Hunting for quantum-classical crossover in condensed matter problems Nobuyuki Yoshioka1 2 3 Tsuyoshi Okubo4 3Yasunari Suzuki5 3Yuki Koizumi5and Wataru Mizukami6 7 3 1Department of Applied Physics University of Tokyo

2025-05-08 0 0 8.31MB 55 页 10玖币

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Hunting for quantum-classical crossover in condensed matter problems

Nobuyuki Yoshioka,1, 2, 3, ∗Tsuyoshi Okubo,4, 3, †Yasunari Suzuki,5, 3, ‡Yuki Koizumi,5and Wataru Mizukami6, 7, 3, §

1Department of Applied Physics, University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

2Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research (CPR), Wako-shi, Saitama 351-0198, Japan

3JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan

4Institute for Physics of Intelligence, University of Tokyo,

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

5NTT Computer and Data Science Laboratories, Musashino 180-8585, Japan

6Center for Quantum Information and Quantum Biology,

Osaka University, 1-2 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

7Graduate School of Engineering Science, Osaka University,

1-3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan.

The intensive pursuit for quantum advantage in terms of computational complexity has further

led to a modernized crucial question: When and how will quantum computers outperform classical

computers? The next milestone is undoubtedly the realization of quantum acceleration in practical

problems. Here we provide a clear evidence and arguments that the primary target is likely to be

condensed matter physics. Our primary contributions are summarized as follows: 1) Proposal of

systematic error/runtime analysis on state-of-the-art classical algorithm based on tensor networks;

2) Dedicated and high-resolution analysis on quantum resource performed at the level of executable

logical instructions; 3) Clariﬁcation of quantum-classical crosspoint for ground-state simulation to be

within runtime of hours using only a few hundreds of thousand physical qubits for 2d Heisenberg and

2d Fermi-Hubbard models, assuming that logical qubits are encoded via the surface code with the

physical error rate of p= 10−3. To our knowledge, we argue that condensed matter problems oﬀer

the earliest platform for demonstration of practical quantum advantage that is order-of-magnitude

more feasible than ever known candidates, in terms of both qubit counts and total runtime.

INTRODUCTION

When and how will quantum computers outperform

classical computers? This pressing question drove the

community to perform random sampling in quantum de-

vices that are fully susceptible to noise [1–3]. We an-

ticipate that the precedent milestone after this quantum

transcendence is to realize quantum acceleration for prac-

tical problems. In this context, an outstanding question

remains: in which problem next? This encompasses re-

search across a range of ﬁelds, including natural science,

computer science, and, notably, quantum technology.

Research on quantum acceleration is predominantly fo-

cused on two areas: cryptanalysis and quantum chem-

istry. In the realm of cryptanalysis, there has been a sub-

stantial progress since Shor introduced a polynomial time

quantum algorithm for integer factorization and ﬁnding

discrete logarithms [4–7]. Gidney et al. have estimated

that a fully fault-tolerant quantum computer with 20 mil-

lion (2×107) qubits could decipher a 2048-bit RSA cipher

in eight hours, and a 3096-bit cipher in approximately

a day [7]. This represents an almost hundred-fold en-

hancement in the the spacetime volume of the algorithm

compared to similar eﬀorts, which generally require sev-

eral days [5,6]. Given that the security of nearly all

asymmetric cryptosystems is predicated on the classical

intractability of integer factoring or discrete logarithm

ﬁndings [8,9], the successful implementation of Shor’s

algorithm is imperative to safeguard the integrity of mod-

ern and forthcoming communication networks.

fig1_target_scaling_v4

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"!""!!""#

O(103)O(104)O(105)O(106)

#Qubits

O(Hours) O(Hours) O(Days) O(Days)

Runtime

Runtime Random

Circuit

Cond-mat

Physics

Quantum

Chemistry Factoring

fig1_target_scaling_v4

!"!""!#

"!""!!""#

O(103)O(104)O(105)O(106)

#Qubits

O(Hours) O(Hours) O(Days) O(Days)

Runtime

Runtime Random

Circuit

Cond-mat

Physics

Quantum

Chemistry Factoring

(107)

Problems

#Qubits

Runtime

O(Hours)

O(Days)

O(Hours)

(106)

(105)

(104)

FIG. 1. Schematic diagram for scaling of computational re-

source required to achieve quantum advantage using fault-

tolerant quantum computers.

The potential impact of accelerating quantum chem-

istry calculations, including ﬁrst-principles calculations,

is immensely signiﬁcant as well. Given its broad applica-

tions in materials science and life sciences, it is noted that

computational chemistry, though not exclusively quan-

tum chemistry, accounts for 40% of HPC resources in

the world [10]. Among numerous benchmarks, a notable

target with signiﬁcant impact is quantum advantage in

simulation of energies of a molecule called FeMoco, found

in the reaction center of a nitrogen-ﬁxing enzyme [11].

According to the resource estimation that employs the

state-of-the-art quantum algorithm, calculation of the

ground-state energy of FeMoco requires about four days

on a fault-tolerant quantum computer equipped with four

million (4 ×106) physical qubits [12]. Additionally, Go-

ings et al. conducted a comparison between quantum

computers and the contemporary leading heuristic clas-

arXiv:2210.14109v3 [quant-ph] 26 Feb 2024

sical algorithm for cytochrome P450 enzymes, suggesting

that the quantum advantage is realized only in compu-

tations extending beyond four days [13].

A practical quantum advantage in both domains has

been proposed to be achievable within a timescale of days

with millions of physical qubits. Such a spacetime volume

of algorithm may not represent the most promising initial

application of fault-tolerant quantum computers. This

paper endeavors to highlight condensed matter physics

as a novel candidate (See Fig. 1). We emphasize that

while models in condensed matter physics encapsulate

various fundamental quantum many-body phenomena,

their structure is simpler than that of quantum chemistry

Hamiltonians. Lattice quantum spin models and lattice

fermionic models serve as nurturing grounds for strong

quantum correlations, facilitating phenomena such as

quantum magnetism, quantum condensation, topologi-

cal order, quantum criticality, and beyond. Given the

diversity and richness of these models, coupled with the

diﬃculties of simulating large-scale systems using classi-

cal algorithms, even with the most advanced techniques,

it would be highly beneﬁcial to reveal the location of

the crosspoint between quantum and classical comput-

ing based on runtime analysis.

Our work contributes to the community’s knowledge

in three primary ways: 1) Introducing a systematic anal-

ysis method to estimate runtime to simulating quantum

states within target energy accuracy using the extrap-

olation techniques, 2) Conducting an end-to-end run-

time analysis of quantum resources at the level of ex-

ecutable logical instructions, 3) Clearly identifying the

quantum-classical crosspoint for ground-state simulation

to be within the range of hours using physical qubits on

the order of 105. To the best of our knowledge, this sug-

gests the most imminent practical and feasible platform

for the crossover.

We remark that there are some works that assess the

quantum resource to perform quantum simulation on

quantum spin systems [14,15], while the estimation is

done solely regarding the dynamics; they do not involve

time to extract information on any physical observables.

Also, there are existing works on phase estimation for

Fermi-Hubbard models [16,17] that do not provide es-

timation on the classical runtime. In this regard, there

has been no clear investigation on the quantum-classical

crossover prior to the current study that assesses end-to-

end runtime.

RESULTS

Our argument on the quantum-classical crossover is

based on the runtime analysis needed to compute the

ground state energy within desired total energy accu-

racy, denoted as ϵ. The primal objective in this section

is to provide a framework that elucidates the quantum-

classical crosspoint for systems whose spectral gap is con-

stant or polynomially-shrinking. In this work, we choose

Bond dimension

Elapsed time [s]

2d - Heisenberg ( )

J2= 0.5

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EE0

FIG. 2. Elapsed time scaling of DMRG algorithm in J1-J2

Heisenberg model at J2= 0.5 with lattice size 10 ×10. Al-

though the simulation itself does not reach ϵ= 0.01, learning

curves for diﬀerent bond dimensions ranging from D= 600

to D= 3000 collapse into a single curve, which implies the

adequacy to estimate runtime according to the obtained scal-

ing law. All DMRG simulations are executed using ITensor

library [18].

two models that are widely known due to their pro-

foundness despite the simplicity: the 2d J1-J2Heisenberg

model and 2d Fermi-Hubbard model on a square lattice

(see the Method section for their deﬁnitions). Meanwhile,

it is totally unclear whether a feasible crosspoint exists

at all when the gap closes exponentially.

It is important to keep in mind that condensed matter

physics often entails extracting physical properties be-

yond merely energy, such as magnetization, correlation

function, or dynamical responses. Therefore, in order

to assure that expectation value estimations can done

consistently (i.e. satisfy N-representability), we demand

that we have the option to measure the physical observ-

able after computation of the ground state energy is done.

In other words, for instance the classical algorithm, we

perform the variational optimization up to the desired

target accuracy ϵ; we exclude the case where one calcu-

lates less precise quantum states with energy errors ϵi≥ϵ

and subsequently perform extrapolation. The similar re-

quirement is imposed on the quantum algorithm as well.

Runtime of classical algorithm

Among the numerous powerful classical methods avail-

able, we have opted to utilize the DMRG algorithm,

which has been established as one of the most pow-

erful and reliable numerical tools to study strongly-

correlated quantum lattice models especially in one di-

mension (1d) [19,20]. In brief, the DMRG algorithm per-

forms variational optimization on tensor-network-based

ansatz named Matrix Product State (MPS) [21,22]. Al-

though MPS is designed to eﬃciently capture 1d area-law

entangled quantum states eﬃciently [23], the eﬃcacy of

DMRG algorithm allows one to explore quantum many-

body physics beyond 1d, including quasi-1d and 2d sys-

tems, and even all-to-all connected models, as considered

in quantum chemistry [24,25].

A remarkable characteristic of the DMRG algorithm is

its ability to perform systematic error analysis. This is

intrinsically connected to the construction of ansatz, or

the MPS, which compresses the quantum state by per-

forming site-by-site truncation of the full Hilbert space.

The compression process explicitly yields a metric called

“truncation error,” from which we can extrapolate the

truncation-free energy, E0, to estimate the ground truth.

By tracking the deviation from the zero-truncation result

E−E0, we ﬁnd that the computation time and error typ-

ically obeys a scaling law (See Fig. 2for an example of

such a scaling behavior in 2d J1-J2Heisenberg model).

The resource estimate is completed by combining the ac-

tual simulation results and the estimation from the scal-

ing law. [See Sec. S2 in Supplementary Materials (SM)

for detailed analysis.]

We remark that it is judicious to select the DMRG

algorithm for 2d models, even though the formal com-

plexity of number of parameters in MPS is expected

to increase exponentially with system size N, owing

to its intrinsic 1d-oriented structure. Indeed, one

may consider another tensor network states that are

designed for 2d systems, such as the Projected Entan-

gled Pair States (PEPS) [26,27]. When one use the

PEPS, the bond dimension is anticipated to scale as

D=O(log(N)) for gapped or gapless non-critical sys-

tems and D=O(poly(N)) for critical systems [28–30] to

represent the ground state with ﬁxed total energy accu-

racy of ϵ=O(1) (it is important to note that the former

would be D=O(1) if considering a ﬁxed energy den-

sity). Therefore, in the asymptotic limit, the scaling on

the number of parameters of the PEPS is exponentially

better than that of the MPS. Nonetheless, regarding the

actual calculation, the overhead involved in simulating

the ground state with PEPS is substantially high, to the

extent that there are practically no scenarios where the

runtime of the variational PEPS algorithm outperforms

that of DMRG algorithm for our target models.

Runtime of quantum algorithm

1. Overview of quantum resource estimation

Quantum phase estimation (QPE) is a quantum algo-

rithm designed to extract the eigenphase ϕof a given

unitary Uby utilizing ancilla qubits to indirectly read

out the complex phase of the target system. More con-

cretely, given a trial state |ψ⟩whose ﬁdelity with the k-th

QFT

†

|x1⋯xm⟩

Hamiltonian

Simulation

State

Preparation

System

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| GSi

Ancilla

FIG. 3. Schematic description of quantum circuit for the QPE

algorithm to compute the eigenenergy of the ground state of

quantum many-body system.

eigenstate |k⟩of the unitary is given as fk=∥⟨k|ψ⟩∥2, a

single run of QPE projects the state to |k⟩with probabil-

ity fk, and yields a random variable ˆ

ϕwhich corresponds

to a m-digit readout of ϕk.

It was originally proposed by Ref. [31] that eigenener-

gies of a given Hamiltonian can be computed eﬃciently

via QPE by taking advantage of quantum computers to

perform Hamiltonian simulation, e.g., U= exp(−iHτ).

To elucidate this concept, it is beneﬁcial to express the

gate complexity for the QPE algorithm as schematically

shown in Fig. 3as

C∼CSP +CHS +CQFT†,(1)

where we have deﬁned CSP as the cost for state prepa-

ration, CHS for the controlled Hamiltonian simulation,

and CQFT†for the inverse quantum Fourier transfor-

mation, respectively (See Sec. S4 in SM). The third

term CQFT†is expected to be the least problematic with

CQFT†=O(log(N)), while the second term is typically

evaluated as CHS =O(poly(N)) when the Hamiltonian

is, for instance, sparse, local, or constituted from poly-

nomially many Pauli terms. Conversely, the scaling of

the third term CSP is markedly nontrivial. In fact, the

ground state preparation of local Hamiltonian generally

necessitates exponential cost, which is also related to the

fact that the ground state energy calculation of local

Hamiltonian is categorized within the complexity class

of QMA-complete [32,33].

2. State preparation cost

Although the aforementioned argument seems rather

formidable, it is important to note that the QMA-

completeness pertains to the worst-case scenario. Mean-

while, the average-case hardness in translationally invari-

ant lattice Hamiltonians remains an open problem, and

furthermore we have no means to predict the complex-

ity under speciﬁc problem instances. In this context, it

is widely believed that a signiﬁcant number of ground

states that are of substantial interest in condensed mat-

ter problems can be readily prepared with a polynomial

cost [34]. In this work, we take a further step to ar-

gue that the state preparation cost can be considered

negligible as CSP ≪CHS for our speciﬁc target models,

(b)

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(a)

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<latexit sha1_base64="kk417oG8AKOKxSaFveZjr8e2J+g=">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</latexit>

Lx=2,✏f=0.15

<latexit sha1_base64="W1WGdKWdT1YjrODDIfY2//U1l6Y=">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</latexit>

Lx=4,✏f=0.15

<latexit sha1_base64="sJ62PjugJDGgREM6CuXaZOVf+5U=">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</latexit>

Lx=6,✏f=0.15

<latexit sha1_base64="IGNu8nvP9SW7NvRmNirRj6PCjhg=">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</latexit>

Lx=6,✏f=0.30

<latexit sha1_base64="qiLVDp4oTL/EOif/w6otq9aqMqA=">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</latexit>

Lx=2,✏f=0.30

<latexit sha1_base64="iIJlxIMcWHXNlKALtADxwy4rO+Y=">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</latexit>

Lx=4,✏f=0.30

<latexit sha1_base64="4J4UNPHJXDulKZxSMsIUpt619+0=">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</latexit>

✏f

<latexit sha1_base64="IUElKQUzYugX4Unwj/9jZwasp3I=">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</latexit>

=0,Linear

<latexit sha1_base64="Q745N81WKHfFoM8Sz2L4wXxRGIk=">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</latexit>

=1,B

<latexit sha1_base64="stcv4pjJnxNukzt2XiO0cp15t8M=">AAACgXichVHLLgRBFD3ae7wGGwmLiQkhkcmdiSBEImwsvQaJkUl1K3SmX+mumYSJjaUfsLAiEcSKX7DxAxY+QSxJbCzc6elEENxKVZ06dc+tU1W6Z5mBInqs0Wrr6hsam5pjLa1t7R3xzq7VwC36hswaruX667oIpGU6MqtMZcl1z5fC1i25phfmKvtrJekHpuusqD1PbtpixzG3TUMopvLxvlxBeJ6YzowkcrZQu4awyrMH+SqbjycpRWEkfoJ0BJKIYsGNXyCHLbgwUIQNCQeKsQWBgNsG0iB4zG2izJzPyAz3JQ4QY22RsyRnCGYLPO7waiNiHV5Xagah2uBTLO4+KxMYoAe6ohe6p2t6ovdfa5XDGhUvezzrVa308h1HPctv/6psnhV2P1V/elbYxkTo1WTvXshUbmFU9aX945flyaWB8iCd0TP7P6VHuuMbOKVX43xRLp0gxh+Q/v7cP8FqJpUeS40ujiZnZqOvaEIv+jHE7z2OGcxjAVk+9xCXuMGtVqsNa6RlqqlaTaTpxpfQpj4AnpaUJQ==</latexit>

=2,B

<latexit sha1_base64="bGV7S+zoto90wMD/m+sdsbSDKF0=">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</latexit>

=3,B

<latexit sha1_base64="cMYAQzh5KLjbNzY5xiHJ2LyW88U=">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</latexit>

=2,S

<latexit sha1_base64="cRbp2pox5Qp9P1Ah8pIsZtWSjwA=">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</latexit>

=4,S

FIG. 4. Scaling of the ASP time tASP for 2d J1-J2Heisenberg

model with J2=0.5. (a) Scaling with the target inﬁdelity ϵf

for system size of 4 ×4 lattice. The interpolation function

is taken so that the derivative up to κ-th order is zero at

t= 0, tASP. Here we consider the linear interpolation for

κ= 0, and for smoother ones we take Sκand Bκthat are

deﬁned from sinusoidal and incomplete Beta functions, re-

spectively (see Sec. S3 in SM). While smoothness for higher

κensures logarithmic scaling for smaller ϵf, for the current

target model, we ﬁnd that it suﬃces to take s(t) whose deriva-

tive vanishes up to κ= 2 at t= 0, tASP. (b) Scaling with the

spectral gap ∆. Here we perform the ASP using the MPS

state for system size of Lx×Ly, where results for Lx= 2,4,6

is shown in cyan, blue, and green data points. We ﬁnd that

the scaling exhibits tASP ∝1/∆βwith β∼1.5.

namely the gapless spin liquid state in the J1-J2Heisen-

berg model or the antiferromagnetic state in the Fermi-

Hubbard model. Our argument is based on numerical

ﬁndings combined with upper bounds on the complexity,

while we leave the theoretical derivation for scaling (e.g.

Eq. (4)) as an open problem.

For concreteness, we focus on the scheme of the Adia-

batic State Preparation (ASP) as a deterministic method

to prepare the ground state through a time evolution of

period tASP. We introduce a time-dependent interpolat-

ing function s(t) : R7→ [0,1] (s(0) = 0, s(tASP) = 1) such

that the ground state is prepared via time-dependent

Schr¨odinger equation given by

i∂

∂t |ψ(t)⟩=H(t)|ψ(t)⟩,(2)

where H(t) = H(s(t)) = sHf+ (1 −s)H0for the target

Hamiltonian Hfand the initial Hamiltonian H0. We

assume that the ground state of H0can be prepared

eﬃciently, and take it as the initial state of the ASP.

Early studies suggested a suﬃcient (but not necessary)

condition for preparing the target ground state scales as

tASP =O(1/ϵf∆3) [35] where ϵf= 1 − |⟨ψGS|ψ(tASP)⟩|

is the target inﬁdelity and ∆ is the spectral gap. This

has been reﬁned in recent research as

tASP =(O(1

ϵ2

f∆2|log(∆)|ζ) (ζ > 1) [36]

O(1

∆3log(1/ϵf)) [37].(3)

Two conditions independently achieve the optimality

with respect to ∆ and ϵf. Evidently, the ASP algorithm

can prepare the ground state eﬃciently if the spectral

gap is constant or polynomially small as ∆ = O(1/Nα).

For both of our target models, numerous works suggest

that α= 1/2 [38–40], which is one of the most typical

scalings in 2d gapless/critical systems such as the sponta-

neous symmetry broken phase with the Goldstone mode

and critical phenomena described by 2d conformal ﬁeld

theory. With the polynomial scaling of ∆ to be granted,

now we ask what the scaling of CSP is, and how does it

compare to other constituents, namely CHS and C†

QFT.

In order to estimate the actual cost, we have numer-

ically calculated tASP required to achieve the target ﬁ-

delity (See Sec. S3 in SM for details) up to 48 qubits.

With the aim of providing a quantitative way to esti-

mate the scaling of tASP in larger sizes, we reasonably

consider the combination of the upper bounds provided

in Eq. (3) as

tASP =O1

∆βlog(1/ϵf).(4)

Figures 4(a) and (b) illustrate the scaling of tASP con-

cerning ϵfand ∆, respectively. Remarkably, we ﬁnd that

Eq. (4) with β= 1.5 gives an accurate prediction for

2d J1-J2Heisenberg model. This implies that the ASP

time scaling is tASP =O(Nβ/2log(1/ϵf)), which yields

gate complexity of O(N1+β/2polylog(N/ϵf)) under opti-

mal simulation for time-dependent Hamiltonians [41,42].

Thus, CSP proves to be subdominant in comparison to

CHS if β < 2, which is suggested in our simulation. Fur-

thermore, under assumption of Eq. (4), we can estimate

tASP to at most a few tens for practical system size of

N∼100 under inﬁdelity of ϵf∼0.1. This is fairly neg-

ligible compared to the controlled Hamiltonian simula-

tion that requires dynamics duration to be order of tens

of thousands in our target models [43]. This outcome

stems from the fact that the controlled Hamiltonian sim-

ulation for the purpose of eigenenergy extraction obeys

the Heisenberg limit as CHS =O(1/ϵ), a consequence of

time-energy uncertainty relation. This is in contrast to

the state preparation, which is not related to any quan-

tum measurement and thus there does not exist such a

polynomial lower bound.

# -gates

Nph

FIG. 5. Spacetime cost of the phase estimation algorithm,

namely the T-count and physical qubit counts Nph. Here,

we estimate the ground state energy up to target accuracy

ϵ= 0.01 for 2d J1-J2Heisenberg model (J2/J1= 0.5) and

2d Fermi-Hubbard model (U/t = 4), both with lattice size

of 10 ×10. The blue, orange, green, and orange points in-

dicate the results that employ qDRIFT, 2nd-order random

Trotter, Taylorization, and qubitization, where the circle and

star markers denote the spin and fermionic models, respec-

tively. Two ﬂavors of the qubitization, the sequential and

newly proposed product-wise construction (see Sec. S5 for de-

tails), are discriminated by ﬁlled and unﬁlled markers. Note

that Nph here does not account for the magic state factories,

which are incorporated in Fig. 7.

3. Main quantum resource

As we have seen in the previous sections, the dominant

contribution to the quantum resource is CHS, namely

the controlled Hamiltonian simulation from which the

eigenenergy phase is extracted into the ancilla qubits.

Fortunately, with the scope of performing quantum re-

source estimation for the QPE and digital quantum sim-

ulation, numerous works have been devoted to analyz-

ing the error scaling of various Hamiltonian simulation

techniques, in particular the Trotter-based methods [44–

46]. Nevertheless, we point out that crucial questions

remain unclear; (A) which technique is the best prac-

tice to achieve the earliest quantum advantage for con-

densed matter problems, and (B) at which point does the

crossover occur?

Here we perform resource estimation under the follow-

ing common assumptions: (1) logical qubits are encoded

using the formalism of surface codes [47]; (2) quantum

gate implementation is based on Cliﬀord+Tformalism;

Initially, we address the ﬁrst question (A) by compar-

ing the total number of T-gates, or T-count, across vari-

ous Hamiltonian simulation algorithms, as the applica-

tion of a T-gate involves a time-consuming procedure

known as magic-state distillation. Although not neces-

sarily, this procedure is considered to dominate the run-

time in many realistic setups. Therefore, we argue that

T-count shall provide suﬃcient information to determine

the best Hamiltonian simulation technique. Then, with

the aim of addressing the second question (B), we further

perform high-resolution analysis on the runtime. We in

particular consider concrete quantum circuit compilation

with speciﬁc physical/logical qubit conﬁguration compat-

ible with the surface code implemented on a square lat-

tice.

Let us ﬁrst compute the T-counts to compare the state-

of-the-art Hamiltonian simulation techniques: (random-

ized) Trotter product formula [48,49], qDRIFT [46], Tay-

lorization [50–52], and qubitization [41]. The former two

commonly rely on the Trotter decomposition to approxi-

mate the unitary time evolution with sequential applica-

tion of (controlled) Pauli rotations, while the latter two,

dubbed as “post-Trotter methods,” are rather based on

the technique called block-encoding, which utilize ancil-

lary qubits to encode desired (non-unitary) operations on

target systems (See Sec. S5 in SM). While post-Trotter

methods are known to be exponentially more eﬃcient in

terms of gate complexity regarding the simulation accu-

racy [50], it is nontrivial to ask which is the best practice

in the crossover regime, where the prefactor plays a sig-

niﬁcant role.

We have compiled quantum circuits based on exist-

ing error analysis to reveal the required T-counts (See

Sec. S4,S6,S7 in SM). From results presented in Ta-

ble I, we ﬁnd that the qubitization algorithm provides the

most eﬃcient implementation in order to reach the tar-

get energy accuracy ϵ= 0.01. Although the post-Trotter

methods, i.e., the Taylorization and qubitization algo-

rithms require additional ancillary qubits of O(log(N))

to perform the block encoding, we regard this overhead

as not a serious roadblock, since the target system itself

and the quantum Fourier transformation requires qubits

of O(N) and O(log(N/ϵ)), respectively. In fact, as we

show in Fig. 5, the qubitization algorithms are eﬃcient

at near-crosspoint regime in physical qubit count as well,

due to the suppressed code distance (see Sec. S9 in SM

for details).

We also mention that, for 2d Fermi-Hubbard model,

there exists some specialized Trotter-based methods that

improve the performance signiﬁcantly [16,17]. For in-

stance, the T-count of the QPE based on the state-or-

the-art PLAQ method proposed in Ref. [17] can be es-

timated to be approximately 4 ×108for 10 ×10 system

under ϵ= 0.01, which is slightly higher than the T-count

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Huntingforquantum-classicalcrossoverincondensedmatterproblemsNobuyukiYoshioka,1,2,3,∗TsuyoshiOkubo,4,3,†YasunariSuzuki,5,3,‡YukiKoizumi,5andWataruMizukami6,7,3,§1DepartmentofAppliedPhysics,UniversityofTokyo,7-3-1Hongo,Bunkyo-ku,Tokyo113-8656,Japan2TheoreticalQuantumPhysicsLaboratory,RIKENClusterforP...

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Hunting for quantum-classical crossover in condensed matter problems Nobuyuki Yoshioka1 2 3 Tsuyoshi Okubo4 3Yasunari Suzuki5 3Yuki Koizumi5and Wataru Mizukami6 7 3 1Department of Applied Physics University of Tokyo.pdf

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Hunting for quantum-classical crossover in condensed matter problems Nobuyuki Yoshioka1 2 3 Tsuyoshi Okubo4 3Yasunari Suzuki5 3Yuki Koizumi5and Wataru Mizukami6 7 3 1Department of Applied Physics University of Tokyo

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