Variational quantum simulation of critical Ising model with symmetry averaging Troy J. Sewell1 2Ning Bao3and Stephen P. Jordan4 2 1Joint Center for Quantum Information and Computer Science College Park MD 20742

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Variational quantum simulation of critical Ising model with symmetry averaging

Troy J. Sewell,1, 2, ∗Ning Bao,3and Stephen P. Jordan4, 2

1Joint Center for Quantum Information and Computer Science, College Park, MD, 20742

2University of Maryland, College Park, MD, 20742

3Computational Science Initiative, Brookhaven National Lab, Upton, NY, 11973

4Microsoft, Redmond, WA 98052

Here, we investigate the use of deep multi-scale entanglement renormalization (DMERA) circuits

as a variational ansatz. We use the exactly-solvable one-dimensional critical transverse-ﬁeld Ising

model as a testbed. Numerically exact simulation of the quantum circuit ansatz can in this case be

carried out to hundreds of qubits by exploiting eﬃcient classical algorithms for simulating matchgate

circuits. We ﬁnd that, for this system, DMERA strongly outperforms a standard QAOA-style ansatz,

and that a major source of systematic error in correlation functions approximated using DMERA

is the breaking of the translational and Kramers-Wannier symmetries of the transverse-ﬁeld Ising

model. We are able to reduce this error by up to four orders of magnitude by symmetry averaging,

without incurring additional cost in qubits or circuit depth. We propose that this technique for

mitigating systematic error could be applied to NISQ simulations of physical systems with other

symmetries.

I. INTRODUCTION

The multi-scale entanglement renormalization ansatz

(MERA) tensor networks represent quantum states on

addimensional lattice using a d+ 1 dimensional rep-

resentation. The extra dimension can be interpreted as

scale, with slices of the network along this dimension cor-

responding to successively coarse grained states. In one

dimension, the tree-like MERA tensor network can repre-

sent ground states of critical systems, reproducing poly-

nomially decaying correlation functions and logarithmic

scaling of subsystem entanglement entropy [1–3]. In the

case where the local tensors are fermionic Gaussian uni-

taries, the networks can viewed as a wavelet transform on

fermion operators and can be rigorously shown to sup-

port good approximations of local free-fermion ground

states [4–6].

The local unitary structure of a MERA tensor network

may be interpreted as a quantum circuit which introduces

further UV degrees of freedom scale-by-scale to entangle

qubits of the target ground state. However, on a quan-

tum computer it is more natural to allow increased cir-

cuit depth rather than local bond dimension as a means

of increasing expressivity of an ansatz, giving rise to a

class of quantum circuits known as DMERA [7] which

could be used as a variational circuit for ground state

preparation. Of particular interest for near-term appli-

cation is that local observables and correlation functions

of ground states prepared by DMERA circuits feature an

inherent resilience to local noise stemming from circuit

topology [7]. It is also possible to prepare subregions

of DMERA states on quantum computers much smaller

than the total system size, which may help to leverage the

capabilities of small quantum devices [8]. This has been

demonstrated on an ion-trap quantum computer, where

∗tjsewell@umd.edu

the critical ground state subregion can be prepared as a

robust ﬁxed-point state of a local quantum channel de-

rived from a DMERA quantum circuit [9].

MERA tensor networks can be contracted in polyno-

mial time classically. However, the polynomial scaling

with bond dimension is quite severe, e.g. O(χ8) for 1D

and O(χ16) for 2D using the schemes proposed in [2].

Thus direct execution of MERA on quantum computers

can yield large polynomial speedups, and can addition-

ally serve as a method for preparing initial states in the

context of a quantum algorithm for simulating quantum

dynamics.

This work investigates the feasibility of multi-scale cir-

cuits for variational ground state preparation on quan-

tum computers in the fashion of a variational quantum

eigensolver (VQE) [10, 11], or more generally an ansatz

circuit for some variational quantum algorithm (VQA)

[12], where some family of parameterized circuits are min-

imized according to some Hamiltonian energy in hopes

that a good approximation of the ground state can be

found. Another MERA-inspired quantum circuit ansatz

for variational ground state preparation was studied in

Ref. [13], however, the circuit ansatz studied in the

present work more directly follows the constructions of

Refs. [4, 5, 7].

Conformal ﬁeld theories and the use of renormalization

are of central importance to the study of quantum ﬁeld

theories, which may generally be viewed as deformations

from critical ﬁxed points of a renormalization group ﬂow.

The prospect to simulate conformal ﬁeld theories with

critical lattice models and including the use of renormal-

ization theory is seen as an important step in the ability

to handle more general study of ﬁeld theory simulations

on quantum computers [14]. Ground states of critical sys-

tems feature large correlation lengths which are challeng-

ing to reproduce using a more locally entangled ansatz

such as matrix product states or short depth quantum

circuits. Scale invariance, however, makes these states

somewhat simpler to describe using a multi-scale ansatz

arXiv:2210.15053v2 [quant-ph] 29 Apr 2023

which may incorporate this symmetry directly into the

circuit parameterization. Gapped states may be better

able to be represented by an ansatz with local correla-

tions, but may also be prepared using a multi-scale cir-

cuit with parameters that diﬀer between scales, and using

fewer scales overall due to the lack of long range correla-

tions.

We benchmark the viability of the DMERA ansatz for

variational state preparation by numerically optimizing

for approximate ground states of the critical Ising spin

chain in one dimension. We are able to ﬁnd high ﬁdelity

ground state approximations using relatively low circuit

depth Dof each scaling transformation. The ansatz

states and local energy density converge to their exact

values exponentially with D, with relative error in the

energy density below 10−8for D= 6. Key features

of critical ground states in one dimension such as poly-

nomially decaying correlation functions and logarithmic

scaling subsystem entanglement entropy are found on av-

erage. See Fig. 1 for a representation of the multi-scale

variational circuit with D= 4.

Symmetry-averaging is also used to improve the sys-

tematic error of local observable expectation values. This

can be implemented in hybrid variational schemes by us-

ing classical post processing to average observables which

are related by symmetries which are explicitly broken in

the circuit ansatz. In the case of the critical Ising model

spatial translation and Kramiers-Wannier symmetry are

used, although we also note that these symmetries are

still approximately reproduced in the ansatz states them-

selves, with smaller variance over these symmetry groups

for increasing D. Similar ideas have been studied in the

context of Refs. [15, 16], where symmetry projection is

used in post-processing to improve broken symmetries of

ansatz states.

II. CIRCUIT CONSTRUCTION

We consider states which double the number of qubits

with each scaling transformation, interleaving new qubits

in the zero state between qubits from each of the previous

layers. After `scaling circuits we have a state on L= 2`

qubits. Our goal is to prepare an approximation to the

ground state of a corresponding critical Ising Hamilto-

nian. Each transformation is a quantum circuit U(θ)`of

depth D, parameterized in terms of angles θ.

|ψ`+1i=U`(θ)|ψ`i⊗|0i⊗2`.(1)

Approximate translation and scale invariance of the state

are imposed by having each scale transformation be a

periodic brickwork circuit invariant under translation by

two sites, and constraining each scaling transformation

circuit at diﬀerent layers to be made up of the same gates,

albeit acting on diﬀerent numbers of qubits. The approx-

imate translation symmetry reduces the total number of

variational parameters from scaling with the total num-

ber of gates, O(DL), to only with total circuit depth

(a)

(b)

FIG. 1:

(a): State preparation by successive application of

D= 4 scale transformation circuits U`(θ) with new

qubits initialized in the state |0i. Colors show the

identity of local gate parameters imposed to replicate

approximate translation and scaling symmetry.

(b): Relative error in average entanglement entropy of

Nqubit sybsystems for an L= 256 qubit state, which is

exponentially small in D. The subsystem entropy scales

logarithmically with Nfor critical ground states in one

dimension, with DMERA circuits featuring an excess of

entropy for D < 3 and a small entropy deﬁcit for D > 3.

O(Dlog L). The approximate scale symmetry further

reduces the number of circuit parameters down to only

O(D), the depth of a single scaling transformation cir-

cuit. (See ﬁg 1).

These symmetries cannot be imposed exactly due to

the discrete nature of the gates and scaling transforma-

tions, so the exact symmetries are broken by the ansatz

state, yet retained as approximate symmetries. Devia-

tions in local observables are exponentially small in D

(a)

(b)

FIG. 2:

(a): Relative error of variational ﬁxed-point ground

state energy for channels of depth D, using the inﬁnite

volume energy density of −4/π. Trend lines showing the

exponential scaling of energy density error for our

ansatz (solid) as well as those from the analytic wavelet

construction of [5](dashed) and the numerically

optimized non-Gaussian MERA in [3] (dotted).

(b): Normalized state inﬁdelity 1 − F1/L of states

prepared by depth Dscaling transformation for ground

states of various system sizes L.

Both quantities appear to decay exponentially with D

with a coeﬃcient of approximately −4.89.

but constitute a major source of error for experimentally

realistic values of D,e.g. D= 1,2,...,6 as studied here.

However, as shown in Fig. 4, the error in two-point corre-

lation functions related by Kramiers-Wannier symmetry

are nearly out of phase with each other. Consequently,

this source of error can be reduced by averaging expecta-

tion values over this known symmetries. This symmetry

averaging lowers error in the correlation functions by ap-

proximately two orders of magnitude, yielding relative er-

ror below 10−7for correlation functions using the D= 6

state. This improvement is consistent across both of the

example Hamiltonians considered in this work (eq. 3 and

6), as shown in Figs. 4 and 5.

In [4–6], wavelet transformations on fermionic modes

are used to construct approximations to free fermion

ground states. These constructions use knowledge of the

underlying physics to analytically construct the desired

wavelet and achieve accuracy exponential in the order of

the wavelet (corresponding to circuit depth D). In this

work we instead determine the variational parameters in

our DMERA ansatz by numerical optimization. Inspired

by the free-fermion MERA construcions of [4] we take

u(x, y) = 





cos(x) 0 0 sin(x)

0 cos(y) sin(y) 0

0−sin(y) cos(y) 0

−sin(x) 0 0 cos(x)





(2)

as the local gates in our circuit, where the two variational

parameters specify the rotation on the odd and even par-

ity subspaces of the pair of qubits. Because we use real-

valued parity-conserving gates, the anstatz states will al-

ways be parity-even states with time-reﬂection symme-

try.

Similarly to [5], we ﬁnd that variationally optimized

parameters achieve substantially more accurate energy

densities than analytically constructed parameters, even

though the latter become exact in the limit of DMERA

of inﬁnite depth or MERA of inﬁnite bond dimension.

III. MODEL AND VARIATIONAL

OPTIMIZATION

We focus on the transverse-ﬁeld Ising model at criti-

cality, namely

HI=−

j=1

XjXj+1 +Zj.(3)

This spin chain is known to be well described by a

conformal ﬁeld theory with c= 1/2 at low energies, but is

also integrable due to the Jordan Wigner duality relating

it to the following free fermion model

H=i

j=1

γjγj+1,(4)

where γjare Majorana operators. In this convention

there are 2LMajorana operators for Lspatial sites, with

operators γ2jand γ2j−1together comprise the fermion at

spatial site j. At the critical point the coupling between

Majorana operators at the same spatial site and neigh-

boring sites are equal, which is sometimes referred to has

the ”half-shift symmetry”, and allows for the simple de-

scription of the Hamiltonian in Eqn. (4). For the spin

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VariationalquantumsimulationofcriticalIsingmodelwithsymmetryaveragingTroyJ.Sewell,1,2,NingBao,3andStephenP.Jordan4,21JointCenterforQuantumInformationandComputerScience,CollegePark,MD,207422UniversityofMaryland,CollegePark,MD,207423ComputationalScienceInitiative,BrookhavenNationalLab,Upton,NY,119734...

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Variational quantum simulation of critical Ising model with symmetry averaging Troy J. Sewell1 2Ning Bao3and Stephen P. Jordan4 2 1Joint Center for Quantum Information and Computer Science College Park MD 20742

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