Thermal State Preparation via Rounding Promises

2025-05-06 0 0 1.25MB 43 页 10玖币

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Patrick Rall1, Chunhao Wang2, and Pawel Wocjan3

1IBM Quantum, MIT-IBM Watson AI Lab, Cambridge, Massachusetts 02142, USA

2Department of Computer Science and Engineering, Pennsylvania State University

3IBM Quantum, Thomas J Watson Research Center, Yorktown Heights, New York 10598, USA

A promising avenue for the preparation of Gibbs states on a quantum computer

is to simulate the physical thermalization process. The Davies generator describes

the dynamics of an open quantum system that is in contact with a heat bath.

Crucially, it does not require simulation of the heat bath itself, only the system we

hope to thermalize. Using the state-of-the-art techniques for quantum simulation

of the Lindblad equation, we devise a technique for the preparation of Gibbs states

via thermalization as speciﬁed by the Davies generator.

In doing so, we encounter a severe technical challenge: implementation of the

Davies generator demands the ability to estimate the energy of the system unam-

biguously. That is, each energy of the system must be deterministically mapped

to a unique estimate. Previous work showed that this is only possible if the system

satisﬁes an unphysical ‘rounding promise’ assumption. We solve this problem by

engineering a random ensemble of rounding promises that simultaneously solves

three problems: First, each rounding promise admits preparation of a ‘promised’

thermal state via a Davies generator. Second, these Davies generators have a

similar mixing time as the ideal Davies generator. Third, the average of these

promised thermal states approximates the ideal thermal state.

Patrick Rall: patrickjrall@ibm.com

Chunhao Wang: cwang@psu.edu

Pawel Wocjan: pawel.wocjan@ibm.com

Accepted in Quantum 2023-10-03, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.01670v2 [quant-ph] 4 Oct 2023

Contents

1 Introduction 3

2 Preliminaries 7

2.1 Gibbs states and Davies generators ......................... 8

2.2 Rounding promises .................................. 10

3 Algorithm overview 12

4 Averaging together promised thermal states 14

4.1 The Left-Right POVM ................................ 15

4.2 Coarse-grained rounding promises .......................... 19

4.3 Ensemble analysis .................................. 21

5 Lindblad dynamics on the promised subspace 24

6 Implementing Lindblad dynamics 28

7 Some open questions 33

8 Acknowledgements 34

A Glossary 34

B Polynomial construction 35

C The Approximate Lindbladian 39

Accepted in Quantum 2023-10-03, click title to verify. Published under CC-BY 4.0. 2

1 Introduction

Motivation. Preparing Gibbs states is a major task for quantum computers. There are

several reasons for this. First, the Gibbs state is one of the most important states of mat-

ter. For quantum models comprised of many locally interacting particles, it describes a wide

range of physical situations, relevant to condensed matter physics, high energy physics, quan-

tum chemistry [Alh22]. Therefore, to use the quantum computer as a universal simulator

of quantum systems, it is desirable to be able to prepare Gibbs states. Second, for general

Hamiltonians, the Gibbs state is a crucial ingredient in some quantum algorithms such as

those for solving semideﬁnite programs [VAGGdW20,BKL+19,vAG19,GSLW19] and for

training quantum Boltzman machines [KW17]. Third, the problem of estimating the quan-

tum partition function, which is connected to the problem of approximately preparing the

Gibbs state, plays an important role in quantum complexity theory [BCGW21].

Background and overview of prior work. There are three main approaches to preparing

Gibbs states.

The ﬁrst one is a Grover-based approach in which an initial state is mapped onto a

certain puriﬁcation of the Gibbs state at inverse temperature β(see [PW09,CS17]). The

resulting running time is dictated by the overlap between the initial and target states, which

is exponentially small in the system size. The Gibbs state can be approximately prepared in

time on the order of qD/Zβ, where Ddenotes the dimension of the quantum system and

Zβthe partition function at inverse temperature β. In [HMS+22] a quantum algorithm is

presented for preparing a puriﬁcation of the Gibbs state for the Hamiltonian H1=H0+Vat

inverse temperature βstarting from a puriﬁcation of the thermal state of H0.

The second approach is based on Davies generators, which is a diﬀerential equation that

describes how nature thermalizes a quantum system to its thermal equilibrium [Dav76,Dav79].

Davies generators are special cases of Lindbladians [Lin76,BP02], which describe the most

general continuous-time Markovian dynamics of an open quantum system, i.e., a quantum

system that is weakly coupled to the environment and the dynamics of the environment are

fast enough so that the information only ﬂows from the system to the environment while no

information is ﬂowing back to the system. The method in [CB21] simulates a Davies generator

by attaching a heat bath and simulating time evolution on the joint system while repeatedly

refreshing the bath. Their algorithm in some sense follows the original derivation of the Davies

generator as the limit of such joint system-bath Hamiltonian time evolution. When the system

Hamiltonian satisﬁes the Eigenstate Thermalization Hypothesis (ETH), they show that the

implemented quantum map not only converges to the desired Gibbs state, but also does so in

polynomial time.

The third approach is based on the quantum Metropolis algorithm [TOV+11]. This

approach also avoids the exponential scaling when the system Hamiltonian satisﬁes ETH

[SC22,CB21].

The advantages of the second approach are two fold. First, for every quantum system that

thermalizes fast in nature, it is expected that our algorithms can also prepare its corresponding

Gibbs state eﬃciently without suﬀering from the exponential dependence on the number of

qubits. Second, this approach ﬁts well for many physics-motivated applications. For example,

we can use our algorithm to prepare a “partially thermalized” (in the natural thermalization

Accepted in Quantum 2023-10-03, click title to verify. Published under CC-BY 4.0. 3

process) thermal state, which might be of interest in some scenarios.

Main result. The present work examines how to approximately prepare Gibbs states for

arbitrary system Hamiltonians by simulating time evolutions according to carefully engineered

Lindbladians. These Lindbladians are derived from an ideal Davies generator having the Gibbs

state as unique ﬁxed point.

There are some similarities but also important diﬀerences to the work [CB21]. Obviously,

both are based on Davies generators. In contrast, we do not approximate Lindblad evolution

with the help of the Hamiltonian evolution of the system and a bath, a method with which

it is provably impossible to achieve linear scaling in evolution time (see [CW17]). Instead,

we rely on a method for directly simulating Lindbladian time evolution speciﬁed by any

jump operators. Using this method can lead to a reduction in resources: speciﬁcally, we

achieve linear scaling in the mixing time tmix. In addition, a direct Lindblad simulation

approach avoids the complication of dealing with the dynamics of the bath and the interaction

Hamiltonians. We also seek to prove that our quantum map approximates the Gibbs state

for any system Hamiltonian, that is, we do not need to make an assumption such as ETH.

The quantum algorithms for simulating Lindbladian time evolution in [CW17,LW22] as-

sume that the jump operators have been suitably encoded. Unfortunately, it is not possible to

construct jump operators of a Davies generator due to inherent imperfections of energy esti-

mation of general Hamiltonians. However, we show how to construct a family of Lindbladians

from the given Davies generators such that simulating them with the help of the simulation

algorithms in [CW17,LW22] and taking the average of the resulting quantum states provides

a good approximation of the Gibbs state. This is formulated in more detail in the theorem

statement below.

Theorem 1 (Main result – informal statement).Assume we are given block encodings of the

Hamiltonian H, coupling operators Sα, and a ﬁlter function G. With appropriately chosen

Sα, these give rise to a Davies generator Lthat has the Gibbs state ρβfor inverse temperature

βas a unique ﬁxed point. Assume that after time tmix the time evolved state exp(tmixL)(σ0)

is ε-close to the Gibbs state ρβfor any initial state σ0.

We engineer a certain family of 2rmany Lindbladians ˜

L(j)from the above Davies generator

L. These Lindbladians have a new ‘attenuated’ mixing time tmix, and their jump operators

can be encoded eﬃciently with imperfect energy estimation. This makes it possible to simulate

their time evolutions expt˜

L(j). We prove that the average

2rX

exp˜

tmix ˜

L(j)(σ0) (1)

is ε-close to the Gibbs state ρβfor any initial state σ0. Furthermore, we show how to imple-

ment the time evolution according to ˜

L(j)to arbitrarily small failure probability δL, and that

the total number of invocations of the block encoding of the Hamiltonian is bounded by:

O˜

tmix ·γ−1·β3ε−7·polylog(˜

tmix/δL)·log2(β/ε),(2)

where γis an attenuation coeﬃcient that aﬀects the attenuated mixing time ˜

tmix.

Accepted in Quantum 2023-10-03, click title to verify. Published under CC-BY 4.0. 4

Remark. After the ﬁrst version of this manuscript was made public, recent work [CKBG23]

resolved an open question raised in this manuscript (see Section 7) on utilizing the block

encoding of the form Pj|j⟩ ⊗ Lj. Using the Theorem III.1 of [CKBG23] together with slight

adaptation of the simulation algorithm in [LW22], the complexity of our algorithm can be

improved to ˜

O(˜

tmix ·γ−1·βε−2). We also note that the additional factor of log2(β/ε)can be

removed via existing techniques from [Ral21].

In Section 5we give some numerical experiments indicating that the attenuated mixing

time ˜

tmix is on the order of the original mixing time tmix for suitably chosen γ−1.

We formulate in Section 7some open research questions whose solution could lead to

improvements of our current methods.

Technical overview. As mentioned above, the implementation of jump operators of a

Davies generator requires perfect energy estimation. More precisely, what we mean by perfect

is that energy estimation would have to be unambiguous: for each energy of the Hamiltonian, it

must yield a unique energy estimate. Unfortunately, energy estimation unavoidably produces

superpositions of diﬀerent energy estimates for general Hamiltonians: if |ψ⟩is an eigenstate

of the Hamiltonian with eigenvalue λ, energy estimation implements a map:

|ψ⟩ 7→ |ψ⟩ ⊗ α|˜

λ1⟩+β|˜

λ2⟩(3)

where ˜

λ1and ˜

λ2are two diﬀerent estimates of λ. This superposition of estimates cause

constant-size errors in quantum algorithms implementing Davies generators, and must be

eliminated. With perfect estimation, the superposition on the second register is not present

and there is always a unique estimate.

It is possible to construct an ‘approximate Davies generator’ from an energy estimation

algorithm that produces superpositions of estimates. However, the resulting dynamics no

longer correspond to the Davies generator of any particular Hamiltonian, making it challenging

to analyze. To our knowledge, no method exists for rigorously proving the accuracy of an

algorithm based on such an approximate Davies generator. We highlight some of the challenges

of this task in Appendix C.

It was shown in [Ral21] that perfect energy estimation is possible when the Hamiltonian

satisﬁes a ‘rounding promise’ assumption. The rounding promise prohibits the Hamiltonian

from having any eigenvalues that induce a superposition of estimates. However, such an

assumption on the Hamiltonian is extremely unphysical and will not be satisﬁable in practice.

The main technical idea of the present work is to shift the notion of a rounding promise away

from the Hamiltonian itself, but rather to a family of states. The basic idea is very simple:

if a quantum state has no support on any eigenstates whose eigenvalues have a superposition

of estimates, then it is as if the Hamiltonian did not have such eigenvalues. This family of

states is deﬁned by a ‘promised subspace’.

Since perfect energy estimation is possible on a promised subspace, implementation of

Davies generators is possible as well. While the analysis involves a wide variety of error

parameters, we ﬁnd that all of them admit a mathematically rigorous treatment.

Our construction relies on just one assumption: that we can construct coupling operators

for each promised subspace that ensure that the Davies generator converges in a reason-

able amount of time. We give numerical evidence that projecting a coupling operator into a

Accepted in Quantum 2023-10-03, click title to verify. Published under CC-BY 4.0. 5

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ThermalStatePreparationviaRoundingPromisesPatrickRall1,ChunhaoWang2,andPawelWocjan31IBMQuantum,MIT-IBMWatsonAILab,Cambridge,Massachusetts02142,USA2DepartmentofComputerScienceandEngineering,PennsylvaniaStateUniversity3IBMQuantum,ThomasJWatsonResearchCenter,YorktownHeights,NewYork10598,USAApromisingav...

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