Super-resolution of Greens functions on noisy quantum computers Diogo Cruz and Duarte Magano Instituto de Telecomunica c oes Portugal and

2025-05-02 0 0 623.35KB 12 页 10玖币

侵权投诉

Super-resolution of Green’s functions on noisy quantum computers

Diogo Cruz and Duarte Magano

Instituto de Telecomunica¸c˜oes, Portugal and

Instituto Superior T´ecnico, Universidade de Lisboa, Portugal

(Dated: December 20, 2023)

Quantum computers, using eﬃcient Hamiltonian evolution routines, have the potential to simu-

late Green’s functions of classically-intractable quantum systems. However, the decoherence errors

of near-term quantum processors prohibit large evolution times, posing limits to the spectrum res-

olution. In this work, we show that Atomic Norm Minimization, a well-known super-resolution

technique, can signiﬁcantly reduce the minimum circuit depth for accurate spectrum reconstruc-

tion. We demonstrate this technique by recovering the spectral function of an impurity model from

measurements of its Green’s function on an IBM quantum computer. The reconstruction error

with the Atomic Norm Minimization is one order of magnitude smaller than with more standard

signal processing methods. Super-resolution methods can facilitate the simulation of large and pre-

viously unexplored quantum systems, and may constitute a useful non-variational tool to establish

a quantum advantage in a nearer future.

The simulation of quantum systems beyond the capa-

bilities of classical computers is one of the oldest and

most important promises of quantum computing. Since

the seminal ideas of Manin [1] and Feynman [2], quan-

tum algorithms have been designed for Hamiltonian evo-

lution [3], estimating eigenvalues [4], preparing ground

states [5, 6], computing transition probabilities [7], and

studying equilibrium properties [8]. More recently, there

has been a growing interest on quantum algorithms for

calculating Green’s functions [9–18]. These dynamic pair

correlation functions play a central role in quantum the-

ory, with applications in chemistry [19], condensed mat-

ter [20], and high-energy physics [21].

Unfortunately, the Green’s functions of many-body

strongly-correlated systems are notoriously hard to com-

pute classically [22]. Leveraging quantum computers’

capacity to store and time-evolve large wave functions,

Refs. [9–14] propose measuring Green’s functions in

the real time domain. Other approaches exploit the

Lehmann’s representation [14–16] and the continued frac-

tion representation [17, 18] of Green’s functions. Other

works [13–17] consider variational methods, which are

better suited to the current era of noisy intermediate-

scale quantum (NISQ) computers than Hamiltonian evo-

lution or phase estimation-based algorithms. However,

it is diﬃcult to characterize the computational scaling

of such variational algorithms, since in general they may

be oﬄoading the hardness of the problem onto the op-

timization step [23]. Moreover, methods relying on the

Lehmann’s representation may need to prepare an expo-

nentially large number of excited states [15].

One important feature of the Green’s functions is that

they hold information about the single-particle excita-

tion spectrum. In particular, the (single-particle) spec-

tral function is proportional to the imaginary part of the

Fourier transform of the retarded Green’s function [24].

In this article, we consider the problem of recovering the

spectral function from real time measurements of Green’s

functions.

The previous literature [9–14] has always approached

the problem essentially the same way. One measures a

Green’s function at diﬀerent times (possibly with multi-

ple runs of a quantum circuit), takes the discrete Fourier

transform of the measurements, and associates its peaks

with the single-particle excitation energies. From Ga-

bor’s uncertainty principle [25], the spectral resolution

δfis inversely proportional to the maximum measured

time tmax, that is, δf≳1/tmax. So, it becomes challeng-

ing for NISQ computers to produce accurate estimates,

as we are in practice restricted to very small time win-

dows due to decoherence errors.

We argue that the aforementioned strategy is not op-

timal for this problem. We know there is a ﬁnite number

of spectral lines, but the Fourier transform does not in-

corporate this assumption. The situation is similar to re-

constructing a sparse signal in the discrete Fourier basis.

In this setting, compressive sensing methods [26], seeking

the sparsest possible representation of the signal in terms

of the said basis, recover the signal with fewer measure-

ments than what we would otherwise expect from the

Nyquist-Shannon sampling theorem [27]. However, for

the same time window, the standard theory of compres-

sive sensing [28] does not guarantee a better spectral line

resolution compared with the discrete Fourier transform.

In essence, we are still discretizing the continuous pa-

rameter space as a ﬁnite grid, while in general the true

spectral lines do not fall into the grid – this is known as

the basis mismatch problem [29].

In a breakthrough in signal processing theory, Refs.

[30, 31] showed that under certain conditions it is possi-

ble to go beyond Gabor’s uncertainty limit for the spec-

tral line estimation problem, a result often referred to

as super-resolution. These methods work directly on the

continuous parameter space, circumventing the limita-

tions imposed by discretization (and so they are also

sometimes referred to as “oﬀ-the-grid” compressive sens-

ing).

Applying the Atomic Norm Minimization technique

developed in [31–34], we show that we can reach a

super-resolution performance for the reconstruction of

arXiv:2210.04919v2 [quant-ph] 19 Dec 2023

the spectral function from measurements of Green’s func-

tions on quantum computers. As a demonstration of

this technique, we recover the spectral function of a

single-impurity model from measurements of the real-

time evolution on one of IBM’s quantum processors.

We benchmark our results against the “naive” discrete

Fourier transform method, showing that only the super-

resolution method provides an accurate reconstruction.

I. A SPECTRAL LINE ESTIMATION PROBLEM

The Green’s function, denoted here as G(t), is a two-

point correlation function that plays a central role in

the theory of many-body systems. Computing it for

strongly-correlated systems with many particles is a very

demanding task for classical computers [22]. In contrast,

if there is an eﬃcient qubit encoding of the fermionic de-

grees of freedom, and if the corresponding Hamiltonian

His eﬃciently simulatable [35], we can approximate G(t)

by evaluating the expected value of suitable observables

on quantum circuits [9–14]. The key routine for these

methods is the eﬃcient implementation of the unitary

exp(−iHt). However, large simulation times will gener-

ally require greater circuit depths, which becomes an ob-

stacle in devices with limited coherence times such as the

ones in the NISQ computers era. Therefore, in practice

we are limited to computing G(t) for very small values of

We can write the Green’s function as

G(t) = −iΘ(t)

l=1

cleiωlt.(1)

for some s∈Nand a suitable choice of positive real am-

plitudes c1, . . . , csand distinct real energies ω1...,ωs.

Closely related to the Green’s function is the spectral

function A(ω), which can be written as ﬁnite sum of

weighted Dirac deltas. Physically, it conveys information

about the single-particle excitation spectrum. From the

values of the c’s and ω’s we can directly infer the weights

and the locations of the poles of the spectral function,

A(ω) =

l=1

clδ(ω−ωl).(2)

In the language of signal processing theory, we say

that we “sample the signal” G(t) at ndiscrete times,

t0, t1, . . . , tmax. Ideally, the samples would constitute

an n-dimensional vector x∗such that x∗

j=G(tj). In

practice, the sampled signal deviates from the exact

Green’s function because of approximation errors, hard-

ware noise, and the fact that expectation values of ob-

servables are estimated with a ﬁnite number of mea-

surements. Then, we say that we record a noisy sig-

nal y=x∗+ϵ, where ϵis additive noise. The spectral

line estimation problem is to recover the amplitudes and

energies {(cl, ωl)}s

l=1 using as few resources as possible.

We mean that we would like to minimize the number of

measurements nand, most importantly in the context of

NISQ quantum simulation, the maximum sampling time,

tmax.

In previous proposals on quantum simulation of the

Green’s function in real time [9–14], the spectral lines

are always recovered via the discrete Fourier transform

of the signal. The discretization of the frequency space

introduces an error in the frequency estimation, δf. With

this approach, both the number of samples, n, and the

maximum sampling time, tmax, scale as 1/δf(the later

being a manifestation of Gabor’s uncertainty principle).

Fortunately, as we will see, it is possible to circumvent

the tmax ∼1/δfscaling by working directly on the con-

tinuous parameter space, entering the so-called super-

resolution regime.

II. ATOMIC NORM MINIMIZATION

We now succinctly introduce the super-resolution

methods developed in [31–34], centered on the concept

of the atomic norm. The key idea is to seek the decom-

position of the signal yinvolving the smallest possible

number of atoms, to be precised in what follows. For

concreteness, choose the sampling times as tj=jwith

j∈ {0, . . . , n −1}. Then, deﬁne the atoms a(f)∈Cnas

a(f)|j:= ei2πf j , j ∈ {0, . . . , n −1}.(3)

The atomic set

A:= {a(f) : f∈[0,1)}(4)

constitutes the building blocks of our signals. After

proper rescaling (see Supplemental Material A), we can

write the noiseless signal, x∗, as a combination of such

atoms. In fact, there is an inﬁnite number of ways to

express x∗as a sum of elements of A. But, in the spirit

of Occam’s razor, we aim for the simplest description of

the signal in terms of this atomic set. This notion is

quantiﬁed in terms of the atomic norm ∥·∥A, deﬁned by

identifying its unit ball with the convex hull of A, de-

noted as conv(A) [32]. For any x∈Cn, deﬁne

∥x∥A:= inf

t>0t:x∈tconv(A)(5)

= inf

ck∈C

fk∈[0,1)

X

k|ck|:x=X

cka(fk).(6)

A decomposition Pkcka(fk) that achieves the inﬁmum

is called an atomic decomposition of x. See Figure 1 for

an illustration of this concept. While it is not immedi-

ately clear from the deﬁnition that solving Equation (5)

is computationally feasible, Refs. [31, 33] show that this

can be reduced to a semideﬁnite program with a reason-

ably eﬃcient solution.

FIG. 1. For any non-empty, compact, and centrally-

symmetric subset Aof some vector space V, we can induce

a norm by identifying the 1-ball with the convex hull of A

[32]. The atomic norm of any element x∈Vis the small-

est dilation factor t≥0 such that x∈t·conv(A). In this

Figure (considering a two-dimensional vector space), the red

line represents an arbitrary atomic set A, the blue region is

conv(A), and the yellow region is the t-ball of ∥·∥A.

Having measured a noisy signal y, we produce an esti-

mate ˆ

x(y) of x∗as

x(y) = arg min

x∈Cn1

2∥y−x∥2

2+τ∥x∥A,(7)

where τ > 0 is a suitably chosen regularization parameter

[33]. Eﬀectively, by solving Equation (7) we are denoising

the measured signal. This method can be seen as a gener-

alization of LASSO regression, with the l1norm replaced

by our atomic norm. Note that, while our atomic set A

is inﬁnite, the l1norm used in LASSO can be induced by

a discrete atomic set – for example, the canonical basis

vectors together with their reﬂections about the origin.

The relevance of the denoised estimate ˆ

x(y) for the

spectral line estimation problem becomes clear in light

of Lagrangian dual theory [33]. For all f∈[0,1), we

deﬁne the dual polynomial Qas

Qy(f) := 1

τ|⟨a(f),y−ˆ

x(y)⟩|,(8)

where ⟨·,·⟩ denotes the real inner product. One can show

that Qis upper bounded by 1. Moreover, we can write

x(y) as a sum of atoms a(f) only with frequencies for

which Qy(f) = 1. In other words, the line spectra is

identiﬁed with the peaks of the dual polynomial, oﬀering

a completely diﬀerent perspective on the problem com-

pared with traditional approaches.

Now the pending question is how close is ˆcl,ˆ

fll

to the true signal cl, fl}l(or, indirectly, how good

an estimate of x∗is ˆ

x(y)). An answer was provided in

Ref. [34] (and later improved by Ref. [36]), asserting

that the reconstruction performance critically depends on

the minimal separation between the frequencies, known

as the frequency gap, ∆f:= minj̸=l|fj−fl|, where | ·

|is understood as the wraparound distance around the

unit circle. Then, under some reasonable assumptions

aH•

e−iHt

•H

X X

FIG. 2. Quantum circuit used to compute the Green’s func-

tion. It is a variation of the single-qubit interferometry

scheme (commonly referred to as the Hadamard test) em-

ployed in Reference [11]. His the Hadamard gate and the g

gate denotes the ground state preparation circuit. As shown

in Supplemental Material E, we can use the fact that we are

only interested in the expected value of an observable deﬁned

on the top qubit to simplify the circuit to only two qubits.

on the error, the distribution of the coeﬃcients, and the

signal-to-noise ratio [36], there is an assignment of τin

Equation (7) for which the estimate is close to the true

signal (with high probability), as long as

tmax ≥2.5/∆f.(9)

This behavior is strikingly diﬀerent from the discrete

Fourier transform. For a given error bound δf, with the

atomic norm minimization method we only need to sam-

ple up to tmax ∼1/∆f, as opposed to the tmax ∼1/δf

required with the discrete Fourier transform. Note that

in order to resolve all the spectral lines we need δf≤∆f

and, in practice, we usually want δf≪∆f. For these

cases, the Atomic Norm Minimization will signiﬁcantly

outperform the discrete Fourier transform.

III. QUANTUM SIMULATIONS

As a demonstration of our technique, We apply the

Atomic Norm Minimization (ANM) method to the single-

impurity model (also known as the Anderson model) [37],

which has received wide attention in the context of quan-

tum simulation for its role in dynamical mean-ﬁeld theory

and its relation with the Hubbard model. Due to quan-

tum hardware limitations, we consider only one fermionic

bath site. For a suitable choice of parameters (corre-

sponding to a ﬁxed point of the dynamical mean-ﬁeld

theory loop [38]), the single-impurity Hamiltonian can

be encoded into the Hilbert space of three qubits with

the Bravyi-Kitaev transformation as (cf. Supplemental

Material B)

H=Z1Z2+ 0.75X2+ 0.37(1 + Za)X1.(10)

Letting Cbe the unitary encoded in the quantum cir-

cuit of Figure 2, the impurity’s Green’s function can be

expressed as

Gimp(t) = −iΘ(t)C†ZaC,(11)

where the expectation value is over the all-zero state.

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

Super-resolutionofGreen’sfunctionsonnoisyquantumcomputersDiogoCruzandDuarteMaganoInstitutodeTelecomunica¸c˜oes,PortugalandInstitutoSuperiorT´ecnico,UniversidadedeLisboa,Portugal(Dated:December20,2023)Quantumcomputers,usingefficientHamiltonianevolutionroutines,havethepotentialtosimu-lateGreen’sfuncti...

展开>> 收起<<

Super-resolution of Greens functions on noisy quantum computers Diogo Cruz and Duarte Magano Instituto de Telecomunica c oes Portugal and.pdf

共12页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Super-resolution of Greens functions on noisy quantum computers Diogo Cruz and Duarte Magano Instituto de Telecomunica c oes Portugal and

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: