Assessment of various Hamiltonian partitionings for the electronic structure problem on a quantum computer using the Trotter approximation

2025-04-29 0 0 748.07KB 13 页 10玖币

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Assessment of various Hamiltonian partitionings for the

electronic structure problem on a quantum computer using

the Trotter approximation

Luis A. Martínez-Martínez1,2, Tzu-Ching Yen1,2, and Artur F. Izmaylov1,2

1Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada

2Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada

Solving the electronic structure problem

via unitary evolution of the electronic

Hamiltonian is one of the promising

applications of digital quantum computers.

One of the practical strategies to implement

the unitary evolution is via Trotterization,

where a sequence of short-time evolutions

of fast-forwardable (i.e. eﬃciently

diagonalizable) Hamiltonian fragments is used.

Given multiple choices of possible Hamiltonian

decompositions to fast-forwardable fragments,

the accuracy of the Hamiltonian evolution

depends on the choice of the fragments.

We assess eﬃciency of multiple Hamiltonian

partitioning techniques using fermionic and

qubit algebras for the Trotterization. Use of

symmetries of the electronic Hamiltonian and

its fragments signiﬁcantly reduces the Trotter

error. This reduction makes fermionic-

based partitioning Trotter errors lower

compared to those in qubit-based techniques.

However, from the simulation-cost standpoint,

fermionic methods tend to introduce quantum

circuits with a greater number of T-gates

at each Trotter step and thus are more

computationally expensive compared to their

qubit counterparts.

1 Introduction

Solving the electronic structure problem using

controlled time evolution of quantum systems is one

of the most promising applications of digital quantum

computers. Ease of mapping fermionic operators

to qubits using one of few existent mapping makes

the second quantized formalism a convenient starting

point. The molecular electronic Hamiltonian with N

single-particle spin-orbitals is

pq=1

hpq ˆa†

pˆaq+

pqrs=1

gpqrsˆa†

pˆaqˆa†

rˆas(1)

where ˆa†

p(ˆap) is the creation (annihilation) fermionic

operator for the pth spin-orbital, hpq and gpqrs are

one- and two-electron integrals [1]. To obtain the

spectrum of Husing digital quantum computers

with error-correcting capability one can use the

quantum phase estimation (QPE) algorithm. QPE

involves unitary evolution generated by the system

Hamiltonian exp[−it ˆ

H]to obtain the auto-correlation

function and then Fourier transform (we will use

atomic units throughout this work, ℏ= 1).

One of the challenges of QPE is that exp[−it ˆ

cannot be implemented straightforwardly on a

digital quantum computer. Within a fault-

tolerant paradigm, several approaches have been

put forward to address this problem: oracle query-

based algorithms [2], quantum walks [3], qubitization

[4], linear combination of unitaries [5] and product

formulas [6]. We will refer to the last approach as

Trotterization throughout this work. Trotterization

starts with partitioning the Hamiltonian to exactly

solvable or fast-forwardable parts, ˆ

Hn’s:

n=1

Hn.(2)

Using the Trotter approximation, exp[−it ˆ

H]can be

expressed as a sequence of e−iˆ

Hnttransformations for

small t

e−it ˆ

n=1

e−it ˆ

Hn+O(t2),(3)

where time propagator operators exp[−it ˆ

Hn]can be

translated to quantum gates easily. Implementation

of exp[−it ˆ

Hn]propagators is straightforward because

there are known unitary transformations ˆ

Vnthat

bring ˆ

Hnto operators diagonal in computational basis

Dn,ˆ

Hn=ˆ

V†

nˆ

Dnˆ

Vn. Due to non-commutativity of

individual ˆ

Hn’s the Trotter error is proportional to

commutator norms ||[ˆ

Hn,ˆ

Hn′]||.

Even though Trotterization exhibits a steeper

scaling of quantum circuit resources with target error

compared to the rest of simulation approaches, it

has two unique advantages compared to the so-called

post-Trotter algorithms [7]: 1) lower overhead and the

number of required ancilla qubits, and 2) the unique

error scaling with the commutativity between the

fast-forwardable Hamiltonian fragments, {ˆ

Hn}. The

second property can be exploited in the simulation

of physical systems where locality of interactions

Accepted in Quantum 2023-08-04, click title to verify. Published under CC-BY 4.0. 1

arXiv:2210.10189v2 [quant-ph] 7 Aug 2023

can be leveraged to reduce the number of non-zero

commutators between Hamiltonian fragments.

Thus, Trotterized Hamiltonian simulation has been

the subject of active research aiming at the reduction

of the complexity of its associated quantum circuits

in diﬀerent physical systems [8,9,10,11,12].

There are several approaches to partitioning

of molecular electronic Hamiltonians into exactly

solvable fragments. Many of these approaches

were developed in the measurement problem [13,

14,15,16,17,18] of the Variational Quantum

Eigensolver (VQE) [19], where the partitioning is

needed to obtain the expectation value of the

Hamiltonian. A few of these partitioning techniques

were considered for the Troterization problem [20,17],

yet diﬀerent partitioning methods have not been

compared systematically.

In this work, we consider Hamiltonian partitioning

methods within two Hamiltonian realizations, which

use fermionic and qubit operators, respectively.

Although both realizations employ exactly solvable

Hamiltonians, the structure of these Hamiltonians

is diﬀerent. In the fermionic case, the origin of

solvability can be traced to the well-known solvability

of hermitian one-electron Hamiltonians, while in the

qubit case, the solvability is based on existence of

Cliﬀord group transformations that diagonalize any

linear combination of commuting Pauli products.

For a Trotterization algorithm, the exponentials

in approximate time evolution operator introduced

in Eq. (3) are decomposed in terms of single-qubit

rotations and Cliﬀord gates. In an error-corrected

algorithm such as surface code, Cliﬀord gates can be

implemented fault-tolerantly. Single-qubit-rotations,

on the other hand, need to be compiled as a series

of gates consisting of Cliﬀord operations and at least

one non-Cliﬀord one, usually taken to be the T

gate, a rotation around the z-axis by π

8[21]. The

implementation of T gates requires a procedure called

magic state distillation [22], which renders T-gate

synthesis the most costly aspect of error correction.

The number of T gates required for a quantum

algorithm is one of the essential eﬃciency determining

parameters in fault-tolerant quantum computations.

Thus, to determine which Hamiltonian partitioning

approach is better, it is reasonable to compare

the number of T gates required for performing the

approximate unitary evolution.

The rest of the article is organized as follows.

In section 2we introduce the techniques used to

analyze our Trotter error results for the diﬀerent

molecular systems and methods. Moreover, the main

fermionic and qubit Hamiltonian partition schemes

employed in our analysis are introduced. In the same

section, we discuss the use of symmetries present in

all Hamiltonian fragments that allow us to ﬁnd tighter

estimations of the Trotter error. Section 3presents

and discusses Trotter errors and T-gate estimates for

diﬀerent partitioning methods. Finally, Section 4

concludes by summarizing our most relevant results

and giving the outlook.

2 Theory

2.1 Upper bounds for Trotter error

Equation (3)is a ﬁrst order Trotter-Suzuki formula.

Higher order Trotter approximations exist for

exp[−it ˆ

H], which feature an error with steeper scaling

with t, at the cost of increasing the number of

products of exponentials [23]. For simplicity in our

analysis, we focus on the error associated to the ﬁrst-

order Trotter formula. Here, the Trotter error can be

written in terms of the spectral norm of the diﬀerence

[11]

||e−it ˆ

H−Y

e−it ˆ

Hn|| ≤ ¯αt2/2,(4)

where

¯α=X

||[ˆ

Hn,X

n′<n

Hn′]||.(5)

To avoid order dependent parameter ¯αone can

use the triangle inequality to upper bound it with a

simpler quantity

||[ˆ

Hn,X

n′<n

Hn′]|| ≤ X

n,n′

||[ˆ

Hn,ˆ

Hn′]|| =α. (6)

For Trotter error analysis it is convenient to

derive an upper bound on the commutator norm

(α) that contains properties of individual fragments.

Such an upper bound can provide an insight on

what properties of fragments deﬁne the success of a

partitioning in lowering α.

Using the triangle inequality

||[ˆ

Hn,ˆ

Hk]|| ≤ 2|| ˆ

Hn|| · || ˆ

Hk|| (7)

provides a loose upper bound. This bound can be

further tightened by considering shifted Hamiltonians

Hi=ˆ

Hi−siˆ

I,(8)

where siis a scalar, and ˆ

Iis the identity operator. The

shift introduced in Eq. (8) leaves the commutators

between Hamiltonian fragments invariant

[ˆ

Hi,ˆ

Hj]=[ˆ

Hi,ˆ

Hj](9)

but changes the spectral norms, || ˆ

Hi|| ̸=|| ˆ

Hi|| by

shifting the spectrum of ˆ

Hiin Eq. (8). A tighter upper

bound for αcan be written as

α= 2 X

i>j

||[ˆ

Hi,ˆ

Hj]|| ≤ 4X

i>j

|| ˆ

Hi|||| ˆ

Hj|| (10)

≤4X

i>j

min

si,sj

|| ˆ

Hi|| · || ˆ

Hj||.(11)

Finding the minimum norm shifts can be done

explicitly as

min

|| ˆ

Hi−siI|| =∆Ei

2,(12)

where

∆Ek= (Emax,k−Emin,k)(13)

Accepted in Quantum 2023-08-04, click title to verify. Published under CC-BY 4.0. 2

is the spectral range for the kth Hamiltonian fragment

and Emax,k (Emin,k) is the maximum (minimum)

eigenvalue of Hk. Then

α≤X

i>j

∆Ei∆Ej=β, (14)

constitutes the tightest upper bound for αamong

those involving triangle inequality for shifted

Hamiltonian fragments, and it involves only spectral

properties of individual fragments.

βupper bound can be further analyzed using

statistical quantities of fragment’s spectral ranges

β=1

2

 X

∆Ei!2

−X

(∆Ei)2

(15)

2 X

∆Ei!2 1−X

ω2

i!,(16)

where positive weights ωiare deﬁned as

ωi=∆Ei

Pk∆Ek

, ωi∈[0,1].(17)

Introducing weights ωiallows us to recognize in the

second factor of Eq. (16) the linearized entropy

SL= 1 −X

ω2

i.(18)

Therefore, by denoting the ﬁrst factor in Eq. (16) as

C=Pi∆Eiwe obtain

β=1

2C2SL.(19)

Reducing both Cand SLwill decrease β. For

reducing Cone needs to reduce ∆Eiof the fragments.

SLcan be decreased by selecting fragments with

unevenly distributed weights.

Finding spectral ranges for exactly solvable

problems is not always straightforward because one

needs to ﬁnd the largest and lowest eigenvalues among

exponentially many. There exists an upper bound

for ∆E/2that is based on L1norm of a coeﬃcient

vector for a linear combination of unitaries (LCU)

decomposition for the corresponding operator [24].

Let us consider an LCU for ˆ

Hi=X

c(i)

kˆ

Vk+diˆ

I,(20)

where ˆ

Vkare some unitaries, c(i)

kand diare

coeﬃcients. Then, there is a following sequence of

inequalities

∆Ei

2≤ || ˆ

Hi−diˆ

I|| ≤ X

|c(i)

k|,(21)

where we used Eq. (12) in the ﬁrst inequality and

the triangle inequality with accounting for || ˆ

Vk|| = 1

in the second one. This upper bound suggests that

fragments with lower LCU decomposition L1norms

can be better candidates for reducing the Trotter

error.

2.2 T-gate estimates

Time-energy uncertainty principle requires

propagation for longer time to reach higher accuracy

in the energy estimation. The Trotter error bound in

Eq. (4) for a single Trotter step can be extended to

multiple steps, Ns

ϵ=||e−iT ˆ

H− Y

e−it ˆ

Hn!Ns

|| ≤ αT 2

,(22)

where T=Nstis the simulation time. Here, we

used the fact that the overall error from repeating

the ﬁrst-order Trotter sequence Nstimes accumulates

at most linearly with Ns[25]. The Trotter error α

is an important ﬁgure of merit for the estimation of

quantum resources. The number of Trotter steps to

reach the level of accuracy ϵwhile keeping Tﬁxed is

Ns≤αT 2/ϵ, so lowering αwould allow to take longer

and fewer time-steps to cover T.

The number of T-gates will be proportional to Ns

multiplied by the number of T-gates required for each

time-step. For comparison of diﬀerent partitioning

methods we will only use system independent

characteristics, such as the product αNR, where NR=

PΓ

i=1 N(i)

Ris the number of single-qubit rotations

needed in each Trotter step obtained as a sum over the

number of single-qubit rotations for each Hamiltonian

fragment N(i)

R. The number of single-qubit rotations

with arbitrary angles is proportional to the number of

T gates, where the coeﬃcient of proportionality can

depend on a particular compiling scheme [26,27].

In Appendix Awe develop an analysis of the T-gate

count for Adaptive QPE eigenvalue estimation for a

ﬁxed target error, considering relevant sources of error

in addition to the Trotter approximation. This was

done with the purpose of showing that the ﬁgure of

merit αNRsuﬃces to compare the resource-eﬃciency

of the diﬀerent Hamiltonian decomposition methods

addressed in this work.

2.3 Fermionic partitioning methods

These methods are based on solvability of one-

electron Hamiltonians using orbital rotations, for a

one-electron part of the electronic Hamiltonian we can

write

h1e =X

hpq ˆa†

pˆaq=ˆ

U† X

ϵ(1)

pˆnp!ˆ

U, (23)

where ˆnp= ˆa†

pˆapoccupation number operators,

ϵpare real constants, and ˆ

Uare orbital rotation

transformations

U=Y

p>q

eθpq (ˆa†

pˆaq−ˆa†

qˆap).(24)

Note that ˆnpform a largest commuting subset of

{ˆa†

pˆaq}operators, [ˆnp,ˆnq]=0, and they are directly

mapped to polynomial functions of Pauli-ˆzoperators

by all standard fermion-qubit mappings [28]. Orbital

Accepted in Quantum 2023-08-04, click title to verify. Published under CC-BY 4.0. 3

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AssessmentofvariousHamiltonianpartitioningsfortheelectronicstructureproblemonaquantumcomputerusingtheTrotterapproximationLuisA.Martínez-Martínez1,2,Tzu-ChingYen1,2,andArturF.Izmaylov1,21DepartmentofPhysicalandEnvironmentalSciences,UniversityofTorontoScarborough,Toronto,OntarioM1C1A4,Canada2ChemicalP...

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Assessment of various Hamiltonian partitionings for the electronic structure problem on a quantum computer using the Trotter approximation

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