Suppression of 1fnoise in quantum simulators of gauge theories Bhavik Kumar 1Philipp Hauke 2 3and Jad C. Halimeh4 5 1Department of Physical Sciences Indian Institute of Science Education and Research IISER

2025-04-26 0 0 1.75MB 12 页 10玖币

侵权投诉

Suppression of 1/f noise in quantum simulators of gauge theories

Bhavik Kumar ,1Philipp Hauke ,2, 3 and Jad C. Halimeh 4, 5, ∗

1Department of Physical Sciences, Indian Institute of Science Education and Research (IISER),

Mohali, Knowledge City, Sector 81, Punjab 140306, India

2INO-CNR BEC Center and Department of Physics,

University of Trento, Via Sommarive 14, I-38123 Trento, Italy

3INFN-TIFPA, Trento Institute for Fundamental Physics and Applications, Trento, Italy

4Department of Physics and Arnold Sommerfeld Center for Theoretical Physics (ASC),

Ludwig-Maximilians-Universit¨at M¨unchen, Theresienstraße 37, D-80333 M¨unchen, Germany

5Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, D-80799 M¨unchen, Germany

(Dated: October 14, 2022)

In the current drive to quantum-simulate evermore complex gauge-theory phenomena, it is nec-

essary to devise schemes allowing for the control and suppression of unavoidable gauge-breaking

errors on diﬀerent experimental platforms. Although there have been several successful approaches

to tackle coherent errors, comparatively little has been done in the way of decoherence. By nu-

merically solving the corresponding Bloch–Redﬁeld equations, we show that the recently developed

method of linear gauge protection suppresses the growth of gauge violations due to 1/fβnoise as

1/V β, where Vis the protection strength and β > 0, in Abelian lattice gauge theories, as we show

through exemplary results for U(1) quantum link models and Z2lattice gauge theories. We sup-

port our numerical ﬁndings with analytic derivations through time-dependent perturbation theory.

Our ﬁndings are of immediate applicability in modern analog quantum simulators and digital NISQ

devices.

CONTENTS

I. Introduction 1

II. Linear gauge protection 2

III. 1/f noise and the Bloch–Redﬁeld master equation 3

IV. Results and discussion 3

A. U(1) quantum link model 4

B. Z2lattice gauge theory 6

V. Conclusion and outlook 7

Acknowledgments 7

A. Further details on the derivation of the

Bloch–Redﬁeld master equation 7

B. Perturbation theory 8

C. Supplemental numerical results 9

References 9

I. INTRODUCTION

Quantum simulators are quantum systems imple-

mentable in the laboratory onto which quantum many-

body models of interest can be mapped and studied [1–

4]. Due to its promise as a probe of phenomena relevant

∗jad.halimeh@physik.lmu.de

for high-energy and nuclear physics on easily accessible

table-top quantum devices, and its potential to calculate

time evolution from ﬁrst principles, the quantum simula-

tion of lattice gauge theories [5] has come at the forefront

of research in several ﬁelds ranging from condensed mat-

ter to subatomic physics [6–13]. Thanks to the advent

of high-control and precision synthetic quantum devices,

recent years have seen various groundbreaking quantum-

simulation experiments of gauge theories [14–27].

Of particular interest in this endeavor are gauge the-

ories with both dynamical matter and gauge ﬁelds. The

characteristic property of gauge theories is their gauge

symmetry [28–30], which imposes local constraints that

enforce speciﬁc conﬁgurations of matter and electric

ﬁelds, such as Gauss’s law from quantum electrodynam-

ics. A major issue in quantum simulations is stabilizing

gauge symmetry against gauge-breaking terms that will

unavoidably arise either due to higher orders in the per-

turbative mapping or due to experimental imperfections

[31]. These terms allow for processes driving the sys-

tem dynamics out of the physical gauge sector of Gauss’s

law, in which it should stay in an ideal scenario where

such terms are not present. Even when perturbative in

strength, gauge-breaking terms can be quite detrimental

to gauge-theory quantum simulations, leading to gauge-

noninvariant dynamics that cannot be directly related to

the target model [32–34].

Various methods have been proposed to suppress co-

herent gauge-breaking errors [31,35–54], but there has

been little work done on suppressing incoherent errors

due to decoherence, which can be quite adverse to the

stability of gauge-theory implementations [34,55]. In-

deed, decoherence [56,57] poses a major roadblock to

achieving long evolution times in quantum simulations

arXiv:2210.06489v1 [quant-ph] 12 Oct 2022

of quantum many-body models in general, whose key

properties of quantum entanglement and superposition

are particularly sensitive to interactions with the envi-

ronment. Prominent examples of the detrimental eﬀects

of decoherence on quantum many-body systems include

1/f noise in superconducting quantum interference de-

vices (SQUIDs) that undermines superconducting qubits

[58–63]. Given that superconducting qubits, as well as

other platforms, have been of great recent interest in the

quantum simulation of gauge theories [26,27], suppress-

ing 1/f noise sources using eﬃcient and experimentally

feasible schemes becomes of central importance.

In this work, using exact diagonalization calculations

and time-dependent perturbation theory, we demonstrate

how the principle of linear gauge protection, initially de-

vised to control coherent gauge-breaking errors [51], can

be employed to suppress the growth of the gauge vio-

lations due to incoherent errors with spectral form 1/fβ

(β > 0) as 1/V β, where Vis the protection strength. The

rest of this paper is organized as follows: We brieﬂy re-

view the concept of linear gauge protection in Sec. II, and

1/fβnoise and the corresponding Bloch–Redﬁeld formal-

ism in Sec. III. We present our main numerical results in

Sec. IV. We ﬁnally conclude and provide an outlook in

Sec. V. We include Appendix Afor a derivation of the

Bloch–Redﬁeld equation employed for our analysis, Ap-

pendix Bfor our derivations in time-dependent perturba-

tion theory, in addition to Appendix Cwhere we provide

supplemental numerical results.

II. LINEAR GAUGE PROTECTION

Let us consider an Abelian gauge theory described by

the Hamiltonian ˆ

H0, and whose gauge symmetry is gen-

erated by the operator ˆ

Gj, where jdenotes a site on a

lattice of length L. The gauge invariance of ˆ

H0is encoded

in the commutation relations ˆ

H0,ˆ

Gj= 0,∀j. The set

of gauge-invariant states {|ψi} is deﬁned as the simulta-

neous eigenstates of the generators: ˆ

Gj|ψi=gj|ψi,∀j.

A set of these eigenvalues g= (g1, g2, . . . , gL) over the

volume of the system deﬁnes a unique gauge superselec-

tion sector, the projector onto which is ˆ

Pg. One can

further deﬁne a target or physical gauge superselection

sector gtar = (gtar

1, gtar

2, . . . , gtar

L) in which one wishes to

restrict the dynamics in an experiment, for example.

In experimental implementations of gauge theories, ˆ

is mapped onto the microscopic degrees of freedom of

a quantum simulator. In general, unavoidable gauge

symmetry-breaking errors λˆ

H1at strength λwill arise

in this process either due to higher orders in the per-

turbation theory used to perform the mapping, or in ex-

perimental imperfections in equipment. Even when per-

turbative, these errors can generate gauge violations that

grow as λ2t2over evolution time t, which in turn lead to a

complete departure from faithful gauge-theory dynamics

beyond timescales t∝1/λ [31].

In order to suppress these errors in a controlled way,

the concept of linear gauge protection was introduced in

Ref. [51]. It entails adding the protection term

Vˆ

HG=VX

cjˆ

Gj,(1)

where Vis the protection strength. The sequence cj

can be chosen to be rational and satisfying the condition

Pjcjg−gtar

j= 0 ⇐⇒ gj=gtar

j,∀j. In this case,

the sequence is said to be compliant, and, for a volume-

independent and suﬃciently large V, the gauge violation

is controlled up to times exponential in V[51,64]. Al-

though Vis volume-independent, the sequence cjwould

have to grow (not faster than) exponentially with system

size in order to satisfy the compliance condition. This

renders the compliant sequence somewhat inconvenient

for large-scale gauge-theory quantum simulators such as

those realized in recent cold-atom setups [23,24].

However, reality turns out to be more forgiving, and

even simple noncompliant sequences such as cj= (−1)j

can give excellent protection in the target sector against

gauge errors up to all accessible evolution times in both ﬁ-

nite systems [51] and the thermodynamic limit [65]. This

can be explained through the coherent quantum Zeno ef-

fect [66–69], which guarantees that upon adding the pro-

tection term (1) an eﬀective Zeno Hamiltonian ˆ

HZ=

H0+λˆ

Pgtar ˆ

H1ˆ

Pgtar emerges that faithfully reproduces

the dynamics of the faulty gauge theory ˆ

H0+λˆ

H1+Vˆ

up to timescales linear in Vin a worst-case scenario [51].

For certain gauge theories, the full local generator ˆ

may be too challenging to realize in an experiment [19],

in which case the linear gauge protection as given in

Eq. (1) becomes impractical. Nevertheless, a powerful

workaround exists based on local pseudogenerators ˆ

Wj,

which are identical to the full local generators ˆ

Gjin the

target sector, but not necessarily outside of it [52]. For-

mally, they satisfy the relation

Wj|φi=gtar

j|φi ⇐⇒ ˆ

Gj|φi=gtar

j|φi.(2)

One can then extend the principle of linear gauge protec-

tion to one in terms of the local pseudogenerator, with

protection term

Vˆ

HW=VX

cjˆ

Wj,(3)

where the same rules apply for the sequence cjas in

the case of Eq. (1). Note that even though ˆ

H0com-

mutes with ˆ

Gj, it generally does not commute with

Wj, with the latter associated with a local symmetry

richer than that generated by ˆ

Gj[70]. The result-

ing Zeno Hamiltonian when protecting with Eq. (3) is

HZ=ˆ

Pgtar ˆ

H0+λˆ

H1ˆ

Pgtar , under which the dynam-

ics of the faulty gauge theory ˆ

H0+λˆ

H1+Vˆ

HWcan be

faithfully reproduced up to times at least linear in V[52].

In terms of purely unitary errors, extensive numeri-

cal simulations in exact diagonalization (ED) and inﬁ-

nite matrix product states (iMPS) based on the time-

dependent variational principle [71–73] have shown that

for a compliant or properly chosen noncompliant se-

quence, linear gauge protection in the full local generator

or the local pseudogenerator leads to stabilized gauge-

theory dynamics up to all accessible evolution times with

the gauge violation settling at a timescale ∝1/V into a

plateau of value ∝λ2/V 2[51,52,65,74]. Importantly,

the linear gauge protection terms (1) and (3) are com-

posed of single and two-body terms at most, and they are

local, which renders them experimentally highly feasible.

It is a relevant open question whether linear gauge pro-

tection can be employed to protect against incoherent er-

rors due to noise in an experiment. When left unchecked,

these errors lead to gauge violations growing ∝γt, where

γis the strength of the incoherent errors. Even just slow-

ing down the growth of gauge violations due to them

would be greatly desirable in near-term quantum simu-

lators.

III. 1/f NOISE AND THE BLOCH–REDFIELD

MASTER EQUATION

We focus here on 1/f noise, a decohering process with

a noise power spectrum

S(ω) = γ

|ω|β,(4)

where γis the system-environment coupling strength, ω

is the frequency, and 0 < β < 2. This type of noise

is ubiquitous in nature, especially in condensed matter

systems in quasi-equilibrium (for β≈1) and electronic

equipment, but this signal can also be found in biolog-

ical systems, music, and even in economics [75,76]. In

particular, as mentioned above, it is present in SQUIDs,

which can lead to adverse eﬀects on quantum simulation

platforms based on superconducting qubits [58–63].

Since we a priori know the noise power spectrum of

the environment, we employ the Bloch–Redﬁeld formal-

ism [77,78] to derive a master equation from a micro-

scopic perspective. We consider a system ˆ

HScoupled to

a bath (the environment) ˆ

HBwith the interaction Hamil-

tonian ˆ

HSB =√γPαˆ

Aα⊗ˆ

Bα, where ˆ

Aαand ˆ

Bαare

system and bath operators, respectively, with system-

environment coupling strength γ. In general, the sys-

tem operators ˆ

Aαdo not preserve Gauss’s law. Under

the assumption of weak system-environment coupling, we

obtain a master equation in terms of system operators

and correlation functions that characterize the statistical

properties of the bath.

To obtain the master equation in terms of a noise

power spectrum that can be numerically implemented,

we write the bath correlation function Cαν (τ) =

γTrBhˆ

Bα(t)ˆ

Bν(t−τ)ˆρBi—here, we denote tilde on

quantities written in the interaction picture—in terms of

the spectral function Sαν (ω), after neglecting a small en-

ergy shift arising due to the imaginary part in the Fourier

transform of Cαν (τ):

Sαν (ω)=2Z∞

dτeiωτ Cαν (τ).(5)

Hence, one can show that the ﬁnal form of the Bloch–

Redﬁeld master equation, describing the evolution of the

reduced density matrix for the system, after employing

the Born, Markov, and the secular approximation as de-

tailed in Appendix Acan be written explicitly as,

dtρab(t) = −iωabρab(t) + X

c,d

Rabcdρcd(t),(6)

where Rabcd is the Bloch–Redﬁeld relaxation tensor,

which can be written in matrix form with ˆ

Aαassumed

to be Hermitian for ease of numerical implementation,

Rabcd =−1

αδbd X

Aα

anAα

ncSα(ωcn)

−Aα

acAα

dbSα(ωca) + δac X

Aα

dnAα

nbSα(ωdn)

−Aα

acAα

dbSα(ωdb).(7)

The Redﬁeld tensor contains all the information about

the dissipative processes that arise due to the coupling

of the system with the bath degrees of freedom.

One requirement for the validity of the Bloch–Redﬁeld

approach is the smallness of the Bloch–Redﬁeld decay

rates that describe the eﬀective incoherent coupling be-

tween two eigenlevels iand fagainst the relevant tran-

sition frequencies ωif [79]. The Bloch–Redﬁeld decay

rates, also known as the golden rule rates, are deﬁned

as Γif ∝Pα



Di



Aα



fE



2Sα(ωif ). We checked for the

numerical models we describe throughout our paper that

the condition Γif ωif was always satisﬁed. In par-

ticular, as the system operators ˆ

Aαviolate Gauss’s law,

the relevant incoherent transitions happen on large en-

ergy scales of order V, where the noise spectrum becomes

weak, thus further solidifying our approach for employing

this formalism.

As 1/f noise and other types of decoherence can dras-

tically undermine performance in an experimental setup,

it becomes important to ﬁnd ways that may ameliorate

its eﬀect. Left unchecked, decoherence can lead to a fast

buildup in the gauge violation, which renders the quan-

tum simulation of true gauge-theory dynamics unfaithful

[34,55].

IV. RESULTS AND DISCUSSION

We now present our numerical results on the quench

dynamics of gauge theories subjected to 1/f noise, which

we have computed using the exact diagonalization toolkit

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：

Suppressionof1=fnoiseinquantumsimulatorsofgaugetheoriesBhavikKumar,1PhilippHauke,2,3andJadC.Halimeh4,5,1DepartmentofPhysicalSciences,IndianInstituteofScienceEducationandResearch(IISER),Mohali,KnowledgeCity,Sector81,Punjab140306,India2INO-CNRBECCenterandDepartmentofPhysics,UniversityofTrento,ViaSomm...

展开>> 收起<<

Suppression of 1fnoise in quantum simulators of gauge theories Bhavik Kumar 1Philipp Hauke 2 3and Jad C. Halimeh4 5 1Department of Physical Sciences Indian Institute of Science Education and Research IISER.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！

Suppression of 1fnoise in quantum simulators of gauge theories Bhavik Kumar 1Philipp Hauke 2 3and Jad C. Halimeh4 5 1Department of Physical Sciences Indian Institute of Science Education and Research IISER

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: