Learning many-body Hamiltonians with Heisenberg-limited scaling Hsin-Yuan Huang1 Yu Tong12 Di Fang23 and Yuan Su4

2025-04-29 0 0 757.07KB 43 页 10玖币

侵权投诉

Learning many-body Hamiltonians with

Heisenberg-limited scaling

Hsin-Yuan Huang ∗1, Yu Tong ∗1,2, Di Fang2,3, and Yuan Su4

1Institute for Quantum Information and Matter, California Institute of Technology

2Department of Mathematics, University of California, Berkeley

3Simons Institute for the Theory of Computing, University of California, Berkeley

4Microsoft Quantum

October 7, 2022

Abstract

Learning a many-body Hamiltonian from its dynamics is a fundamental problem in physics. In

this work, we propose the ﬁrst algorithm to achieve the Heisenberg limit for learning an interacting

N-qubit local Hamiltonian. After a total evolution time of O(−1), the proposed algorithm can

eﬃciently estimate any parameter in the N-qubit Hamiltonian to -error with high probability. The

proposed algorithm is robust against state preparation and measurement error, does not require

eigenstates or thermal states, and only uses polylog(−1) experiments. In contrast, the best previous

algorithms, such as recent works using gradient-based optimization or polynomial interpolation,

require a total evolution time of O(−2) and O(−2) experiments. Our algorithm uses ideas from

quantum simulation to decouple the unknown N-qubit Hamiltonian Hinto noninteracting patches,

and learns Husing a quantum-enhanced divide-and-conquer approach. We prove a matching lower

bound to establish the asymptotic optimality of our algorithm.

∗These authors contributed equally to this work.

arXiv:2210.03030v1 [quant-ph] 6 Oct 2022

Contents

1 Introduction 3

2 Main results 5

3 Proof ideas 6

3.1 Reshaping an unknown Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Learning a single-qubit Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3 Learning a few-qubit Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.4 Learning a many-qubit Hamiltonian through divide and conquer . . . . . . . . . . . . . 9

3.5 Characterizing approximation error in reshaping Hamiltonians . . . . . . . . . . . . . . 10

3.6 Establishing a matching lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Outlook 11

A Preliminaries 17

A.1 Notations ............................................ 17

A.2 Low-intersection Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

B Reshaping Hamiltonians using randomization 18

B.1 Decoupling into noninteracting patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

B.2 Isolating the diagonal Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

B.3 Combining the two reshaping procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

C Learning the unknown Hamiltonian after reshaping 22

C.1 Executing quantum experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

C.2 Estimatingthediagonal .................................... 23

C.3 Estimating for all bases and clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

D Deviation from the limiting dynamics in the randomization approach 27

D.1 Thedecouplingerror...................................... 30

D.2 The error in the local dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

E Reshaping Hamiltonians using Trotterization 32

E.1 Decoupling the dynamics using Trotterization . . . . . . . . . . . . . . . . . . . . . . . . 33

E.2 Isolating the diagonal Hamiltonian using Trotterization . . . . . . . . . . . . . . . . . . 34

F Deviation from the limiting dynamics in Trotterization 35

G Lower bound for learning Hamiltonian from dynamics 37

G.1 Model of quantum experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

G.2 Learning task and lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

G.3 ProofofTheorem28...................................... 39

G.3.1 Reduction........................................ 39

G.3.2 TV upper bound for a single experiment . . . . . . . . . . . . . . . . . . . . . . . 40

G.3.3 TV upper bound for many experiments . . . . . . . . . . . . . . . . . . . . . . . 41

G.3.4 Lower bound from TV upper bound . . . . . . . . . . . . . . . . . . . . . . . . . 43

1 Introduction

Learning an unknown Hamiltonian Hfrom its dynamics U(t) = e−iHt is an important problem that

arises in quantum sensing/metrology [1, 2, 3, 4, 5, 6, 7, 8, 9], quantum device engineering [10, 11, 12,

13, 14, 15, 16], and quantum many-body physics [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. In quantum

sensing/metrology, the Hamiltonian Hencodes signals that we want to capture. A more eﬃcient method

to learn Himplies the ability to extract these signals faster, which could lead to substantial improvement

in many applications, such as microscopy, magnetic ﬁeld sensors, positioning systems, etc. In quantum

computing, learning the unknown Hamiltonian His crucial for calibrating and engineering the quantum

device to design quantum computers with a lower error rate. In quantum many-body physics, the

unknown Hamiltonian Hcharacterizes the physical system of interest. Obtaining knowledge of His

hence crucial to understanding microscopic physics. A central goal in these applications is to ﬁnd the

most eﬃcient approach to learning H.

In this work, we focus on the task of learning many-body Hamiltonians describing a quantum

system with a large number of constituents. For concreteness, we consider an N-qubit system. Given

any unknown N-qubit Hamiltonian H, we can represent Hin the following form,

H=X

E∈{I,X,Y,Z}⊗N

λEE, (1)

where λE∈Rare the unknown parameters. The goal of learning the unknown Hamiltonian His hence

equivalent to learning λEfor each N-qubit Pauli operator E. In previous works on learning many-body

Hamiltonians [27, 28, 29, 30, 31, 32, 33, 34, 35, 36], in order to reach an precision in estimating the

parameters λE, the number of experiments and the total time required to evolve the system have a

scaling of at least −2. However, the −2precision scaling is likely not the best-possible scaling for

learning an unknown many-body Hamiltonian Hfrom dynamics.

In quantum sensing/metrology, the scaling of −2for learning an unknown parameter to error is

known as the standard quantum limit. For simple classes of Hamiltonians, such as when Hcontains

only one unknown parameter or when Hdescribes a single-qubit system, one can surpass the standard

quantum limit using quantum-enhanced protocols [1, 3, 7, 37, 38, 39]. The true limit set by the

basic principles of quantum mechanics is known as the Heisenberg limit, which gives a scaling of −1.

Assuming quantum mechanics is true, the Heisenberg limit states that the scaling of the total evolution

time must be at least of order −1. If a protocol uses Jexperiments, where the j-th experiment uses

the unknown Hamiltonian evolution e−iHtj,1, . . . , e−iHtj,Kjfor some time tj,1, . . . , tj,Kj, then the total

evolution time is deﬁned as

j=1

k=1

tj,k.(2)

Other measures of complexity, e.g., the number of experiments, could surpass the −1precision scaling,

but that does not imply that the Heisenberg limit is beaten [37, 39].

There are two well-established quantum-enhanced approaches for achieving the Heisenberg limit in

learning simple Hamiltonians, such as a single-qubit Hamiltonian H=ωZ with unknown parameter ω.

The ﬁrst approach [3, 4, 5] considers evolving a highly-entangled state over `=O(−1) copies of the

system under `copies of the unknown Hamiltonian dynamics U(t)⊗`. The second approach [1, 40, 41]

considers long-time coherent evolution with time t=O(−1) over a single copy of the system. However,

both approaches are challenging to apply in many-body systems with a large system size Nand many

unknown parameters. The diﬃculty stems from the many-body interactions in the Hamiltonian H.

As time tbecomes larger, the entanglement growth in e−itH will cause all the unknown parameters

in Hto tangle with one another. Furthermore, the many-body entanglement can be seen as a form

XZ?

…

≅

Short-time evolution

Long-time evolution

Repeat many times

Initial state preparation

Single-qubit Pauli gate

Single-qubit Clifford gate

Z-basis measurement

Unknown

Hamiltonian

Evolution

(b)

(a)

(c)

Previous algorithms

Our algorithm for achieving the Heisenberg limit

Symbols

Figure 1: Algorithms for learning many-body Hamiltonians. (a) Our algorithm for achieving the

Heisenberg limit −1:We use ideas from quantum simulation to decouple the unknown Hamiltonian

Hinto non-interacting patches, where the unknown local Hamiltonian on each patch has known eigen-

vectors. We then perform long-time coherent evolution for each non-interacting patch to learn the un-

known parameters. One only needs O(polylog(−1)) experiments and a total evolution time of O(−1).

(b) Previous algorithms for achieving the standard quantum limit −2:Previous methods [27, 30, 34, 35]

repeatedly run a short-time evolution under the unknown Hamiltonian Hincoherently for many times.

One needs O(−2) experiments and a total evolution time of O(−2). (c) Symbols: The symbols used

in (a, b). The unknown Hamiltonian evolution is U(t) = e−itH .

of decoherence, which kills the quantum enhancement. To prevent the system from becoming too

entangled, prior work on learning many-body Hamiltonians focuses on a short time t, which loses the

quantum enhancement and obtains at best the standard quantum limit scaling as −2.

In this paper, we propose the ﬁrst learning algorithm to achieve the Heisenberg limit for learning

interacting many-body Hamiltonian. We prove that the proposed algorithm can learn a model of an

unknown N-qubit local Hamiltonian Hafter a total evolution time of

T=O−1log(δ−1)(3)

which is independent of the system size N, such that for any parameter in the unknown N-qubit

Hamiltonian H, the algorithm can estimate the parameter to at most error with probability at

least 1 −δ. The proposed algorithm only uses Opolylog −1log δ−1experiments. Furthermore,

after running the experiments, the classical computational time of the proposed learning algorithm to

estimate all parameters only needs to be of O(Npolylog(−1) log(δ−1)). In quantum sensing/metrology,

the failure probability δis usually considered to be a ﬁxed constant, e.g., 0.01. In this setting, our

algorithm achieves a scaling of O(−1) saturating the Heisenberg limit.

The proposed algorithm is robust against state preparation and measurement (SPAM) errors. To

establish the optimality of the proposed algorithm, we prove a matching lower bound of

T= Ω −1log(δ−1)(4)

for any learning algorithm robust against SPAM error. The lower bound can be seen as an algorithmic

proof of the Heisenberg limit with the failure probability δtaken into account.

Our learning algorithm has the additional advantage of using only single-qubit Cliﬀord gates, and

not requiring eigenstates or thermal states of the Hamiltonian H. The shortcomings are that the total

evolution time Twould have an explicit dependence on Nto achieve −1scaling in learning quantum

systems with all-to-all long-range interactions, and that the precision it can achieve is limited by how

fast we can apply single-qubit Pauli gates.

2 Main results

Given a system size N. We focus on learning an unknown N-qubit Hamiltonian H=PaλaEathat

can be written as a linear combination of few-body terms Ea, where each qubit is acted on by O(1) of

the few-body terms. Such a Hamiltonian can always be written as

a=1

λaEa,(5)

where λ1, . . . , λMare the unknown parameters and S={E1, . . . , EM}⊆{I, X, Y, Z}⊗Nis a subset

of N-qubit Pauli operators. Each Pauli operator Eaacts nontrivially on k=O(1) qubits and each

qubit is acted on by O(1) of the Pauli operators in S. The number of unknown parameters is equal

to M=|S|= Θ(N). We refer to this class of Hamiltonians as low-interaction Hamiltonians following

[27, 34]. This class of Hamiltonians includes geometrically-local Hamiltonians as a special case and is

also referred to as bounded-degree local Hamiltonians [42, 43] in the literature. Following [27, 34], we

assume that Sis ﬁxed and known.

We consider algorithms that can learn from experiments involving the unknown N-qubit Hamiltonian

dynamics U(t) = e−iHt. Each experiment prepares an initial state with an arbitrary number of ancillas,

evolves under an interleaving sequence of unknown Hamiltonian dynamics and controllable quantum

circuit,

VK+1 U(tK)VK. . . V2U(t1)V1(6)

where Kis some integer, t1, . . . , tKare the evolution times, and V1, . . . , VK+1 are the controllable

circuits, and ends with a POVM measurement. This is similar to deﬁnitions considered in [44, 45, 46, 47].

To model SPAM error, we assume that the actual initial state and the actual POVM implemented are

only approximately equal to the ideal initial state and ideal POVM.

We consider a simple set of experiments, where the initial state is a noisy all-zero state |0Ni, each

controllable circuit Vkis a layer of single-qubit Cliﬀord gates, and the POVM is a noisy computational

basis measurement. We refer to these experiments as single-qubit Cliﬀord experiments. We give a learn-

ing algorithm with a rigorous upper bound on the total evolution time, as stated in the theorem below

(more detailed statements can be found in Theorems 15 and 23 (in Sections C.3 and E.2 respectively)

on the number of Cliﬀord gates and experiments needed).

Theorem 1. There is a learning algorithm robust to SPAM error and restricted to single-qubit Cliﬀord

experiments that achieves the following. For any unknown N-qubit low-interaction Hamiltonian H=

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Learningmany-bodyHamiltonianswithHeisenberg-limitedscalingHsin-YuanHuang*1,YuTong*1,2,DiFang2,3,andYuanSu41InstituteforQuantumInformationandMatter,CaliforniaInstituteofTechnology2DepartmentofMathematics,UniversityofCalifornia,Berkeley3SimonsInstitutefortheTheoryofComputing,UniversityofCalifornia,Ber...

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Learning many-body Hamiltonians with Heisenberg-limited scaling Hsin-Yuan Huang1 Yu Tong12 Di Fang23 and Yuan Su4

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