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
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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