltonian contains many body interactions which
are challenging or even impossible to be implemented
using available quantum hardware (see App. B.1). To
circumvent this problem, we consider a natural basis
for the physical states in terms of small loops around
each plaquette, and large electric loops around the
whole lattice. In such a basis, the electric and mag-
netic interactions take a simple form. To conveniently
describe these interactions, we introduce a set of oper-
ators, rotators and strings (see Fig. 1), that are diag-
onal and label the loop basis. As we show in [31], the
Hamiltonian formulated in terms of these operators
can be simulated with current quantum hardware.
With the notation and conventions presented in
Fig. 1, rotators and strings are given by the relations
ˆ
E
(0,0),e
x
=
ˆ
R
1
+
ˆ
R
x
− (ˆq
(1,0)
+ ˆq
(1,1)
),
ˆ
E
(1,0),e
x
=
ˆ
R
2
−
ˆ
R
3
+
ˆ
R
x
,
ˆ
E
(1,0),e
y
=
ˆ
R
1
−
ˆ
R
2
− ˆq
(1,1)
,
ˆ
E
(1,1),e
y
= −
ˆ
R
3
,
ˆ
E
(0,1),e
x
= −
ˆ
R
1
,
ˆ
E
(1,1),e
x
=
ˆ
R
3
−
ˆ
R
2
,
ˆ
E
(0,0),e
y
=
ˆ
R
2
−
ˆ
R
1
+
ˆ
R
y
− ˆq
(0,1)
,
ˆ
E
(0,1),e
y
=
ˆ
R
3
+
ˆ
R
y
, (9)
where the charges ˆq
n
are required by Gauss’ law. An
intuitive way to understand the effect of the charge
terms in Eq. (9) is to consider them as sources of addi-
tional electric strings (whose concrete choice is just a
matter of convention but consistent with Gauss’ law),
as displayed by the green lines in Fig. 1(b). We re-
mark that this becomes evident from the link formula-
tion in App. B.1. In App. B.2 we give an alternative
explanation of the form of Eq. (9).
As mentioned above, rotators and strings automat-
ically preserve Gauss’ law, which can be readily veri-
fied by observing that at any site, the incoming fields
are always balanced by the outgoing ones. Moreover,
by recalling the plaquette operator
ˆ
P
n
in Eq. (2), it
becomes clear why
ˆ
R
i
and
ˆ
R
µ
are a convenient choice
Accepted in Quantum 2021-01-20, click title to verify. Published under CC-BY 4.0. 6
to represent the electric gauge field components. The
operator
ˆ
P
n
increases the anticlockwise quanta of the
electric field circulating in the n-th plaquette. Con-
sequently, it does not act on strings and takes the
form of the lowering operator of the associated ro-
tator. This fact can be formally proven by examin-
ing the raising and lowering operators of rotators and
strings. From the commutation relations of the links
and the relations shown in Eq. (9), it follows that
ˆ
R
i
,
ˆ
P
j
=δ
i,j
ˆ
P
j
,
ˆ
R
x
,
ˆ
U
(0,0),e
x
ˆ
U
(1,0),e
x
=
ˆ
U
(0,0),e
x
ˆ
U
(1,0),e
x
≡
ˆ
P
x
,
ˆ
R
y
,
ˆ
U
(0,0),e
y
ˆ
U
(0,1),e
y
=
ˆ
U
(0,0),e
y
ˆ
U
(0,1),e
y
≡
ˆ
P
y
, (10)
where
ˆ
P
j
, j = 1, 2, 3 is the plaquette operator of pla-
quette j as denoted in Fig. 1. Moreover, we defined
the string lowering operators
ˆ
P
x
≡
ˆ
U
(0,0),e
x
ˆ
U
(1,0),e
x
and
ˆ
P
y
≡
ˆ
U
(0,0),e
y
ˆ
U
(0,1),e
y
.
The magnetic Hamiltonian for the periodic plaque-
tte in Fig. 1(b),
ˆ
H
B
= −
1
2g
2
a
2
ˆ
P
1
+
ˆ
P
2
+
ˆ
P
3
+
ˆ
P
4
+ H.c.
, (11)
is proportional to the sum of four plaquette opera-
tors, while there are only three independent rotators.
The fourth rotator can be written as a combination
of the others, since the effect of lowering (raising) all
other rotators, i.e.
ˆ
R
1
,
ˆ
R
2
and
ˆ
R
3
, amounts to rais-
ing (lowering)
ˆ
R
4
. This relation can be understood
by examining Eq. (9): By lowering all of the three
rotators
ˆ
R
1
,
ˆ
R
2
and
ˆ
R
3
, we manipulate the electric
fields on the links constituting
ˆ
R
4
in exactly the same
way as an increment of the latter would do. As such,
the magnetic Hamiltonian becomes
ˆ
H
B
= −
1
2g
2
a
2
ˆ
P
1
+
ˆ
P
2
+
ˆ
P
3
+
ˆ
P
†
3
ˆ
P
†
2
ˆ
P
†
1
+ H.c.
,
(12)
while, by inserting Eq. (9) into Eq. (1), the electric
term takes the form:
ˆ
H
E
= g
2
(
2
h
ˆ
R
2
1
+
ˆ
R
2
2
+
ˆ
R
2
3
−
ˆ
R
2
(
ˆ
R
1
+
ˆ
R
3
)
i
+
ˆ
R
2
x
+
ˆ
R
2
y
+ (
ˆ
R
1
+
ˆ
R
2
−
ˆ
R
3
)
ˆ
R
x
− (
ˆ
R
1
−
ˆ
R
2
−
ˆ
R
3
)
ˆ
R
y
−
h
ˆq
(1,0)
(
ˆ
R
1
+
ˆ
R
x
)
+ ˆq
(0,1)
(
ˆ
R
2
−
ˆ
R
1
+
ˆ
R
y
)
+ ˆq
(1,1)
(2
ˆ
R
1
−
ˆ
R
2
+
ˆ
R
x
)
i
+ .
ˆq
2
(1,0)
+ ˆq
2
(0,1)
+ 2ˆq
(1,1)
(ˆq
(1,0)
+ ˆq
(1,1)
)
2
)
.
(13)
Once the effective gauge Hamiltonian
ˆ
H
gauge
=
ˆ
H
E
+
ˆ
H
B
has been derived in terms of rotator and
string operators, we must further modify the matter
Hamiltonian
ˆ
H
matter
=
ˆ
H
M
+
ˆ
H
K
[for the descrip-
tion in terms of field operators, see App. B.1]. While
the mass term in Eq. (5) is independent of the gauge
fields, the kinetic contribution has to be rephrased.
The kinetic contribution in Eq. (6) corresponds to
the creation or annihilation of a particle-antiparticle
pair on neighbouring lattice sites and the simultane-
ous adjustment of the electric field on the link in be-
tween. The green lines in Fig. 1(b) mark the fields
ˆ
E
(0,0),e
x
,
ˆ
E
(0,0),e
y
and
ˆ
E
(1,0),e
y
which are automati-
cally adjusted when charges are created. This fact fol-
lows from our arbitrary choice of enforcing the three
Gauss’ law constraints on exactly those links. For any
other link, we require combinations of raising and low-
ering operators
ˆ
P
j
and
ˆ
P
µ
(j = 1, 2, 3 and µ = x, y)
such that the specific link is adjusted, while all others
remain unchanged. As an example, let us consider
the generation of a particle in position (1, 1) and an
antiparticle in (1, 0). This choice implies either that
the electric field
ˆ
E
(1,0),e
y
has to decrease [which is au-
tomatically adjusted through the creation of a charge
string], or that the electric field
ˆ
E
(1,1),e
y
has to in-
crease and hence the rotator
ˆ
R
3
has to decrease. How-
ever, this action changes the electric fields
ˆ
E
(1,1),e
x
,
ˆ
E
(0,1),e
y
and
ˆ
E
(1,0),e
x
as well. To remedy that, we
lower the rotator
ˆ
R
2
, adjusting
ˆ
E
(1,1),e
x
and
ˆ
E
(1,0),e
x
,
and raise the string
ˆ
R
y
to compensate for the change
in
ˆ
E
(0,1),e
y
. Following the same procedure, the rules
for translating the kinetic Hamiltonian of Eq. (6) into
the language of rotators and strings read
ˆ
Ψ
†
(0,0)
ˆ
U
†
(0,0),e
x
ˆ
Ψ
(1,0)
→
ˆ
Ψ
†
(0,0)
ˆ
Ψ
(1,0)
,
ˆ
Ψ
†
(1,0)
ˆ
U
†
(1,0),e
x
ˆ
Ψ
(0,0)
→
ˆ
Ψ
†
(1,0)
ˆ
P
†
x
ˆ
Ψ
(0,0)
,
ˆ
Ψ
†
(1,0)
ˆ
U
†
(1,0),e
y
ˆ
Ψ
(1,1)
→
ˆ
Ψ
†
(1,0)
ˆ
Ψ
(1,1)
,
ˆ
Ψ
†
(1,1)
ˆ
U
†
(1,1),e
y
ˆ
Ψ
(1,0)
→
ˆ
Ψ
†
(1,1)
ˆ
P
†
y
ˆ
P
2
ˆ
P
3
ˆ
Ψ
(1,0)
,
ˆ
Ψ
†
(0,1)
ˆ
U
†
(0,1),e
x
ˆ
Ψ
(1,1)
→
ˆ
Ψ
†
(0,1)
ˆ
P
1
ˆ
Ψ
(1,1)
,
ˆ
Ψ
†
(1,1)
ˆ
U
†
(1,1),e
x
ˆ
Ψ
(0,1)
→
ˆ
Ψ
†
(1,1)
ˆ
P
†
x
ˆ
P
2
ˆ
Ψ
(0,1)
,
ˆ
Ψ
†
(0,0)
ˆ
U
†
(0,0),e
y
ˆ
Ψ
(0,1)
→
ˆ
Ψ
†
(0,0)
ˆ
Ψ
(0,1)
,
ˆ
Ψ
†
(0,1)
ˆ
U
†
(0,1),e
y
ˆ
Ψ
(0,0)
→
ˆ
Ψ
†
(0,1)
ˆ
P
†
y
ˆ
Ψ
(0,0)
. (14)
Inserting these into Eq. (6), we obtain the kinetic con-
tribution to the total Hamiltonian as
ˆ
H
K
= κ
h
ˆ
Ψ
†
(0,0)
( +
ˆ
P
x
)
ˆ
Ψ
(1,0)
+
ˆ
Ψ
†
(0,1)
(
ˆ
P
1
+
ˆ
P
†
2
ˆ
P
x
)
ˆ
Ψ
(1,1)
+
ˆ
Ψ
†
(0,0)
( +
ˆ
P
y
)
ˆ
Ψ
(0,1)
+
ˆ
Ψ
†
(1,0)
( +
ˆ
P
†
2
ˆ
P
†
3
ˆ
P
y
)
ˆ
Ψ
(1,1)
+ H.c.
i
. (15)
In conclusion, with the gauge part
ˆ
H
gauge
of the
Hamiltonian described by Eqs. (12) and (13) and the
matter part
ˆ
H
matter
by Eqs. (5) and (15), the system
is fully characterised.
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The effective Hamiltonian we derive here for a peri-
odic plaquette can be extended to a torus of arbitrary
size [see Sec. 5.2] or to d−dimensional lattices. For the
latter, one chooses operators
ˆ
R
i
that describe the to-
tal electric field circulating around the i-th plaquette.
Furthermore, one defines d operators corresponding
to loops that circulate around the whole lattice (
ˆ
R
x
and
ˆ
R
y
in the two-dimensional case here). The charge
strings are eventually defined by arbitrary paths to
each lattice point starting from the origin, as we show
in Sec. 5.2 for d = 2.
We will use the just derived Hamiltonian to com-
pute the expectation value of the plaquette operator
hi = −
g
2
a
2
V
hΨ
0
|
ˆ
H
B
|Ψ
0
i, (16)
where |Ψ
0
i is the ground state, and V the number
of plaquettes in the lattice, V = 4 in this case. The
expectation value of the operator is defined as a
dimensionless number, which is bounded by ±1 and
proportional to the magnetic energy.
3 Transformation into the magnetic
representation
In the following, we describe a scheme that allows
switching from the so-called electric representation,
where
ˆ
H
E
is diagonal, to the magnetic one, where
ˆ
H
B
is diagonal. Our method is based on the replacement
of the U(1) gauge group with the group Z
2L+1
, and an
accompanying transition from the compact formula-
tion to a completely compact formulation, where both
field degrees of freedom are treated as compact vari-
ables. While this procedure is general, we illustrate it
for the minimal periodic system introduced in Sec. 2.2
and consider generalisations in Sec. 5.
Before presenting the scheme, we discuss the fol-
lowing observations about the considered Hamilto-
nian, that is now reduced to the sum of Eqs. (12)
and (13), while all charges ˆq
n
in Eq. (13) are set
to zero. In particular, the lowering (raising) oper-
ators
ˆ
P
(†)
x
and
ˆ
P
(†)
y
acting on the strings are solely
contained in the now absent kinetic Hamiltonian in
Eq. (15). The total Hamiltonian thus commutes with
ˆ
R
x
and
ˆ
R
y
, i.e. [
ˆ
H
gauge
,
ˆ
R
x
] = [
ˆ
H
gauge
,
ˆ
R
y
] = 0,
and as a consequence the strings become constants
of motion. The dynamics induced by the pure-gauge
Hamiltonian are thus restricted to different subspaces
defined by
ˆ
R
µ
|r
µ
i = r
µ
|r
µ
i, for µ = x, y. Starting in
Sec. 4, we will be interested in a ground state prop-
erty, therefore we restrict ourselves to the subspace
where both strings are confined to the vacuum. The
effective Hamiltonian of this subspace can be readily
obtained by setting
ˆ
R
x
=
ˆ
R
y
= 0 in Eqs. (12) and
(13) which yields
ˆ
H
(e)
=
ˆ
H
(e)
E
+
ˆ
H
(e)
B
,
ˆ
H
(e)
E
= 2g
2
h
ˆ
R
2
1
+
ˆ
R
2
2
+
ˆ
R
2
3
−
ˆ
R
2
(
ˆ
R
1
+
ˆ
R
3
)
i
,
ˆ
H
(e)
B
= −
1
2g
2
a
2
h
ˆ
P
1
+
ˆ
P
2
+
ˆ
P
3
+
ˆ
P
1
ˆ
P
2
ˆ
P
3
+ H.c.
i
,
(17)
where we introduced the superscript (e) to emphasise
is the electric representation.
Since the three rotators possess discrete but infinite
spectra, any numerical approach for simulating the
Hamiltonian in Eq. (17) requires a truncation of the
Hilbert space. In the following, l denotes the cut-off
value which is identified by
ˆ
R
j
|r
j
i = r
j
|r
j
i ∀r
j
= −l, −l + 1, . . . , l. (18)
Thus, the action of the truncated lowering operators
is given as
ˆ
P
j
|r
j
i =
(
|r
j
− 1i, if r
j
> −l
0, if r
j
= −l.
(19)
Note that the total dimension of the Hilbert space is
reduced to (2l + 1)
3
, which is still challenging to sim-
ulate even for relatively small values of l. In partic-
ular, calculations in the weak coupling regime suffer
from this severe limitation and until now, no practical
methods to solve this issue have been available.
Let us now introduce a formulation that allows for
an efficient representation of the Hamiltonian’s eigen-
states in the weak coupling regime, where g 1. It is
based on the exchange of the continuous U(1) group
with the discrete group Z
2L+1
, which provides a dis-
crete basis for the vector potential operators
ˆ
A
n,e
µ
and enables a direct transformation into this dual ba-
sis via a Fourier transform. The approach is moti-
vated by the key observation that, in the electric rep-
resentation, the Hamiltonians of the continuous U (1)
group and the discrete Z
2L+1
group after truncation
(l < L) are equivalent. The group Z
2L+1
consists of
2L + 1 elements, thus the parameter L indicates the
size of the Hilbert space. In particular, the rotators
ˆ
R
j
and lowering (raising) operators
ˆ
P
(†)
j
(j = 1, 2, 3)
take the form
ˆ
R
j
|r
j
i = r
j
|r
j
i ∀r
j
= −L, . . . , L
ˆ
P
j
|r
j
i =
(
|r
j
− 1i, if r
j
> −L
|Li, if r
j
= −L.
(20)
The only difference between the truncated U(1) group
and untruncated Z
2L+1
group is the cyclic property of
the lowering (raising) operator, which transforms |Li
into |−Li (and vice versa). However, after a trunca-
tion of Z
2L+1
with l < L, this property is lost, mean-
ing that Eqs. (19) and (20) correspond to each other
and the two truncated groups become equivalent.
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For now, consider the Hamiltonian which employs
the complete Z
2L+1
group. Importantly, the rela-
tions in Eq. (20) resort to a compact description of
the electric field since the spectra of the rotators
and strings are constrained to the compact interval
[−L, L]. We now introduce the following replacement
rules for these operators,
ˆ
R 7→
2L
X
ν=1
f
s
ν
sin
2πν
2L + 1
ˆ
R
,
ˆ
R
2
7→
2L
X
ν=1
f
c
ν
cos
2πν
2L + 1
ˆ
R
+
L(L + 1)
3
, (21)
which reassemble Fourier series expansions. Crucially,
this replacement is exact, i.e. there is no truncation
of the Fourier series. Employing the fact that the
spectrum of
ˆ
R is discrete and takes integer values,
the periodicity of the trigonometric functions can be
exploited, which allows one to perform a summation
over all coefficients where the sine (cosine) is equiva-
lent. Hence, a finite number of 2L coefficients remain,
which take the form
f
s
ν
=
(−1)
ν+1
2π
ψ
0
2L + 1 + ν
2(2L + 1)
− ψ
0
ν
2(2L + 1)
(22)
f
c
ν
=
(−1)
ν
4π
2
ψ
1
ν
2(2L + 1)
− ψ
1
2L + 1 + ν
2(2L + 1)
.
(23)
Here, ψ
k
(•) is the k-th polygamma function. Let us
further remark that these rules can be extended to
higher powers in the variables
ˆ
R than considered in
(21).
This replacement turns out to be convenient for
the basis transformation explained below. Introduc-
ing the convention |ri = |r
1
i|r
2
i|r
3
i and recalling
Eq. (20), the electric contribution from Eq. (17) reads
ˆ
H
(e)
E
= 2g
2
L
X
r=−L
2L
X
ν=1
f
c
ν
3
X
j=1
cos
2πν
2L + 1
r
j
−f
s
ν
sin
2πν
2L + 1
r
2
2L
X
µ=1
f
s
µ
sin
2πµ
2L + 1
r
1
+ sin
2πµ
2L + 1
r
3
|rihr|. (24)
Note that we use the notation
P
L
r=−L
to indicate
that the sum collects all combinations of r
j
, where
r
j
∈ [−L, L], j = 1, 2, 3 and we neglected the con-
stant energy shifts introduced by Eq. (21). The Z
2L+1
magnetic Hamiltonian
ˆ
H
(e)
B
can be obtained by sub-
stituting the cyclic
ˆ
P
j
of Eq. (20) into Eq. (17).
We are now in a position to perform the switch to
the dual basis. As shown in App. C, for any γ ∈ N,
the discrete Fourier transform
ˆ
F
2L+1
diagonalises the
lowering operators as
ˆ
F
2L+1
ˆ
P
γ
j
ˆ
F
†
2L+1
=
L
X
r
j
=−L
e
i
2π
2L+1
γr
j
|r
j
ihr
j
|. (25)
Hence, by applying the discrete Fourier transform to
the total Hamiltonian we diagonalise the magnetic
contributions, while sacrificing the diagonal structure
in the electric part, i.e.
ˆ
H
(b)
E
= g
2
2L
X
ν=1
f
c
ν
3
X
j=1
ˆ
P
ν
j
+
f
s
ν
2
h
ˆ
P
ν
2
− (
ˆ
P
†
2
)
ν
i
×
2L
X
µ=1
f
s
µ
h
ˆ
P
µ
1
+
ˆ
P
µ
3
i
+ H.c., (26)
and
ˆ
H
(b)
B
= −
1
g
2
a
2
L
X
r=−L
cos
2πr
1
2L + 1
+ cos
2πr
2
2L + 1
+ cos(
2πr
3
2L + 1
)
+ cos
2π(r
1
+ r
2
+ r
3
)
2L + 1
|rihr|. (27)
Note that we introduced the superscript (b), which
refers to the magnetic representation of the Hamilto-
nian. Using this representation, computations in the
weak coupling regime g 1 can be performed effi-
ciently, as a truncation l now chooses a cut-off for the
magnetic field energy. We emphasize that although
we employed the rotator formulation of the Hamil-
tonian, the just presented procedure is likewise valid
for the link formulation utilizing the electric field op-
erators. Indeed, the replacement rules in Eq. (21)
are then formulated in terms of
ˆ
E instead of
ˆ
R and
inserted into the Hamiltonian in Eq. (1). The corre-
sponding magnetic representation is analogously ob-
tained via an application of the Fourier transform.
The parameter L now affects the accuracy of the
simulation. In fact, while L is completely irrelevant in
the electric representation (truncated U(1) and trun-
cated Z
2L+1
are equivalent), it strongly influences the
results derived in the magnetic representation. While
examining the relationship between L and l in more
detail in Sec. 4, we qualitatively discuss our procedure
to simulate the U (1) group with the two representa-
tions of Z
2L+1
in the following. To be more precise,
for any g we might always formulate a sequence of
approximating representations for any quantum state
of the system in the computational basis defined by
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|ri, i.e.,
|ψ
(e)
(g)i =
∞
X
r=−∞
p
U(1)
(g, r)|ri
≈
L
X
r=−L
p
Z
2L+1
(g, r)|ri
≈
l
X
r=−l
p
(e)
(g, r)|ri. (28)
Here, p
(e)
denotes the expansion coefficients in the
electric representation, with the subscript indicating
the group to which they are referring to (no subscript
stands for the truncated Z
2L+1
). The first approxima-
tion in Eq. (28) is due to the transition from U (1) to
the Z
2L+1
group, while the second approximation rep-
resents the truncation from (2L+1)
3
down to (2l+1)
3
states.
The same scheme exists for the magnetic represen-
tation, where the weights p
(b)
(g, r, L) are used for the
state |ψ
(b)
(g, L)i. In this case, however, the choice of
L is important. While the truncated electric repre-
sentation directly corresponds to the truncated and
compact U(1) formulation, the completely compact
formulation employed in the magnetic representation
is crucially affected by the level of discretisation L.
This relation is examined further in Sec. 4.2, where
we study the convergence of the two representations to
U(1) for intermediate values of the coupling g. Hence,
in the remainder of this manuscript we consider the
completely compact formulation for the magnetic rep-
resentation only and resort to the compact formula-
tion for the electric representation, i.e. to Eq. (17) for
the case of pure gauge.
The interplay of the parameters L and l can be in-
tuitively understood by employing a geometrical illus-
tration. In Fig. 2, the black circles represent the con-
tinuous U (1) group, which is approximated by 2L + 1
possible states (blue lines) of the Z
2L+1
group. For
l = L, we faithfully describe the untruncated Z
2L+1
group, and use both the solid and the dashed blue
lines in the figure. By choosing l < L, we select the
states marked with solid blue lines that lie symmet-
rically around |r = 0i and achieve a binned approxi-
mation of any continuous p
U(1)
lying in the grey area.
Furthermore, for any fixed l, the parameter L con-
trols the spread of the available basis states (or bins)
around |0i. Since we are interested in the conver-
gence of the truncated Z
2L+1
to U(1) which occurs
for L → ∞, we only consider the 2l + 1 states that are
important for the dynamics. In particular, we disre-
gard cyclic effects from the lowering operator
ˆ
P that
are a distinctive feature of Z
2L+1
with respect to U (1)
[see Eq. (20)].
As an example of this relationship between l and L,
consider the two distributions drawn from U(1), rep-
resented by the red and green dashed lines in Fig. 2.
Clearly, the combination L = 3, l = 1 is insufficient
Figure 2: Discrete approximation of a continuous distribu-
tion of states in the magnetic representation. The ability
to approximate a state is related to the quotient l/L. For a
given l, L controls the resolution of the approximation, which
is always centred around the vacuum |0i. Black circles rep-
resent the U(1) group, the violet 2L + 1 edged polygon the
Z
2L+1
group. Blue lines (solid and dashed) mark the 2L + 1
states of Z
2L+1
, while only the 2l + 1 states indicated with
the solid lines are kept after truncating. Red and green mark-
ers are pictorial representations of states in U (1) while the
light blue areas correspond to their binned approximation.
to approximate the broad red distribution. Hence,
we increase l to completely cover the target distribu-
tion within the grey shaded area. A reduction of L
is also a practicable option, especially given the situ-
ation where l could not be increased further because
of a lack of computational resources. By raising L
instead, our binned approximation has a higher reso-
lution around the zero state. For the more localised
green state, therefore, it is advantageous to choose
the higher value L = 6 when l = 2 is an available
option. In fact, the combination L = 6, l = 1 leads
to a worse approximation of the green state than the
choice L = 3, l = 1, which is therefore preferable
when l is limited to unity.
To that end, it is clear that the interplay of L and
l represents a crucial point when estimating results
and their error due to the performed discretisation.
Note that the spread of the distribution determines
the value of the free parameter L, while l will, for
all practical purposes, be limited by the amount of
physical resources, e.g. by the memory of a classi-
cal computer or the number of available qubits in a
quantum simulator.
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4 Performance and application of the
new approach
In the previous section we outlined the transforma-
tion from the electric to the magnetic representation,
suited to describe the strong and the weak coupling
limits, respectively. Here, we develop a protocol which
allows for assessing convergence of the truncated rep-
resentations.
First, we qualitatively describe the system’s be-
haviour at different values of the bare coupling, which
will be useful for motivating the protocol. Later, we
consider the non-asymptotic cases, where it is not
known whether a given truncation is sufficient to de-
scribe the considered system with the desired preci-
sion. In our example dealing with a pure gauge U(1)
theory, this happens for g ≈ 1, where both repre-
sentations might be inaccurate. Finally, we discuss
convergence in the weak and strong coupling regimes,
respectively and apply our protocol to estimate the
plaquette expectation value. In particular, we con-
sider the interplay between the parameters l and L,
which plays an important role when g 1.
From now on, we work within a unit lattice spacing,
i.e. a = 1, but emphasise that this represents no
restriction for the following results.
4.1 Phenomenological analysis
As mentioned above, the Hamiltonians in the elec-
tric and magnetic representations are related via the
Fourier transform, i.e. the magnetic and electric
fields are the dual of one another. This relation
consequently holds true for the eigenstates, illustrat-
ing the difficulty of expressing the ground state in
the extremal regimes via the representation in which
the dominant term of the total Hamiltonian is non-
diagonal. For example, in the regime g 1 the
ground state is either determined by the bare vacuum
|0i or a superposition of all basis elements, depend-
ing whether the electric or magnetic representation is
employed (both cases Z
2L+1
, l = L)
|GS
(b)
(g 1, L)i = |0i
=
ˆ
F
−1
2L+1
X
r
1
(2L + 1)
3/2
|ri
=
ˆ
F
−1
2L+1
|GS
(e)
(g 1)i. (29)
In other words, the coefficients p(g 1, r) = (2L +
1)
−3/2
in Eq. (28) represent a uniform distribution
assembling an equally weighted superposition of all
basis states. This demonstrates the issue when trun-
cating the Hilbert space by choosing l < L and mo-
tivates the choice of switching to the magnetic rep-
resentation. Note that in the limit g 1 where the
electric Hamiltonian is dominant, the roles of the two
representations are interchanged.
While it is known that in the limit g → ∞ (g → 0)
the ground state in the electric (magnetic) represen-
tation is the vacuum |0i, it is not clear what happens
when g deviates from these limits. In the following,
we employ perturbation theory to estimate the re-
quired resources in order to describe the system. For
any value of the bare coupling g, we determine the
minimal truncation l and resolution L which suggest
a high agreement with the untruncated and U(1) the-
ory.
Throughout the whole range of g the system’s
ground state is center-symmetric, meaning that
p
(e)
(r) = p
(e)
(−r) in Eq. (28). This follows from the
fact that the total Hamiltonian in Eq. (17) is per-
Hermitian [80] (Hermitian with respect to the sec-
ondary diagonal; higher excited states can also be
center-antisymmetric). One can hence infer that the
spread of the distribution |p
(e)
(g, r)| in the electric
representation is centred around |0i and decreases
with g, again motivating the developed basis trans-
formation of the Hamiltonian. Equivalently, the same
holds in the magnetic representation where the center-
symmetric ground state becomes less localised by in-
creasing g.
Employing the magnetic representation, we esti-
mate the influence of the electric Hamiltonian with
perturbation theory. For g → 0, the unperturbed
ground state is |GS
(b)
(g = 0, L)i = |0i, while the first
order correction |GS
(b)
corr
i takes the form
|GS
(b)
corr
(g, L)i =
l
X
r=−l,
r6=0
hr|
ˆ
H
(b)
E
|0i
E
0
− hr|
ˆ
H
(b)
B
|ri
|ri,
where
hr|
ˆ
H
(b)
E
|0i
E
0
− hr|
ˆ
H
(b)
B
|ri
.
g
4
(2L + 1)
4
|r|
4
. (30)
Here, we require L 1 for the inequality, and
we introduced the unperturbed ground state energy
E
0
= −4/g
2
. Note that the chosen truncation l de-
termines the maximal length of r as |l| =
√
3l, while
the population in the states |ri is proportional to
(gL/|r|)
8
. The upper bound on the population in
each |ri allows one to determine the part p
r>l
of the
population that is left is out by the truncation at l,
which yields p
r>l
∝ g
8
L
8
/l
5
. Hence, in order to cover
the whole distribution by our truncation, we require
l
5
> g
8
L
8
such that large |r| states which are not
covered by the truncation are only marginally pop-
ulated. Respectively, if the truncation l is fixed, we
infer that a resolution change L ∝ g
−1
is required.
In fact, if g
8
L
8
/|r|
8
takes large values for all r, the
chosen discretisation is not able to capture the spread
of the true distribution, i.e. one would encounter the
situation illustrated in the first row of Fig. 2.
This can be intuitively understood by observing
that the transition amplitudes between the ground
state and states |ri induced by
ˆ
H
(b)
E
are suppressed
by the respective energy gap, i.e. the denominator in
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Eq. (30). The gap itself is controlled by L, which is a
direct consequence of Eq. (27).
The analogous calculation for the electric represen-
tation yields l > g
−2
/
√
3 which is independent of L.
Here, the energy gap is not affected by L and hence
deviations from the continuum result have to be asso-
ciated with a truncation l which is insufficient for the
state one aims to approximate.
Concluding, decreasing (increasing) g requires ad-
ditional computational resources in the electric (mag-
netic) representation.
4.2 Fidelity and convergence of the two repre-
sentations
This section is devoted to a convergence analysis,
which examines the agreement between the two
representations. Although we have developed a
scheme that allows one to represent, discretise and
truncate the Hamiltonian in the weak coupling
regime, the optimal choice of the parameter L is not
clear a priori. Clearly, l should usually be chosen
according to the availability of the computational
resources, which then determines the most suitable L
depending on the bare coupling. Furthermore, there
is an uncertainty regarding which representation to
choose if one is not explicitly considering one of the
extremal regimes, g 1 and g 1.
We first develop a criteria to estimate the agree-
ment of the two representations. Therefore, we em-
ploy their relation via a unitary transformation and
define the Fourier fidelity F
f
with respect to the same
state derived in both representations, e.g. an eigen-
states belonging to the same eigenvalue of some ob-
servable, such as the ground state derived in both
(truncated) representations for a fixed value of g. We
write F
f
as
F
f
(l) = max
L>l
hψ
(b)
(L, l)|
ˆ
F(L, l)|ψ
(e)
(l)i
2
, (31)
where the Fourier transform
ˆ
F(L, l) =
1
√
2L + 1
l
X
k,j=−l
e
i
2π
2L+1
jk
|jihk|
⊗3
(32)
is truncated, i.e. the indices of the sums are limited
by ±l instead of ±L
1
. Due to the truncation, the
features captured in both states are not necessarily
equivalent which results in low values of the Fourier
fidelity. Vice versa, high values indicate that – for the
considered state – the representations are equivalent
1
Note that this is a consequence of the truncated Hilbert
space. However, the operator in (32) not being unitary is no
limitation, since the states in Eq. (31) could be embedded in
the Hilbert space required by the full Fourier transform. Here,
the coefficients for each basis element c
n
with n > l are set to
zero in both representations.
and yield the same result. Note that this further sug-
gest that the result is close to the hypothetical one
derived within the untruncated theory, since the uni-
fication of both representations nearly covers the total
Hilbert space
2
.
Clearly, a low Fourier fidelity is not the only deci-
sive criteria whether a derived result is robust against
changes in l or L, especially in the extremal regimes
of the bare coupling where the truncation effects of
the non-appropriate representation are severe. We
thus employ the so called sequence Fidelity F
s
, which
measures the overlap of the same state (in the cho-
sen representation) derived within successive values
of truncations l − 1 and l,
F
(µ)
s
(l, L) =
l−1
X
r=−l+1
hψ
(µ)
(l − 1, L)|rihr|ψ
(µ)
(l, L)i.
(33)
Here, µ = e, b indicates considered representation
while L is only present in the magnetic case (in the
electric representation we can use the truncated U(1)
model). Since the truncated models converge to the
untruncated U (1) model in the limit l → ∞, high
values of F
s
indicate, under a suitable assumption,
that the chosen truncation l is able to capture the
whole distribution of the wave vector (as for the
case l = 2, L = 3 in Fig. 2). Such a conclusion can
not be drawn in the case where the distribution is
multimodal with disjoint fractions that lay outside
the covered space. Then, the sequence fidelity yields
high values for subsequent values of l but would not
for larger differences of the considered truncation.
Nevertheless, this represents a common issue present
in all approaches employing truncation techniques
that lack the exact true solution.
Let us now return to the ground state of the
pure gauge model. Due to their diagonal forms
the electric (magnetic) representation yields more
accurate results in the strong (weak) coupling regime,
however there is no intuition for the intermediate
regime g ≈ 1. We hence calculate the Fourier fidelity
to obtain an indicator whether results obtained via
the different representations at finite values of l are
compatible, that is, whether the chosen truncation is
enough to capture the local and non-local properties
of the state vector. Fig. 3(a) illustrates the Fourier
infidelity 1 − F
f
(l) of the ground state for different
values of g
−2
. The global maximum of F
f
arises
in from the compromise of having truncation l and
resolution L big enough to both contain and resolve
the details of the state’s distribution. For example, in
Fig. 2, it becomes clear that an increase in resolution
L reduces the available domain to accommodate a
distribution with too high spread if l is not increased
2
Recall that under the Fourier transform, local features are
transformed into global ones and vice versa, e.g. a Gaussian is
transformed into a Gaussian with inverse width.
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Figure 3: Convergence analysis of the basis representa-
tions. In (a), the Fourier infidelity in the intermediate region
is decreasing with l as the whole wave function can be cap-
tured by the truncation. The sequence infidelities in (b) and
(c) illustrate convergence to the U (1) theory and the freezing
effect respectively. The values of L optimizing the sequence
fidelities of (c) are displayed in (d). Here, freezing is detected
by curves similar to the black dashed lines.
accordingly. This relation is the origin of the kink in
the Fourier fidelity appearing for larger l in Fig. 3(a),
from where the decrease in the Fourier fidelity is
solely attributed to an increase of the resolution.
Note that for l = 10 we exceed a fidelity of 99.99%.
In the remainder of this section, we will focus on the
strong and weak coupling regime where the Fourier
fidelity is no meaningful quantity due to the inability
to express the state within a truncated basis in both
representations. For the electric representation, the
sequence Fidelity has a simple interpretation (L is
absent here) as it quantifies the overlap between the
ground state obtained within different truncations.
Since the energy spectrum is fixed and does not
depend upon L, a unit value of F
(e)
s
(l) implies that
the considered state is unaffected by an increase
in l. This suggests that higher truncations do not
improve the result and that the model converged to
the untruncated U(1) ground state, which can be
further motivated by examining the behaviour of
1 − F
(e)
s
(l) in Fig. 3(b). As expected, in the strong
coupling regime the sequence Fidelity approaches
unity, indicating convergence to the untruncated
model, where it is helpful to recall that the ground
state in this limit is given by a single basis state,
|0i. Approaching the intermediate regime g ≈ 1,
F
(e)
s
(l) reduces to a l-dependent constant value,
which indicates that the truncation is insufficient to
describe all features of the ground state appropriately.
In the magnetic representation, the situation is sub-
stantially more complicated, since the approximation
of the continuous U(1) group with the discrete Z
2L+1
group introduces the intricate interplay of l and L.
As mentioned above, higher values of L allow for a
better local approximation of the state which comes
at the expense of the tails, which are cut off if l is
too small (see Fig. 2). In terms of the sequence fi-
delity F
(b)
s
(l, L), this implies that for each value of l
there exists a unique optimal value L
opt
of L. Let us
stress that this is only true for the ground state of the
pure gauge theory considered here. In a more general
setting, possibly including matter and higher excited
states, F
(b)
s
(l, L) might have multiple optimal values
of L.
Another complication is given by the fact that
L
opt
does not necessarily corresponds to the global
maximum of the sequence fidelity. In particular, a
freezing effect can occur for highly localised distribu-
tions, where the resolution L is insufficient to cap-
ture any of its features. Consequently |ψ
(b)
(l, L)i and
|ψ
(b)
(l + 1, L)i are practically the same state and thus
yield high values of the sequence fidelity in Eq. (33).
In the scenario examined here, the freezing mecha-
nism can be observed in the weak coupling regime,
where the ground state is highly localised around |0i.
If L is too small, i.e. the bin belonging to the latter
state is to wide, all population is accumulated there
and the state does not change if g is decreased while
L is kept constant. However, it is possible to iden-
tify the freezing effect by an educated interpretation
of F
(b)
s
.
Fig. 3(c) illustrates that the sequence infidelity
1 − F
(b)
s
(l, L
opt
) in both regimes saturates at a l-
dependent value. Analogous to the electric repre-
sentation, it saturates in the strong coupling regime
(g 1), however the saturation for weak coupling
stems from the limited ability to approximate a con-
tinuous approximation with a fixed number of discrete
levels. To be more precise, for every l the optimal
L
opt
is chosen as the best compromise of resolution
around |0i and a proper representation of the tails of
the distribution. In Sec. 4.1, we demonstrated that
L
opt
increases as g is decreased, which we now con-
firm numerically in Fig. 3(d) (see App. F for more
details). Note that as soon as L increases, it does so
as L ∼ g
−1
, supporting the perturbative results be-
fore. Physically speaking, L increases with g
−1
since
the spread of the population distribution in the mag-
netic representation decreases, and thus more resolu-
tion nearby the state |0i is required.
The black dashed line in Fig. 3(c) corresponds to
the global maximum of F
(b)
s
(l = 1, L) for all g
−2
. It
does not saturate and vanishes in the limit g
−2
→ ∞.
Comparison with the black dashed line in Fig. 3(d),
which indicates that L
opt
≡ l + 1 reveals this as a
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characteristic of the mentioned freezing effect.
Concluding, both the Fourier and the sequence fi-
delities in Eqs. (31) and (33) are two tools to assess
the convergence of and agreement between the two
representations. While the sequence fidelity must be
applied in the extremal regimes, the Fourier fidelity
yields a valuable quantification of the combined ca-
pabilities of the two representations for intermediate
values of the bare coupling.
4.3 Estimation of hi
We now apply the tools developed in Sec. 4.2 to calcu-
late the expectation value hi as defined in Eq. (16).
The value of hi with respect to the system’s ground
state is an important quantity in LGTs, as it can be
related to the running of the coupling [31].
In the absence of dynamical matter, the total
Hamiltonian solely consists of the two gauge field con-
tributions. Therefore, we may determine a value g
m
separating the regimes where either of the respective
representations is advantageous.
Let g
m
be the value of g for which the Fourier
fidelity in Eq. (31) is maximal with respect to the
ground state, i.e.
F
g
m
(l) = max
L>l
g
hGS
(b)
(L, l, g)|
ˆ
F(L, l)|GS
(e)
(l, g)i
.
(34)
Since the electric (magnetic) representation shows
exceeding performance in the strong (weak) coupling
regime, we can assume that for a given truncation l,
the best approximation is achieved by considering the
electric representation in the range g ∈ [g
m
, ∞) and
the magnetic one for g ∈ [0, g
m
] (compare also Sec. 4.2
and Fig. 3).
Fig. 4 shows hi for various truncations, derived
both in the electric [panel (a)] and magnetic [panel
(b)] representation. In the latter, we obtained the
L
opt
values that have been used for plotting via the
sequence fidelity as described above. From here, the
true curve as it would be obtained from the untrun-
cated U(1) theory can be estimated via the asymp-
totic values of the different representations when the
truncation l is increased, since in the limit l → ∞ both
representations converge to the full theory. We exem-
plify such an estimation with the inset in Fig. 4(a),
that contains the results for different l at g
−2
= 10.
The convergence can be clearly observed, and both
representations yield the same result up to the fourth
decimal at l = 10 (hi = 0.9572 ± 0.0001). Note that
this convergence is not necessarily monotonic. How-
ever, in the extremal regimes, we observe that the
expectation value of increases with the truncation
l when employing the electric representation, while it
decreases with the magnetic one, for which we will
provide analytical arguments in App. D.
To summarize this section, we recall that a naive
approximation of U (1) with
2L+1
(with L fixed)
Figure 4: Estimation the plaquette operator. Panel (a)
displays the obtained curves in the electric representation,
where the line styles correspond to different values of the
truncation l. For the magnetic representation in panel (b),
each point has been obtained via the optimisation of the
sequence fidelity over L. We stress the considerably higher
resource requirements (l) of the electric representation for
calculations in the regime g
−2
> 1. The inset in (a) shows
the values for the different representations for all values of l
shown here when g
−2
= 10.
leads to dramatically increasing computational costs
when working on a wide range of g-values. As ex-
plained intuitively in Sec. 3, the problem originates
from the fact that
2L+1
converges not uniformly but
pointwise to U(1). For fixed resolution L and fixed
computational resources l, there is always a coupling
g small enough such that the Z
2L+1
description dis-
plays freezing and hence cannot approximate the U (1)
continuum physics accurately. This can be under-
stood by noting that the magnetic field Hamiltonian
is gapless in both the continuum theory and in the
U(1)-lattice description, but gapped in the
2L+1
-
formulation. For fixed L and decreasing g, the off-
diagonal elements in the Hamiltonian
ˆ
H
E
decrease
with respect to the energy gap in
ˆ
H
B
(as explained
in more detail in Sec. 4.1). If the energy in the sys-
tem becomes comparable to the gap, the difference
between
2L+1
and the true gauge group U(1) be-
comes noticeable, which leads to the freezing effect
(see Fig. 3). Crucially, working with a value of L
suitable for the regime g 1 will lead to exploding
computational costs, i.e. will require very large val-
ues of l, in the intermediated coupling regime g ≈ 1
to capture the relevant physics there. Our solution
to this problem is the dynamical adjustment of the
parameter L with the coupling g, that allows us to
approximate U (1) well for a wide range of couplings
while including only a minimal number of states in
our simulation (see Fig. 2).
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5 Generalisations: Dynamical matter
and arbitrary torus
In the following, we extend the results presented in
Sec. 4 by including staggered fermions in the numer-
ical simulations. In particular, we show that mat-
ter does not introduce any fundamental complication
for the completely compact formulation introduced in
Sec. 2. Moreover, to pave the way for further devel-
opments in the field, we derive the Hamiltonian for an
arbitrary number of plaquettes on a torus with matter
and periodic boundary conditions, and explain how to
include static charges.
5.1 Including dynamical charges
Since the completely compact formulation only affects
the gauge fields, the inclusion of matter is straightfor-
ward. Recall first the electric Hamiltonian in Eq. (13)
and the substitution rules in Eq. (21). By using the
relations for the Fourier transform derived in App.
C, the magnetic representation of the electric term in
Eq. (13) is found to be
ˆ
H
(b)
E
= g
2
2L
X
ν=1
(
f
c
ν
"
ˆ
K
ν
1
+
ˆ
K
ν
2
+
ˆ
K
ν
3
+
ˆ
K
ν
x
+
ˆ
K
ν
y
2
#
+f
s
ν
"
2L
X
µ=1
f
s
µ
1
2
ˆ
L
µ
2
ˆ
L
ν
1
+
ˆ
L
ν
3
−
1
4
ˆ
L
µ
x
ˆ
L
ν
1
+
ˆ
L
ν
2
−
ˆ
L
ν
3
+
1
4
ˆ
L
µ
y
ˆ
L
ν
1
−
ˆ
L
ν
2
−
ˆ
L
ν
3
+i
ˆq
(1,0)
2
ˆ
L
ν
1
+
ˆ
L
ν
x
+i
ˆq
(0,1)
2
ˆ
L
ν
2
−
ˆ
L
ν
1
+
ˆ
L
ν
y
+i
ˆq
(1,1)
2
2
ˆ
L
ν
1
−
ˆ
L
ν
2
−
ˆ
L
ν
x
+g
2
ˆq
2
(1,0)
+ ˆq
2
(0,1)
+ 2ˆq
(1,1)
[ˆq
(1,0)
+ ˆq
(1,1)
]
2
. (35)
For the sake of clarity, we defined the shorthand no-
tations
ˆ
K
ν
j
=
ˆ
P
ν
j
+ (
ˆ
P
†
j
)
ν
and
ˆ
L
ν
j
=
ˆ
P
ν
j
− (
ˆ
P
†
j
)
ν
. (36)
The magnetic field Hamiltonian
ˆ
H
(b)
B
remains the
same as in Eq. (27), since it does not involve fermionic
terms. However, the kinetic Hamiltonian in Eq. (15)
is modified in the presence of matter, yielding
ˆ
H
(b)
K
= κ
L
X
r=−L
h
ˆ
Ψ
†
(0,0)
1 + e
−i
2π
2L+1
r
x
ˆ
Ψ
(1,0)
+
ˆ
Ψ
†
(0,1)
e
−i
2π
2L+1
r
1
+ e
i
2π
2L+1
(r
2
−r
x
)
ˆ
Ψ
(1,1)
+
ˆ
Ψ
†
(0,0)
1 + e
−i
2π
2L+1
r
y
ˆ
Ψ
(0,1)
+
ˆ
Ψ
†
(1,0)
1 + e
i
2π
2L+1
(r
2
+r
3
−r
y
)
ˆ
Ψ
(1,1)
+H.c.
i
|rihr|. (37)
In order to simulate fermionic matter, we recall the
Jordan-Wigner transformation [81]
ˆ
Ψ
†
n
7→
Y
l<n
(iˆσ
l
z
)ˆσ
n
−
, (38)
where the vectorial relation l < n is defined by
(0, 0) < (0, 1) < (1, 1) < (1, 0) to satisfy the
fermionic commutation relations. In higher dimen-
sions, it might however be useful to consider alterna-
tive approaches, such as fermionic projected entan-
gled pair states or the elimination of the fermionic
matter [82, 83]. While we do not insert these equa-
tions into Eq. (37), we remark that the mass Hamil-
tonian in Eq. (5) is simplified to
ˆ
H
M
=
m
2
ˆσ
1
z
− ˆσ
2
z
+ ˆσ
3
z
− ˆσ
4
z
, (39)
which is independent of the chosen representation.
Since these simulations are costly, i.e. the di-
mension of the truncated Hilbert space is given by
2
4
(2l + 1)
5
(four charges, three rotators and two
strings), we estimate the plaquette expectation value
employing a harsh truncation of l = 2, while fixing L
to the optimal values L
opt
found in Sec. 4.2 for the
pure gauge case. This is a reasonable assumption,
since strings and fermionic matter only play a role in
the intermediate regime. Therefore, we can recover
our previous results for the pure gauge theory in the
strong and weak coupling limits, and focus our at-
tention to the differences where g ≈ 1. We further
introduce the mass and kinetic energy parameters as
m = κ = 10.
In Fig. 5, we display the ground state expecta-
tion value hi as a function of g
−2
, together with
the Fourier infidelity 1 − F
f
(l). While the asymp-
totic regimes g 1 and g 1 show no qualitative
difference if compared to the pure gauge case, the sit-
uation changes in the intermediate regime. There are
novel features in both the electric and magnetic repre-
sentations, such as the appearance of a negative dip.
Nevertheless, we stress that conclusions drawn from
this plot have to be taken with care, as the employed
truncation limits the Fourier fidelity below 90%.
Concluding, we demonstrated that our method is
suitable to tackle simulations with matter, and can be
scaled up to more complex systems. A detailed analy-
sis of novel effects and an accompanying study of the
Accepted in Quantum 2021-01-20, click title to verify. Published under CC-BY 4.0. 15
Figure 5: Plaquette expectation value in the pres-
ence of dynamical charges. Panel (a) displays the ex-
pectation value for l = 2 and (b) the Fourier fidelity derived
in this case. The red dashed line in (a) corresponds to results
derived in the magnetic representation, while the solid line is
a result of the electric representation. For all curves we set
m = κ = 10.
convergence is beyond the scope of this manuscript
and left for future works.
5.2 Hamiltonian for an arbitrary torus and
charges
Here, we generalise the Hamiltonian of the minimal
system considered in Sec. 2.2 to any two-dimensional
lattice with periodic boundary conditions. As shown
in Fig. 6, we extend the strategy used above to a torus
of size (N
x
, N
y
). By removing redundant DOF, we ob-
tain the effective Hamiltonian in terms of two strings
and N
x
N
y
− 1 rotators. As before, we indicate each
plaquette with its bottom-left site n = (n
x
, n
y
), where
n
x
(n
y
) = 0, . . . , N
x
− 1 (N
y
− 1). In addition, the ro-
tator associated with the plaquette n is denoted by
ˆ
R
n
, and the two strings by
ˆ
R
x
and
ˆ
R
y
(see Fig. 6).
This leads to N
x
N
y
pairwise expressions for the elec-
tric fields,
ˆ
E
n,e
x
= δ
n
y
,0
ˆ
R
x
+
ˆ
R
n
−
ˆ
R
n−e
y
+ ˆq
n,x
,
ˆ
E
n,e
y
= δ
n
x
,0
ˆ
R
y
+
ˆ
R
n−e
x
−
ˆ
R
n
+ ˆq
n,y
, (40)
where δ
n,m
= 1 for n = m, and zero otherwise. More-
over, ˆq
n,x
and ˆq
n,y
are the electric field’s corrections
due to the presence of dynamical charges, in accor-
dance with Gauss’ law. Since there are several ways
to implement Gauss’ law, a possible choice for ˆq
n,x
and ˆq
n,y
is (see the green lines in Fig. 6)
ˆq
n,x
= −
N
x
−1
X
r
x
=n
x
+1
N
y
−1
X
r
y
=0
δ
n
y
,0
ˆq
r
,
ˆq
n,y
= −
N
x
−1
X
r
x
=0
N
y
−1
X
r
y
=n
y
+1
δ
r
x
,n
x
ˆq
r
, (41)
where ˆq
n
is the charge operator as defined in Eq. (7).
Note that also in this general case it is convenient to
explicitly fix one of the rotators to zero, for instance
ˆ
R
(0,N
y
−1)
= 0.
Moving to the kinetic term, we employ the string
convention presented in Eq. (41), which yields the re-
placement rules (for details, see App. B.3)
ˆ
Ψ
†
n
ˆ
U
†
n,e
x
ˆ
Ψ
n+e
x
7→
ˆ
Ψ
†
n
ˆ
P
−δ
n
x
,N
x
−1
x
n
y
−1
Y
r
y
=0
ˆ
P
(n
x
,r
y
)
ˆ
Ψ
n+e
x
,
ˆ
Ψ
†
n
ˆ
U
†
n,e
y
ˆ
Ψ
n+e
y
(42)
7→
ˆ
Ψ
†
n
ˆ
P
†
y
N
x
−1
Y
r
x
=n
x
N
y
−1
Y
r
y
=0
ˆ
P
(r
x
,r
y
)
δ
n
y
,N
y
−1
ˆ
Ψ
n+e
y
.
From the above equations and from Eq. (40), we can
calculate the components of the gauged Hamiltonian
in the rotator and string basis as
ˆ
H
E
=
g
2
2
X
n
ˆ
E
2
n,e
x
+
ˆ
E
2
n,e
y
,
ˆ
H
B
= −
1
2g
2
a
2
Y
n6=(0,N
y
−1)
ˆ
P
n
+
X
n6=(0,N
y
−1)
ˆ
P
n
+ H.c.,
ˆ
H
K
= κ
X
n
ˆ
Ψ
†
n
ˆ
Ψ
n+e
x
ˆ
P
−δ
n
x
,N
x
−1
x
n
y
−1
Y
r
y
=0
ˆ
P
(n
x
,r
y
)
+
ˆ
Ψ
n+e
y
ˆ
P
†
y
N
x
−1
Y
r
x
=n
x
N
y
−1
Y
r
y
=0
ˆ
P
(r
x
,r
y
)
δ
n
y
,N
y
−1
+ H.c.,
ˆ
H
M
= m
X
n
(−1)
n
x
+n
y
ˆ
Ψ
†
n
ˆ
Ψ
n
. (43)
We note that the kinetic term contains string terms
that depend on the choice of the background strings
(see Fig. 6). Each of such terms can be simulated
digitally with a number of gates that scales linearly
with the system size. We further remark that these
equations are also valid for particles following bosonic
statistics [84], where the charge operator for site n is
defined as
ˆq
n
= q
ˆ
Ψ
†
n
ˆ
Ψ
n
. (44)
Accepted in Quantum 2021-01-20, click title to verify. Published under CC-BY 4.0. 16
Figure 6: Periodic torus with charges. We extend the con-
struction of the periodic plaquette to a generic torus. We
fix the rotator at (0, N
y
− 1) to zero and choose the links’
corrections to the electric field introduced by charges in ac-
cordance with the green dotted line. In particular, for any
charge ˆq
r
, we connect the origin to the site r by moving first
horizontally and then vertically.
In this case, the only required modification concerns
the mass Hamiltonian, which becomes
ˆ
H
M
= m
X
n
ˆ
Ψ
†
n
ˆ
Ψ
n
. (45)
Importantly, employing the relations derived in App.
C directly allows for the transformation between the
electric and magnetic representations.
As a final remark, we highlight that it is possible
to include static background charges into the descrip-
tion by keeping the
ˆ
Q
n
operators in Eq. (3). These
operators will then appear in the electric Hamiltonian,
accompanying their corresponding dynamical charge
ˆq
n
.
6 Conclusions and outlook
We developed a new strategy for studying gauge the-
ories. Our method is suited for simulations of funda-
mental particle interactions in all coupling regimes on
near-term quantum computers (see [31]), as well as on
classical devices. As a testbed, we applied our method
to the lattice Hamiltonian formulation of QED in 2+1
dimensions.
The key insight is the approximation of the contin-
uous U (1) gauge group with finite truncations of the
Z
2L+1
group, where L ∈ N can be arbitrarily large and
is scaled with the value of the bare coupling g. This
strategy allows us to work with fixed computational
resources, i.e. including only a constant number of
states in our simulation, for any value of g. At weak
couplings we truncate the gauge fields in the mag-
netic representation of Z
2L+1
, while the truncation
is performed in the electric representation for strong
coupling.
Depending on the regime, this solution offers to
choose the representation with the smaller trunca-
tion error. We benchmarked this novel regularisation
scheme by computing the expectation value of the pla-
quette operator on a small periodic lattice, with and
without dynamical matter, and estimated the accu-
racy of the computation.
Since our methods allows us to work at all values
of g and therefore at arbitrarily small values of
the lattice spacing a, it provides the perspective to
access, in principle, non-perturbative physics close to
the continuum limit.
Quantum simulations:
With regard to simulations of LGTs, there are two
different lines of work. The first research line stud-
ies lattice models in their own right. Lattice gauge
theories are for example relevant in condensed matter
physics or can be interesting per se. The second line of
research considers lattice gauge theories with the aim
to study the underlying continuum theory which de-
scribes for example fundamental particle interactions
and the standard model. For simulations of the lat-
ter, it is indeed of crucial importance that one is able
to take the continuum limit of a lattice theory. In the
field of quantum simulations, this challenge is mostly
unanswered.
So far, the only practical route to approach the
continuum limit were analog quantum simulators us-
ing infinite degrees of freedom to represent the gauge
fields. Neutral atoms in optical lattices offer a very
good solution in this respect, as the gauge field
can be represented as a spinor condensate while the
charges are identified with moving fermions, and the
gauge-matter interaction by spin changing collisions
[46, 50, 85–88]. A basic building block [41] and a one-
dimensional proof-of-principle experiment [42] that
exploit some of these ingredients have been already
performed. Beyond one dimension, such approaches
– although very promising – are fundamentally lim-
ited since magnetic (plaquette-type) interactions are
realised via higher-order interactions. This results in
low effective coupling strengths and thus in extremely
challenging experimental requirements.
While the analog approach based on bosonic de-
grees of freedom outlined above has the potential
to achieve the continuum limit, the experimental re-
alisation of two-dimensional theories involving mag-
netic terms is currently out of reach experimentally.
This type of interaction is easier to realise digitally
[47, 48, 51] and in qubit-based platforms [49, 54], such
as trapped ions, Rydberg atoms and superconduct-
Accepted in Quantum 2021-01-20, click title to verify. Published under CC-BY 4.0. 17
ing qubits. These simulation strategies however, cur-
rently lack the feature to reach the continuum limit.
Our new framework provides a route towards
reaching the continuum limit in quantum simulators
on different platforms and therefore opens a new
perspective for meaningful simulations that address
physical (i.e. continuum) phenomena that can be
related to experiments in high energy physics. It will
be interesting to explore the use of interacting chains
of spins larger than spin-1/2 [89] to simulate gauge
fields and to investigate the use of ultracold fermions,
which will allow one to simulate fermions on a
system that naturally displays the right quantum
statistics. As a first step towards proof-of-concept
demonstration using our new method, we show how
to apply our scheme to current ion-based quantum
hardware [31].
Tensor network calculations:
Resource minimisation is especially relevant for
classical simulations based on tensor network states.
Our regularisation scheme can also be reinterpreted
as a coupling dependent variational ansatz for the
gauge-field states, where the ratio between the trun-
cation parameter and the dimension of the discrete
group l/L is optimised. With this perspective in
mind, our scheme could be combined with other
variational approaches, e.g. with the method put
forward in [90], with the aim to extend the contin-
uum limit calculations beyond one dimension or for
addressing toy models of high-energy physics, such
as CP (N − 1) theories [91].
Extensions to higher-dimensional non-Abelian
gauge groups:
We note that both the minimal Hamiltonian formu-
lation (in which redundant degrees of freedom have
been removed) and our new regularisation scheme
can be extended to higher dimensions and to non-
Abelian gauge theories, also beyond the Hamiltonian
approach. The solutions we proposed here are inter-
esting for Lagrangian-based formalisms like the ten-
sor approach [92, 93] or for the development of novel
Monte Carlo approaches to avoid the sign problem
[94, 95]. From a geometric perspective, once U(1) is
identified with S
1
, and the magnetic vacuum with the
north pole [see Fig. 2], our regularisation scheme at
weak couplings can be interpreted as a lattice discreti-
sation of a circle around the north pole. One could ex-
ploit the map of SU (N) groups to higher dimensional
spheres considered in [94, 95] and repeat a similar pro-
cedure. An alternative discretisation of non-Abelian
groups for classical and quantum simulations has also
been considered in [52, 53, 96, 97].
In conclusion, our work opens new perspectives for
resource efficient Hamiltonian-based simulations and
provides a concrete step in the ‘quantum way’ to non-
perturbative phenomena in high energy physics.
7 Acknowledgements
We thank Raymond Laflamme, Peter Zoller, and
Rainer Blatt for fruitful and enlightening discussions.
This work has been supported by the Transformative
Quantum Technologies Program (CFREF), NSERC
and the New Frontiers in Research Fund. JFH ac-
knowledges the Alexander von Humboldt Foundation
in the form of a Feodor Lynen Fellowship. CM ac-
knowledges the Alfred P. Sloan foundation for a Sloan
Research Fellowship. AC acknowledges support from
the Universitat Aut`onoma de Barcelona Talent Re-
search program, from the Ministerio de Ciencia, Ino-
vaci´on y Universidades (Contract No. FIS2017-86530-
P), from the the European Regional Development
Fund (ERDF) within the ERDF Operational Program
of Catalunya (project QUASICAT/QuantumCat),
and from the the European Union’s Horizon 2020 re-
search and innovation programme under the Grant
Agreement No. 731473 (FWF QuantERA via QT-
FLAG I03769).
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A Dimensions of the (2+1) dimen-
sional QED Hamiltonian
This appendix provides the dimensional analysis of
the Hamiltonians used throughout this work. For the
sake of simplicity and to avoid problems when switch-
ing to the rotator formulation in Sec. 2.2, we aim for
using dimensionless gauge field and matter operators.
Using natural units c = ~ = 1 results in the following
relation of units,
[length] = [time] = [energy]
−1
= [mass]
−1
. (46)
The transition to the continuum limit is set by
a
2
X
n
→
Z
d
2
x (a → 0), (47)
where we denote the lattice spacing by a and label
all lattice points with a vector n as introduced in
Sec. 2.1. Since a is a length, [a] = mass
−1
. In
the following, we discuss each part of the total
Hamiltonian separately, keeping in mind that its
dimension [H] = [energy].
Electric Hamiltonian
The electric energy is given by
ˆ
H
E
= a
2
X
n
˜g
2
2
ˆ
J
2
n,e
x
+
ˆ
J
2
n,e
y
, (48)
where the term in the sum has the units of a
two-dimensional energy density, [energy]/[length]
2
=
[energy]
3
. We now rescale the electric field operators
ˆ
J
n,ˆe
µ
as a
3
ˆ
J
2
=
ˆ
E
2
and absorb the remaining units
into g = ˜g/
√
a. This yields [g
2
] = [energy] and
ˆ
H
E
=
g
2
2
X
n
ˆ
E
2
n,e
x
+
ˆ
E
2
n,e
y
, (49)
Accepted in Quantum 2021-01-20, click title to verify. Published under CC-BY 4.0. 20
where the field operators
ˆ
E
n,e
µ
are dimensionless.
Magnetic Hamiltonian
The plaquette operators are dimensionless since they
are constructed from the field creation operators,
which themselves are dimensionless. Note that
ˆ
U = e
iag
ˆ
A
, (50)
where the product ag
ˆ
A needs to be dimensionless
(allowing for a valid series representation). There-
fore, the dimension of the gauge potential is fixed by
[
ˆ
A] =
p
[mass]. Recalling Eq. (2), the
ˆ
P
n
operator
can be expressed as
ˆ
P
n
= exp
(
ia
2
g
ˆ
A
n,e
x
−
ˆ
A
n+e
y
,e
x
a
−
ˆ
A
n,e
y
−
ˆ
A
n+e
x
,e
y
a
!)
. (51)
Thus, in order to obtain the required continuum limit
where
ˆ
H
B
converges to
R
dx
2
F
µν
F
µν
/4, we define
ˆ
H
B
= −
1
2g
2
a
2
X
n
ˆ
P
n
+
ˆ
P
†
n
a
4
, (52)
where the the desired second order of
ˆ
P
†
+
ˆ
P is pro-
portional to a
4
. The denominator hence ensures its
survival and thus the correct continuum limit. This
yields
ˆ
H
B
= −
1
2g
2
a
2
X
n
ˆ
P
n
+
ˆ
P
†
n
, (53)
which is consistent with [g
2
] = [mass] since we require
that [1/(g
2
a
2
)] = [mass] = [energy].
Mass Hamiltonian
We have that
ˆ
H
M
= Ma
2
X
n
(−1)
n
x
+n
y
ˆ
ψ
†
n
ˆ
ψ
n
, (54)
where
ˆ
ψ =
ˆ
φ/a is the fermion field and M represents
the bare mass. Hence, the Hamiltonian reduces to
ˆ
H
M
=
m
a
2
a
2
X
n
(−1)
n
x
+n
y
ˆ
φ
†
n
ˆ
φ
n
= m
X
n
(−1)
n
x
+n
y
ˆ
φ
†
n
ˆ
φ
n
, (55)
where the operators
ˆ
φ
n
are dimensionless and
[M] = mass.
Kinetic Hamiltonian
Following the arguments for the mass and magnetic
Hamiltonians, i.e. replacing
ˆ
ψ =
ˆ
φ/a, we arrive at
ˆ
H
K
= K
X
n
X
µ=x,y
(
ˆ
φ
†
n
ˆ
U
n,e
µ
ˆ
φ
n+e
µ
+ H.c.), (56)
where K = 1/(2a).
Rescaling of the fermionic field operators
Note that it is possible to redefine
ˆ
φ =
ˆ
Ψ/
√
α, where
α is dimensionless and
ˆ
Ψ is a rescaled fermion field.
By doing so,
ˆ
H
M
=
M
α
X
n
(−1)
n
x
+n
y
ˆ
Ψ
†
n
ˆ
Ψ
n
, (57)
and
ˆ
H
K
=
K
α
X
n
X
µ=x,y
(
ˆ
Ψ
†
n
ˆ
U
n,e
µ
ˆ
Ψ
n+e
µ
+ H.c.), (58)
where we might define the new effective mass m =
M/α and effective kinetic energy scale κ = K/α valid
for the rescaled fermionic fields as employed in the
main text. Moreover, this rescaling implies a rescaling
of the charge operator, which we define as
ˆq
n
= Q||
ˆ
φ
†
n
ˆ
φ
n
||
max
×
ˆ
φ
†
n
ˆ
φ
n
||
ˆ
φ
†
n
ˆ
φ
n
||
max
−
2
[1 − (−1)
n
x
+n
y
]
!
= q
ˆ
Ψ
†
n
ˆ
Ψ
n
−
2
[1 − (−1)
n
x
+n
y
]
, (59)
where q = Q||
ˆ
Ψ
†
n
ˆ
Ψ
n
||
max
= Q/α ∈ Z and q = 1 in the
main text. Note that this operator is dimensionless as
well, since it is required to be of the same dimension
as the electric field
ˆ
E.
B Hamiltonian in the link formalism
and link-to-rotator translation rules
B.1 Effective Hamiltonian for a minimal lattice
in the link formulation
As explained in Sec. 2, there are several ways to
express the Hamiltonian of one or more plaquettes.
In the main text, we employed electric loops for
parametrizing the physical states and defining the
corresponding operators, rotators and strings. These
represent a natural choice since the rotator’s lower-
ing operators are directly identified with the plaquette
operators
ˆ
P
n
[see Eq. (2)]. Here, we derive the Hamil-
tonian of a lattice containing four sites and staggered
fermions in terms of the links
ˆ
E
n,e
µ
. In particular, we
choose three out of the eight electric fields on the links,
and express them in terms of the others to minimise
the number of degrees of freedom. In this appendix,
we consider the compact U(1) formulation of the QED
lattice Hamiltonian reviewed in Sec. 2.1. The com-
pletely compact Z
2L+1
QED formulation for both the
electric and the magnetic representation can be ob-
tained following the procedure outlined in Sec. 2.2.
Generalisations to multiple plaquettes are straightfor-
ward.
Accepted in Quantum 2021-01-20, click title to verify. Published under CC-BY 4.0. 21
Employing Eq. (3), we can directly assess Gauss’
laws in Fig. 1(b) which yield
ˆ
E
(0,0),e
x
+
ˆ
E
(0,0),e
y
−
ˆ
E
(1,0),e
x
−
ˆ
E
(0,1),e
y
= ˆq
(0,0)
,
ˆ
E
(0,1),e
x
+
ˆ
E
(0,1),e
y
−
ˆ
E
(1,1),e
x
−
ˆ
E
(0,0),e
y
= ˆq
(0,1)
,
ˆ
E
(1,1),e
x
+
ˆ
E
(1,1),e
y
−
ˆ
E
(0,1),e
x
−
ˆ
E
(1,0),e
y
= ˆq
(1,1)
,
ˆ
E
(1,0),e
x
+
ˆ
E
(1,0),e
y
−
ˆ
E
(0,0),e
x
−
ˆ
E
(1,1),e
y
= ˆq
(1,0)
.
(60)
Only three of these constraints are independent since
charge conservation requires ˆq
(0,0)
= −ˆq
(0,1)
− ˆq
(1,1)
−
ˆq
(1,0)
. Expressing the arbitrarily chosen electric field
operators
ˆ
E
(1,0),e
x
,
ˆ
E
(0,1),e
y
and
ˆ
E
(1,1),e
y
in terms of
the others, we write the constrained electric Hamilto-
nian as
ˆ
H
E
=
g
2
2
n
ˆ
E
2
(0,0),e
x
+
ˆ
E
2
(0,0),e
y
+
ˆ
E
2
(0,1),e
x
+
ˆ
E
2
(1,0),e
y
+
ˆ
E
2
(1,1),e
x
+
h
ˆ
E
(0,0),e
y
−
ˆ
E
(0,1),e
x
+
ˆ
E
(1,1),e
x
+ ˆq
(0,1)
2
+
h
ˆ
E
(0,1),e
x
−
ˆ
E
(1,1),e
x
+
ˆ
E
(1,0),e
y
+ ˆq
(1,1)
2
+
h
ˆ
E
(0,0),e
x
+
ˆ
E
(0,1),e
x
−
ˆ
E
(1,1),e
x
+ ˆq
(1,1)
+ ˆq
(1,0)
2
o
. (61)
Since Gauss’ law affects the electric term only, the
changes in the magnetic, mass, and kinetic contribu-
tions to the total Hamiltonians are trivial. Note the
natural appearance of the dynamical charges, which
can be interpreted as sources of the electric field. Im-
portantly, since
ˆ
E
(1,0),e
x
,
ˆ
E
(0,1),e
y
and
ˆ
E
(1,1),e
y
are no
longer dynamical variables, the corresponding raising
and lowering operators become identities, leading to
ˆ
H
B
=
1
2g
2
a
2
n
ˆ
U
(0,0),e
x
ˆ
U
(1,0),e
y
ˆ
U
†
(0,1),e
x
ˆ
U
†
(0,0),e
y
+
ˆ
U
(0,0),e
y
ˆ
U
†
(1,1),e
x
ˆ
U
†
(1,0),e
y
+
ˆ
U
(1,1),e
x
+
ˆ
U
(0,1),e
x
ˆ
U
†
(0,0),e
x
+ H.c.
o
,
ˆ
H
M
=m
n
ˆ
Ψ
†
(0,0)
ˆ
Ψ
(0,0)
−
ˆ
Ψ
†
(0,1)
ˆ
Ψ
(0,1)
+
ˆ
Ψ
†
(1,1)
ˆ
Ψ
(1,1)
−
ˆ
Ψ
†
(1,0)
ˆ
Ψ
(1,0)
o
,
ˆ
H
K
=κ
n
ˆ
Ψ
†
(0,0)
ˆ
U
(0,0),e
x
ˆ
Ψ
(1,0)
+
ˆ
Ψ
†
(0,0)
ˆ
U
(0,0),e
y
ˆ
Ψ
(0,1)
+
ˆ
Ψ
†
(0,1)
ˆ
U
(0,1),e
x
ˆ
Ψ
(1,1)
+
ˆ
Ψ
†
(1,0)
ˆ
U
(1,0),e
y
ˆ
Ψ
(1,1)
+
ˆ
Ψ
†
(1,1)
ˆ
U
(1,1),e
x
ˆ
Ψ
(0,1)
+
ˆ
Ψ
†
(0,1)
ˆ
Ψ
(0,0)
+
ˆ
Ψ
†
(1,0)
ˆ
Ψ
(0,0)
+
ˆ
Ψ
†
(1,1)
ˆ
Ψ
(1,0)
+ H.c.
o
. (62)
B.2 An intuitive picture of the rotator and link
operator relation
The representation of each rotator or string in terms
of the link operators
ˆ
E
n,e
µ
(µ = x, y) can be inter-
preted as an analogue to Kirchhoff’s loop law in elec-
trical circuits. As shown in Fig. 1(a) the electric field
ˆ
E
n,e
x
(
ˆ
E
n,e
x
) between vertices n and n + e
x
(n +e
y
)
is oriented along the positive e
x
(e
y
) direction. Each
rotator or string operator is then defined as the sum
of the link contributions along it. The sign of a contri-
bution is positive, if the rotator or string is oriented
with the positive field direction, or negative, if ori-
ented opposingly. For example, it holds that
ˆ
R
1
= E
(0,0),e
x
+ (−1)(ˆq
(1,0)
+ ˆq
(1,1)
) +
ˆ
E
(1,0),e
y
+ (−ˆq
(1,1)
) + (−E
(0,1),e
x
) + (−E
(0,0),e
y
)
+ ˆq
(1,0)
, (63)
where we explicitly marked opposing directions with
a minus sign. Using the defining equations for the
remaining operators and removing all redundant de-
grees of freedom yields an inverted form of Eq. (9),
which reads
ˆ
R
1
= −
ˆ
E
(0,1),e
x
,
ˆ
R
2
= −
ˆ
E
(0,1),e
x
−
ˆ
E
(1,0),e
y
− ˆq
(1,1)
,
ˆ
R
3
= −
ˆ
E
(1,1),e
y
ˆ
R
x
= −
ˆ
E
(0,0),e
x
+
ˆ
E
(0,1),e
x
+ ˆq
(1,0)
+ ˆq
(1,1)
,
ˆ
R
y
=
ˆ
E
(0,1),e
y
+
ˆ
E
(1,1),e
y
. (64)
B.3 Gauge field creation in the rotator and
string picture
In the following, we illustrate the formulation of
the kinetic Hamiltonian Eq. (6) for a general two-
dimensional plane in the rotator and string formula-
tion. In particular, the mapping involves transforma-
tions of the type
ˆ
ψ
†
n
ˆ
U
†
n,e
µ
ˆ
ψ
n+e
µ
7→
ˆ
ψ
†
n
f
n,e
µ
n
ˆ
P ,
ˆ
P
†
o
ˆ
ψ
n+e
µ
, (65)
where µ = x, y and f
n,e
µ
is a function of both rotator
and string operators. Depending on the particle pair
to be created or annihilated, there are four distinct
rules for building up the functions f
n,e
µ
. These are
shown in the four corresponding panels displayed in
Fig. 7, where we use a yellow (green) circle to indicate
an (anti)fermion, and light blue arrows for the corre-
sponding link in which the electric field is required to
raise [
ˆ
U
†
n,e
µ
in Eq. (65)]. Employing the notation pre-
sented in the legend of Fig. 7, we can directly build
the functions f
n,e
µ
in Eq. (65) and recover the rules
presented in Eq. (42).
As an example, consider Fig. 7(a), which describes
all cases in which a pair is created at locations n and
n + e
x
, with n
x
6= N
x
− 1. If n
y
= 0, the electric
field is automatically corrected by the chosen charge
strings that ensure Gauss’ law (dotted green and yel-
low lines in Fig. 7; see Sec. B.1 and Sec. 5.2), im-
plying f
(n
x
,0),e
x
= . Otherwise (n
y
6= 0), to in-
crease the electric field
ˆ
E
n,e
x
on the link between the
two charges, we lower the rotator
ˆ
R
(n
x
,n
y
−1)
below
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Figure 7: Pair creation on a periodic two-dimensional lattice. The four panels describe the creation of a gauge field (blue
arrow) in x direction [(a) and (b)], and in y direction [(c) and (d)]. The particles (green and yellow circles) imply the creation
of electric fields on the links marked with the corresponding dashed arrows. Only in (c) these contributions are enough to
create the gauge field while maintaining gauge-invariance in all other links. Otherwise, the gauge field has to be created
by annihilating the plaquette operators marked with red circular arrows, which also counteracts the action of the particles’
charges. Note, that the field on a link is effectively unchanged if it is modified by an equal number of arrows in both directions,
which in (b) and (d) requires the introduction of the strings (grey and pink) when the particles are created on the boundary
condition of the lattice.
by applying
ˆ
P
(n
x
,n
y
−1)
. This, however, affects the
electric fields on all other links forming
ˆ
R
(n
x
,n
y
−1)
.
While the vertical ones are taken care of by the charge
strings, the bottom link is not (unless n
y
= 1). As
graphically explained in Fig. 7(a), to increase only
the desired link, we can lower all rotators
ˆ
R
(n
x
,r
y
)
with r
y
= 0, ..., n
y
− 2 below, yielding f
(n
x
,n
y
),e
x
=
Q
n
y
−1
r
y
=0
ˆ
P
(n
x
,r
y
)
.
The remaining panels in Fig. 7 further illustrate the
cases of pairs that are connected by vertical links and
pairs that are created on links closing the periodic
boundary conditions, where we require the strings
ˆ
R
x,y
. By following the same procedure above, it is
then possible to determine the functions f
n,e
µ
for all
allowed choices of n and µ = x, y, yielding Eq. (42).
C Diagonalisation of the magnetic
gauge field Hamiltonian
The magnetic Hamiltonian
ˆ
H
B
is composed of
the lowering and raising operators
ˆ
P
j
,
ˆ
P
†
j
, j =
1, 2, . . . , N − 1, where N is the total number of pla-
quettes. In the Z
2L+1
group, these operators are the
so-called cyclant matrices, which can be diagonalised
exactly. Before truncation, the lowering operators are
defined according to Eq. (20),
ˆ
P
j
= |L
j
ih−L
j
| +
L
j
X
r
j
=−L
j
+1
|r
j
− 1ihr
j
|. (66)
For the sake of simplicity, we drop the index j and
note that the procedure is equivalent for all subsys-
tems. The spectrum of the lowering operators is
ω
k
= e
−i
2π
2L+1
k
, (67)
while the corresponding eigenvectors are given by
v
k
=
1
√
2L + 1
(ω
L
k
, ω
L−1
k
, . . . , ω
−L
k
)
T
, (68)
with k = −L, . . . , L. Hence,
ˆ
U is diagonalised by the
matrix
V
†
= (v
−L
, v
−L+1
, . . . , v
L
)
=
1
√
2L + 1
L
X
µ,ν=−L
e
−i
2π
2L+1
µν
|µihν|
≡
ˆ
F
†
2L+1
, (69)
which is the discrete Fourier transform. Hence, it is
straightforward to show that
ˆ
F
2L+1
ˆ
P
γ
ˆ
F
†
2L+1
=
L
X
r=−L
exp
−i
2π
2L+1
γr
|rihr|, (70)
where γ ∈ . Moreover, for any N − 1 ≥ J ∈ N, we
have that
ˆ
F
2L+1
"
J
O
j=1
ˆ
P
γ
j
#
ˆ
F
†
2L+1
=
L
X
r=−L
exp
−i
2π
2L+1
γr
|rihr|. (71)
Here, we use r = (r
1
, r
2
, . . . , r
J
)
T
and γ =
(γ
1
, γ
2
, . . . , γ
J
)
T
, while we waived to denote that the
Fourier transform is now understood as the product
of the Fourier transforms in the separate N −1 spaces.
Note that, in particular (
ˆ
P
γ
)
†
=
ˆ
P
−γ
and therefore:
ˆ
F
2L+1
J
O
j=1
ˆ
P
γ
j
±
J
O
j=1
ˆ
P
−γ
j
ˆ
F
†
2L+1
= 2
L
X
r=−L
(
cos(
2πrγ
2L+1
)|rihr| for +
−i sin(
2πrγ
2L+1
)|rihr| for −
. (72)
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Let us finally remark that these relations hold inter-
changeably for
ˆ
P in the rotator formalism and
ˆ
U for
the electric fields as introduced in Sec. 2.1.
D Asymptotic behaviour of the ground
state expectation value of
In this appendix, we describe the behaviour of the
ground state expectation value hi both in the electric
and magnetic representation. We look at the extremal
regimes and study truncation effects.
We indicate the average value of the observable
in the electric (magnetic) representation with h
(e)
i
(h
(b)
i). Recall that = −g
2
ˆ
H
B
/4, and consider
the electric representation first. Here,
ˆ
H
(e)
B
is com-
posed of the Hermitian operators
ˆ
P
i
+
ˆ
P
†
i
(i = 1, 2, 3),
and of
ˆ
P
1
ˆ
P
2
ˆ
P
3
+ (
ˆ
P
1
ˆ
P
2
ˆ
P
3
)
†
, where the latter is the
replacement for
ˆ
P
4
+
ˆ
P
†
4
(see Sec. 2.2). Hence, we
can bound the spectrum of −
ˆ
H
B
as −hψ|
ˆ
H
B
|ψi ≤
4hψ|(
ˆ
P
i
+
ˆ
P
†
i
)|ψi/(2g
2
) = 2λ
max
/g
2
. Importantly, this
holds for all i since the operators describe the rota-
tors which represent identical systems. In the regime
where g 1, −2λ
max
/g
2
corresponds to the ground
state energy and consequently λ
max
/2 to h
(e)
i. Now,
as long as the Hamiltonian is not truncated, i.e. l = L,
we have that
ˆ
P
i
+
ˆ
P
†
i
is a circulant matrix [98] and
thus its eigenvalues are given as
ξ
j
= 2 cos
2πj
2L + 1
, j = 0, 1, . . . , 2L. (73)
Consequently, λ
max
= 2 and thus h
(e)
(g = 0)i = 1.
However, as soon as the Hamiltonian is truncated,
ˆ
P
i
+
ˆ
P
†
i
is a tridiagonal Toeplitz matrix [98] with eigen-
values
ξ
j
= 2 cos
π(j + 1)
2l + 2
, j = 0, 1, . . . , 2l. (74)
This yields −hψ|
ˆ
H
B
|ψi ≤ 4 cos
π
2l+2
, which results
in a monotonic increase of h
(e)
(g = 0)i with respect
to an increase of l. The continuous limit is found for
l → ∞. Since the electric representations is based on
the truncated U(1) group, we conclude that for g 1
it approximates the true U(1) value from below.
Consider now the magnetic representation. The
curves in Fig. 4 suggest that h
(b)
i is monotonically
decreasing over the whole range of g when l is in-
creased. In the following, we qualitatively motivate
this behaviour in the strong coupling regime. When
g 1, h
(b)
i can be understood as a Riemann sum
of the discrete eigenvalues of
ˆ
H
(b)
B
, weighted with
the probabilities corresponding to the different basis
states |ri. Assume for now l = L. Then, the ground
state emerges as the equal superposition of all states
(as in Eq. (29), where we considered the electric rep-
resentation for g 1). The weights of the Riemann
sum are thus all equal to (2L + 1)
−3
, meaning that
h
(b)
i =
L
X
r=−L
1
(2L + 1)
3
"
X
i
cos
2π
2L + 1
r
i
+ cos
2π
2L + 1
(r
1
+ r
2
+ r
3
)
= 0. (75)
Under the assumption that the distribution of the
ground state’s coefficients p
(b)
(g 1, r, L) (see
Sec. 4.1) remains uniform in r for l < L, one can show
that this value is larger than zero for any truncation.
In particular,
h
(b)
i ∼ 3(2l + 1)
3
sin
2π(L − l)
2L + 1
−→
l→L
0
+
. (76)
However, the distribution p
(b)
(g 1, r, L) is not uni-
form when l < L, but remains center-symmetric. We
numerically show in App. E that removing higher
levels increases the amplitudes of states with low
values of |r|, and hence enhances positive contribu-
tions of the Riemann sum. In other words, trun-
cation not only removes the negative contributions
stemming from large values of |r| [they appear for
r
i
> (2L + 1)/4, see Eq. (75)], but also emphasises
positive contributions. While the first process is anal-
ogous to the decrease of the spread when decreasing
g, the second is solely originating from the truncation
and we therefore conclude that for any L the approxi-
mated value of h
(b)
i is always larger than the one ob-
tained from a hypothetical exact diagonalisation em-
ploying U(1). Qualitatively, this effect emerges from
the removal of the cyclic property present in the low-
ering operator
ˆ
P (see Eq. (20) and App. E).
E Truncation effects in the strong cou-
pling regime
In this appendix, we sketch the treatment of the trun-
cation of the cyclic Z
2L+1
group. We consider the
strong coupling regime, but employ the magnetic rep-
resentation of the Hamiltonian. In the limit g → ∞,
the electric term, which is composed of operators
ˆ
P
j
,
j = 1, 2, 3 [see Eq. (20)], is dominant. As such, we
ignore the magnetic Hamiltonian in the following and
employ the following ansatz to obtain the truncated
electric field Hamiltonian
H
(b)
E,truncated
=
ˆ
H
(b)
E,untruncated
−
ˆ
V . (77)
We hence define the operator
ˆ
V to study the effects
of the truncation.
In the untruncated
2L+1
formulation we can de-
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Figure 8: Transformation of the ground state distribution
after truncation. Both panels correspond to l = 7 and
L = 8, the x-axis labels the amplitudes belonging to |ri.
Compared to (a), (b) shows the effect when only the elements
corresponding to the cyclic property of Z
2L+1
are removed.
compose any
ˆ
P
j
into four terms,
ˆ
P
j
= |L
j
ih−L
j
| +
L
X
r
j
=l+1
|r
j
− 1ihr
j
|
+
−l
X
r
j
=1−L
|r
j
− 1ihr
j
|
+
l
X
r
j
=1−l
|r
j
− 1ihr
j
|
≡
ˆ
V
0
j
+
ˆ
P
0
j
, (78)
where
ˆ
P
0
j
=
P
l
r
j
=1−l
|r
j
− 1ihr
j
| and
ˆ
P
0
j
is the rest.
Notice that
ˆ
P
0
j
is the truncated operator as defined
in Eq. (19). We now have to collect all contributions
from
ˆ
V
0
j
in the electric Hamiltonian
ˆ
H
(b)
E
of Eq. (26).
In particular,
ˆ
V can be found from
ˆ
H
(b)
E
by applying
the rules
P
j
+ P
†
k
7→
ˆ
V
0
j
+ (
ˆ
V
0
k
)
†
,
P
j
P
k
7→
ˆ
V
0
j
(
ˆ
V
0
k
)
†
+
ˆ
V
0
j
(
ˆ
P
0
k
)
†
+
ˆ
P
0
j
(
ˆ
V
0
k
)
†
. (79)
Note that
ˆ
V has as well a diagonal contribution given
by
3
X
j=1
L
X
r
j
=l+1
|r
j
ihr
j
| + |−r
j
ih−r
j
|, (80)
stemming from the ν = 0 terms in Eq. (26).
The result of a numerical procedure to find the el-
ements of ground state is shown in Fig. 8(a). We
Figure 9: Sequence infidelity for g
−2
= 100 and l =
2, 3, . . . , 10. The minimum corresponds to the value of L
with the best compromise between available domain and res-
olution. Note that this minimum is not always the global
one.
denote the amplitudes of a state obtained with the
untruncated Hamiltonian with the red dashed line.
Clearly, the truncation shifts population from the high
|r| states to lower ones. Note that each |r| can appear
multiple times. A crucial fact is shown in Fig. 8(b),
where we removed only the cyclic elements from the
operators P
j
, i.e., the first terms in Eq. (78). This
cyclic property is the reason why the distributions
p
(b)
(r) of the ground state’s coefficients (see Sec. 4.1)
are uniform in the untruncated case. Hence, their re-
moval has a large impact on the derived results.
F Numerical determination of L
opt
The sequence fidelity calculated in Sec. 4.2 for the
ground state of the pure gauge QED Hamiltonian
involves and optimization over L. We plot the se-
quence infidelity varying the parameters l and L for
g
−2
= 100 in Fig. 9. For any value of l, a kink
is clearly visible, corresponding to L = L
opt
. No-
tice that this kink does not always correspond to a
global minimum, as can be seen from the points char-
acterised by l = 2. As discussed in the main text,
this is the signature of the freezing effect. In fact,
the global minimum is found for L = l + 1 = 3 (i.e.
its minimal value), where the resolution is insufficient
for both l = 1 and l = 2 to capture the distribution
of the untruncated ground state. By increasing g
−2
,
the position of the kink is shifted to higher values of
L. In particular, L
opt
takes its minimal allowed value
in the strong coupling regime, and starts to increase
at g
−2
≈ 5 [see Fig. 3(d)]. This follows from the
fact that, approaching the weak coupling regime, the
distribution of the untruncated U(1) ground state be-
comes more and more localised, and the tails less im-
portant. Note that the value L
opt
can be determined
by a greed search, starting at the lowest allowed value
of L.
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