Towards a Quantum Simulation of Nonlinear Sigma Models with a Topological Term

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IPPP/22/68

MITP-22-079

Towards a Quantum Simulation of Nonlinear Sigma Models with a Topological Term

Jack Y. Araz ,1, ∗Sebastian Schenk ,1, 2, †and Michael Spannowsky 1, ‡

1Institute for Particle Physics Phenomenology, Durham University,

South Road, Durham DH1 3LE, United Kingdom

2PRISMA+Cluster of Excellence & Mainz Institute for Theoretical Physics,

Johannes Gutenberg-Universit¨at Mainz, 55099 Mainz, Germany

We determine the mass gap of a two-dimensional O(3) nonlinear sigma model augmented with a

topological θ-term using tensor network and digital quantum algorithms. As proof of principle, we

consider the example θ=πand study its critical behaviour on a quantum simulator by examining

the entanglement entropy of the ground state. We conﬁrm that the quantum theory is massless in

the strong-coupling regime, in agreement with analytical results. However, we also highlight the

limitations of current quantum algorithms, designed for noisy intermediate-scale quantum devices,

in the theory simulation at weak coupling. Finally, we compare the performance of our quantum

algorithms to classical tensor network methods.

I. INTRODUCTION

Nonlinear sigma models have often been a test bed for

exploring the intricate relationship between high-energy

physics and condensed matter systems. In particular,

the O(3) nonlinear sigma model in two dimensions ex-

hibits various non-trivial features of quantum ﬁeld theo-

ries (QFTs). These include asymptotic freedom and a dy-

namical generation of a strong scale [1], or the emergence

of instantons and merons [2,3]. It furthermore shares

many of these features with four-dimensional Yang-Mills

theory [4].

In two dimensions, the O(3) nonlinear sigma model

allows for non-trivial θ-vacua. Their presence has two

critical implications concerning the structure of the un-

derlying QFT. On the one hand, physically, they can dra-

matically change the fundamental properties of the quan-

tum theory. In general, the latter exhibits diﬀerent fea-

tures within diﬀerent topological sectors. For instance,

the nonlinear sigma model with a topological term of

θ= 2πS is closely related to a quantum spin chain with

spin S[5]. This observation can be exploited to obtain

the mass gap of the quantum theory in diﬀerent topo-

logical sectors (see, e.g., [6]). For instance, it is well

known that the nonlinear sigma model without a topo-

logical term, θ= 0, is gapped [7–11]. In strong contrast,

the same theory with θ=πis massless at the quan-

tum level for any value of the coupling constant [12]. In

this scenario, a conformal ﬁeld theory (CFT) describes

the underlying quantum theory’s critical behaviour by

the vanishing mass gap. In the strong-coupling regime of

the nonlinear sigma model, the latter is equivalent to the

Wess-Zumino-Novikov-Witten CFT, with central charge

c= 1 [12–18]. Similarly, the central charge of the CFT

should approach c= 2 in the weak-coupling limit [12].

On the other hand, in practice, the computational

∗jack.araz@durham.ac.uk

†schenkse@uni-mainz.de

‡michael.spannowsky@durham.ac.uk

treatment of the QFT is altered. In particular, a nu-

merical investigation is drastically hampered due to the

so-called sign problem. Any θ-term that renders the ac-

tion imaginary will turn the path integral into a highly

oscillatory object. A precise evaluation of the latter will

typically fail due to numerical cancellations between con-

tributions of diﬀerent complex phases. In fact, this is the

source of the sign problem and is one of the signiﬁcant

drawbacks in applying Monte Carlo techniques to non-

trivial QFTs, such as quantum chromodynamics on the

lattice (see, e.g., [19]). However, in some special situa-

tions, these problems can be bypassed to a certain degree,

notably in nonlinear sigma models [20–25].1

Tensor Network (TN) methods have been designed

to overcome the sign problem. They have successfully

addressed various lattice gauge theory questions [27–

30] and, in particular, also nonlinear sigma models at

θ= 0 [31] and θ=π[32]. Although TNs constitute

a promising approach to studying QFTs on the lattice,

they also suﬀer from several shortcomings. For instance,

it tensors of large dimensions2are typically required to

approximate the underlying quantum theory suﬃciently.

Additionally, despite the availability of various contrac-

tion algorithms, simulating TNs for lattice QFTs be-

yond two-dimensional spacetimes becomes computation-

ally highly challenging. This is mainly due to the attempt

to solve a quantum system with a classical approach

where the quantum nature has been compensated by ex-

tensive resources. More precisely, the structure of en-

tanglement is captured with large auxiliary dimensions.

Quantum computing (QC) techniques may allow us to

overcome this issue by eliminating the need for extensive

resources by intrinsically retaining the quantum nature

of the problem.

The simulation of QFTs on quantum devices has

been discussed in pioneering studies [33,34] in which

1For a recent assessment of θ-vacua and their lattice regularization

in these models, see [26].

2Technically speaking, the bond dimensions of the TN are often

large, O(≥103).

arXiv:2210.03679v2 [quant-ph] 30 Mar 2023

various quantum algorithms have been proposed, such

as quantum Monte Carlo, Hamiltonian simulation, and

imaginary-time evolution algorithms. The Hamiltonian

simulation, similar to TN approaches, allows quantum

algorithms to avoid the sign problem mentioned above.

However, it is essential to note that these techniques have

their own limitations. Due to the limited number of

“noise free” qubits or qubits to control the error, the em-

bedding of the Hamiltonian is signiﬁcantly restricted. For

scenarios with inﬁnite Hilbert space dimensions, Hamil-

tonian truncation is especially crucial [35,36].

In particular, nonlinear sigma models have been in-

vestigated in spin-lattice systems [37] and with quantum

computing techniques [38,39]. In the latter case, it is

demonstrated that nonlinear sigma models can have a

qubit-eﬃcient description through a fuzzy-sphere repre-

sentation. Although this allows for a Hamiltonian trun-

cation that is suitable for quantum time evolution algo-

rithms, it is not prone to a straightforward generalisation

to nonlinear sigma models that feature a richer structure.

In this work, we go beyond this limitation and augment a

two-dimensional nonlinear sigma model with a topologi-

cal θ-term. In particular, using quantum-gate simulators,

we aim to study the entanglement entropy of the vacuum

to investigate the critical behaviour and determine the

mass gap of the quantum theory.

In addition to the simulation of QFTs, both TN-

inspired quantum circuits and conventional TN methods

have also been used in various machine-learning applica-

tions. It has been shown that their relation with quantum

many-body systems can be used to achieve more inter-

pretable networks [40–43].

This work is organised as follows. In Section II, we re-

view the Hamiltonian formulation of the O(3) nonlinear

sigma model at θ=πin terms of angular momentum

variables on a one-dimensional spin chain. Section III

shows how these can be embedded on a quantum com-

puter. In particular, we use this approach to compute

the bipartite entanglement entropy associated with the

half chain and we conﬁrm the vanishing mass gap of the

theory in Section IV. Finally, we brieﬂy summarise our

results and conclude in Section V.

II. HAMILTONIAN FORMULATION OF

NONLINEAR SIGMA MODELS

The ﬁeld content of a general O(3) nonlinear sigma

model is given by a real vector ﬁeld nthat takes values

on a sphere, S2, i.e. it is normalised to n2= 1. In a two-

dimensional Euclidean spacetime, the associated action

is commonly written as

S=1

2g2Zd2x(∂µn)2.(1)

Here, gis the dimensionless coupling constant, and we

consider Euclidean coordinates τand x. At the classical

level, the vector ﬁeld is massless. This remains true to

all orders in perturbation theory. Nevertheless, it can be

shown that the quantum theory is gapped [7–11].

As the two-dimensional O(3) nonlinear sigma model

admits instanton solutions (see, e.g., [2,3]), it is feasible

to consider an additional topological term in this theory.

Along these lines, we can distinguish ﬁnite-action ﬁeld

conﬁgurations by their topological charge, Q=Rd2x ρQ,

where

ρQ=1

4πabcna∂xnb∂τnc(2)

is the topological charge density. These ﬁeld conﬁgura-

tions, in turn, contribute a ﬁnite θ-term to the action,

Sθ=1

2g2Zd2x(∂µn)2+iθQ . (3)

Since the topological charge is an integer, the action is

2π-periodic with respect to θ. In the following, we aim to

investigate the topological term’s eﬀect on the quantum

theory’s mass gap. For concreteness, let us focus on the

case θ=πin the following. As we will see momentarily,

this choice allows for a simple Hamiltonian formulation

of the QFT. We will comment on the general case later

in this work.

For a numerical investigation of the two-dimensional

O(3) nonlinear sigma model using quantum algorithms,

a suitable Hamiltonian formulation of the former is

needed [7,44,45]. Somewhat fortunately, this allows us

to treat the two-dimensional theory from an eﬀectively

one-dimensional perspective as follows. First, one con-

siders the theory on a discrete spatial axis while keeping

the time coordinate continuous at the same time. In this

scenario, the time derivative of the kinetic term can be

identiﬁed with an angular momentum per lattice site.

Therefore, a one-dimensional chain of coupled quantum

rotors can describe the two-dimensional ﬁeld theory. In

particular, for the nonlinear sigma model at θ=π, the

Hamiltonian can be written as [12]

H=1

2β

k=1

k+β

N−1

k=1

nknk+1 .(4)

Here, we set the lattice spacing in the spatial direction

to a= 1 and make the replacement β= 1/g2. In

this setup, Lkdenotes a (modiﬁed) quantum mechani-

cal angular momentum operator acting on the k-th site

of the spin chain.3While this operator acts on the local

Hilbert space at each site, the second term of the Hamil-

tonian corresponds to the interactions of neighbouring

sites, which we will characterise momentarily. Note that,

at this stage, we impose open boundary conditions to

keep the notation simple. This is why the summation

3We again remark that this spin chain fully characterises the two-

dimensional nonlinear sigma model. This is due to the choice of

Hamiltonian variables (see, e.g., [45]).

of the second term is terminated at the N-th site. In

practice, we will later use periodic boundary conditions

in the simulation.

Before we continue, let us make a few essential re-

marks on the Hamiltonian formulation (4), closely follow-

ing [12]. Naively, the angular momentum operator repre-

sents a particle moving on a unit sphere with coordinate

n. However, it turns out that, in our scenario, the topo-

logical θ-term for θ=πcorresponds to a vector poten-

tial A(n). Physically, the latter is sourced by a magnetic

monopole located at the centre of the unit sphere. That

is, the particle on the sphere is moving in the monopole

potential. The angular momentum operator in the po-

sition space representation is modiﬁed accordingly, such

that it takes the form L=n×(−i∇ − A)−n. There-

fore, ﬁnally, we can systematically construct a suitable

basis for the local Hilbert space at each site of the spin

chain using monopole harmonics. At the same time, this

clearly prevents a straightforward generalisation of this

method to arbitrary θ(as the Hamiltonian formulation is

closely related to the quantised monopole background).

A. Constructing the local Hilbert space

Within our approach, we are interested in the low-lying

excitations in the spectrum of the Hamiltonian (4). That

means, quantum mechanically, we ﬁrst need to determine

the local Hilbert space associated with each site of the

spin chain. As the angular momentum operator is the

generator of rotations, it is intuitive to use an eigenbasis

of the former. It therefore acts on the local Hilbert space

at the k-th site as

k|qlmik=l(l+ 1) |qlmik,

k|qlmik=m|qlmik,(5)

where Lz

kdenotes the z-component of Lk. Following

our earlier discussion, we have introduced the monopole

charge q, which, in our example, takes the value q= 1/2.4

In this case, lis a positive half-integer, l= 1/2,3/2, . . .,

and the projection quantum number takes values m=

−l, . . . , l. Similarly, a discussion on how the operators

nkact on the local Hilbert space at each site can be found

in Appendix A. For more details on this basis, we refer

the reader to [46,47].

Finally, the multiparticle state of the spin chain, in this

basis, is then schematically characterised by the tensor

product

|ψi=

k=1

αk|qlmik,(6)

with coeﬃcients αk. In this basis, we characterise all

necessary matrix elements of the operators belonging to

the Hamiltonian Hin Appendix A.

4Note that the scenario q= 0 reduces to the well-known quantum

mechanical angular momentum basis.

B. Truncating the local Hilbert space

It is evident from Eq. (5) that the spectrum of the an-

gular momentum operator L2is not bounded from above.

Therefore, the local Hilbert space associated with each

site is inﬁnite-dimensional, as we expect from a generic

QFT perspective. Our approach only applies to ﬁnite

vector spaces, so we must use a suitable truncation for

each local Hilbert space. In practice, this means that we

only consider quantum states up to a speciﬁc (maximal)

orbital angular momentum quantum number lmax. For

simplicity, we choose the same truncation for the local

Hilbert space at each site. Each Hilbert space is then of

dimension

dim H=

lmax

l=1/2

(2l+ 1) = lmax(lmax + 2) + 3

4.(7)

In principle, the truncation lmax is a free parameter of our

approach, which has to be treated with care as it may ne-

glect signiﬁcant parts of the Hilbert space if chosen too

small. For instance, crucially, the truncation violates the

nonlinear constraint on the vector ﬁeld, n2= 1. Never-

theless, we expect these complications to be negligible for

suﬃciently large lmax, which has to be carefully checked

throughout the simulation. For a more detailed discus-

sion of this, see [32] (and for the scenario θ= 0 also

[31]).

In principle, for the case of nonlinear sigma models,

one could also impose a diﬀerent truncation scheme of

the local Hilbert spaces. One particular example is based

on the fuzzy sphere, inspired by noncommutative geome-

try [38,48]. Here, the space of continuous function of nis

replaced by the ﬁnite-dimensional algebra of generators

of SU (2), ni→Ji. As such, it is not straightforward

to generalise this representation to include a topologi-

cal θ-term. Luckily, it will turn out that the smallest

truncation that we can simulate on a quantum device

is easily constructed in our truncation scheme. A re-

cent systematic, detailed discussion of the truncation of

bosonic QFTs can be found in [39].

III. QUANTUM SIMULATION

In the following, we brieﬂy highlight our computa-

tional methods to investigate the two-dimensional non-

linear sigma model featuring a topological θ-term. We

focus on the implementation of a suitable quantum cir-

cuit. Furthermore, in practice, throughout the rest of

this work, we impose periodic boundary conditions.

A. Implementation of the quantum circuit

A Variational Quantum Eigensolver (VQE) [49–51] al-

lows for a ﬂexible ansatz to determine the ground or ex-

cited states of a given Hamiltonian. It uses a quantum de-

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IPPP/22/68MITP-22-079TowardsaQuantumSimulationofNonlinearSigmaModelswithaTopologicalTermJackY.Araz,1,SebastianSchenk,1,2,yandMichaelSpannowsky1,z1InstituteforParticlePhysicsPhenomenology,DurhamUniversity,SouthRoad,DurhamDH13LE,UnitedKingdom2PRISMA+ClusterofExcellence&MainzInstituteforTheoreticalPhy...

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