Perturbative boundaries of quantum advantage real-time evolution for digitized 4lattice models Robert Maxton and Yannick Meurice

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Perturbative boundaries of quantum advantage: real-time

evolution for digitized λφ4lattice models

Robert Maxton and Yannick Meurice

1Department of Physics and Astronomy,

The University of Iowa, Iowa City, IA 52242, USA

(Dated: March 13, 2023)

Abstract

The real time evolution of quantum ﬁeld theory models can be calculated order by order in

perturbation theory. For λφ4models, the perturbative series have a zero radius of convergence

which in part motivated the design of digitized versions suitable for quantum computing. In

agreement with general arguments suggesting that a large ﬁeld cutoﬀ modiﬁes Dyson’s reasoning

and improves convergence properties, we show that the harmonic digitizations of λφ4lattice ﬁeld

theories lead to weak coupling expansions with a ﬁnite radius of convergence. Similar convergence

properties are found for strong coupling expansions. We compare the resources needed to calculate

the real-time evolution of the digitized models with perturbative expansions to those needed to

do so with universal quantum computers. Unless new approximate methods can be designed to

calculate long perturbative series for large systems eﬃciently, it appears that the use of universal

quantum computers with digitizations involving a few qubits per site has the potential for more

eﬃcient calculations of the real-time evolution for large systems at intermediate coupling.

arXiv:2210.05493v2 [quant-ph] 9 Mar 2023

I. INTRODUCTION

Performing ab-initio real-time calculations for quantum ﬁeld theory and in particular

quantum chromodynamics (QCD) would have a signiﬁcant impact for theoretical predictions

related to hadron collider experiments [1]. As methods based on probabilistic importance

sampling rely on Euclidean time formulations, real-time evolution for lattice QCD cannot

be performed with classical computers for system sizes comparable to those used for static

problems. For this reason, the use of universal quantum computers [1–42] or analog quantum

simulations using cold atoms [43–58] has become a very active area of research. In this

context, roadmaps [1, 59–62] to implement sequences of models of increasing complexity

and dimension have been considered, leading to physically relevant calculations on rapidly

evolving Noisy Intermediate Scale Quantum (NISQ) platforms [63].

In one of the ﬁrst articles discussing real-time evolution and study of scattering processes

using universal quantum computers [4] for λφ4theories, it is stated correctly that traditional

calculations of quantum ﬁeld theory scattering amplitudes rely on perturbation theory and

that even at weak coupling, the perturbative series are not convergent. In other words,

including higher-order contributions beyond a certain point makes the approximation worse.

However, in order to set up a quantum computation, ﬁnite discretizations of the non-

compact scalar ﬁeld φare used [4, 6, 11, 12, 23, 26, 64] (we call this process “digitization”),

which implicitly introduces a ﬁeld cutoﬀ. It has been argued [65, 66] that cutting oﬀ the

large ﬁeld contributions in λφ4theories aﬀects the instability at negative λinvoked by Dyson

[67] (see Ref. [68] for more literature on the subject), and results in modiﬁed perturbative

series that are expected to converge to values exponentially close to exact. Consequently, the

question of using perturbative methods in an eﬃcient way needs to be revisited for digitized

models.

Several questions need to be answered for digitized models: 1) do the perturbative series

converge? 2) are analytic continuations possible? 3) assuming positive answers for 1) and 2),

what are the computational resources needed to perform reasonably accurate calculations?

In this article we address these questions for lattice versions of λφ4ﬁeld theory. We start

with the standard ﬁeld-continuous formulation of lattice λφ4in the local ﬁeld basis, where

φxis diagonal and the interactions among the ﬁelds at diﬀerent lattice sites xare limited

to neighboring sites. For ﬁnite local digitizations of these models, general results guarantee

[2] that quantum computers could deal eﬃciently with real-time evolution. Several methods

have been used to digitize the local ﬁeld variables [4, 6, 11, 12, 23, 26, 64], of which two

are discussed here. In the ﬁrst, the eigenvalues are equally spaced between ±φmax for some

ﬁeld cutoﬀ φmax and the conjugate momentum is non-trivial. In the other [12, 64], called

the “harmonic” basis, the ﬁeld and its conjugate momentum have eigenvalues which are

the zeros of the Hermite polynomial Hnmax (x) for some nmax which denotes the size of the

ﬁnite local Hilbert space at each lattice site. In the harmonic basis, the standard algebraic

manipulations involving creation and annihilation operators hold, with the exception that

a†|nmax −1i= 0. Because of the simplicity of the conjugate momentum operator and the

preservation of most of the algebraic relations involving creation and annihilation opera-

tors, the harmonic basis provides easy reformulations of the standard perturbative tools for

digitized models and will be used almost exclusively hereafter. The construction of the har-

monic basis is closely related to the Gaussian quadrature method of integration. With nmax

sampling points, the integration of polynomials of 2nmax −1 order remains exact and orthog-

onality relations are preserved. It seems clear that by taking nmax large enough, we recover

the integration with continuous and non-compact ﬁelds as in the standard formulation .

With a universal quantum computer, we use nqqubits per lattice site and in the following

we take nmax = 2nq. In view of the limitations of current NISQ devices, we are inclined to

consider economical situations with small nq. The matching with the target model may not

be perfect, however if the symmetries are preserved, we expect that in the limit where the

lattice is small compared to the correlation length, universal properties of the model will be

preserved. For recent discussions related to this question see e.g., Refs. [14, 27, 29, 30, 34, 38–

40, 42].

In the following, we discuss the cases nq= 2, 3, 4 and 5 and show that the digitization

leads to converging perturbative series, unlike the original model with continuous and non-

compact ﬁelds. For small systems and small couplings, this allows the practical use of

perturbation theory to calculate the real-time evolution accurately. The main open question

is if this method can be used eﬃciently when the system size increases and the coupling

constant takes arbitrary values.

The article is organized as follows. In Sec. II, we present the lattice models considered

and their digitization. In Sec. III, we describe the calculation of the matrix elements of

the evolution operator in a computational basis using perturbation theory. In Sec. IV, we

discuss the one-site problem which is a single digitized quantum anharmonic oscillator. In

IV A, we present numerical methods to calculate perturbative series and determine their

radius of convergence using the simple case of nmax = 4 with one site which can also be

solved exactly. The complex singularities in the complex λplane for nmax = 8, 16 and 32

are presented in Sec. IV B. The complex singularities appear to stay away from the positive

real axis. We also show that similar methods can be applied in the strong coupling limit. In

Sec. V, we consider a 1+1 dimensional model with four sites and nmax = 4. By increasing

the hopping parameter, the complex singularities start pinching the positive real axis in the

complex λplane in agreement with the existence of a second-order phase transition. Higher

dimensional studies are in principle possible but would require optimizations not considered

here. In Sec. VI, we discuss the calculation of the same matrix elements with a universal

quantum computer. We conclude with a discussion of quantum advantage. In this context,

the recent interest [69–74] in applying tensor network methods to λφ4could also provide

relevant elements for the discussion.

II. MODELS

In this section we introduce the lattice models of λφ4ﬁeld theory considered in the

article. These consist of anharmonic oscillators located at lattice sites with quadratic nearest

neighbor coupling. The target models, before digitization, and the associated terminology

are introduced in Sec. II A. The truncated (digitized) harmonic basis is presented in Sec.

II B. This includes a discussion of the basis where the ﬁeld operators are diagonal. It should

be emphasized that in this “ﬁeld basis of the truncated harmonic formulation” the ﬁeld

eigenvalues are not equally spaced. They are the zeros of some Hermite polynomial. This

ﬁeld basis should not be confused with the other ﬁeld basis mentioned in the introduction,

where the ﬁeld eigenvalues are equally spaced.

A. Target models: lattice φ4Hamiltonians in D−1spatial dimensions

In the following we use a spatial lattice. We use the notation xfor the lattice sites and

efor the D−1 orthogonal unit vectors in positive directions. For instance, for D= 3

space-time dimensions, this represents a square spatial lattice. The Hamiltonian ˆ

Hreads

H=X

Hanh.

x−2κX

x,e

φxˆ

φx+e,(1)

with the local anharmonic part

Hanh.

x=ω(ˆa†

xˆax+1/2) + λˆ

φ4

x.(2)

Here, ωrepresents the energy scale of the unperturbed harmonic oscillators; κrepresents

the hopping energy, the energy scale of the nearest-neighbor coupling; and λdetermines the

strength of the anharmonic component, which in the low-λlimit can be interpreted as the

strength of the coupling between free particle states. With the usual ﬁeld deﬁnition

φx≡1

√2ωˆax+ ˆa†

x,(3)

and the conjugate momentum

ˆπx≡ −irω

2ˆax−ˆa†

x,(4)

we have the standard commutation relations

ˆax,ˆa†

y=δxy,(5)

with the other commutators being zero. The above equations provide the standard Hamil-

tonian formulation of λφ4. We will now introduce a truncation of the local Hilbert spaces.

B. The digitized harmonic basis

We now consider the harmonic digitization of ˆaxand ˆa†

xoperators at a given site x. As

the results are independent of xwe drop the site index in this subsection. In order to get a

ﬁnite-dimensional Hilbert space, we start with the standard

ˆa†|ni=√n+ 1 |n+ 1i,for n= 0, . . . , nmax −2,(6)

but impose

ˆa†|nmax −1i= 0.(7)

In addition, we have the standard relations

ˆa|ni=√n|n−1i,for n= 1, . . . , nmax −1,(8)

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Perturbativeboundariesofquantumadvantage:real-timeevolutionfordigitized4latticemodelsRobertMaxtonandYannickMeurice1DepartmentofPhysicsandAstronomy,TheUniversityofIowa,IowaCity,IA52242,USA(Dated:March13,2023)AbstractTherealtimeevolutionofquantumeldtheorymodelscanbecalculatedorderbyorderinperturbat...

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Perturbative boundaries of quantum advantage real-time evolution for digitized 4lattice models Robert Maxton and Yannick Meurice

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