Parametrized quantum circuit for weight-adjustable quantum loop gas Rong-Yang Sun1 2Tomonori Shirakawa1 2and Seiji Yunoki1 2 3 4 1Computational Materials Science Research Team

2025-05-02 0 0 987.65KB 6 页 10玖币

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Parametrized quantum circuit for weight-adjustable quantum loop gas

Rong-Yang Sun,1, 2 Tomonori Shirakawa,1, 2 and Seiji Yunoki1, 2, 3, 4

1Computational Materials Science Research Team,

RIKEN Center for Computational Science (R-CCS), Kobe, Hyogo, 650-0047, Japan

2Quantum Computational Science Research Team,

RIKEN Center for Quantum Computing (RQC), Wako, Saitama, 351-0198, Japan

3Computational Quantum Matter Research Team,

RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

4Computational Condensed Matter Physics Laboratory,

RIKEN Cluster for Pioneering Research (CPR), Saitama 351-0198, Japan

(Dated: January 30, 2023)

Motivated by the recent success of realizing the topologically ordered ground state of the exactly

solvable toric code model by a quantum circuit on the real quantum device [K. J. Satzinger et al.,

Science 374, 1237 (2021)], here we propose a parametrized quantum circuit (PQC) with the same

real-device-performable optimal structure to represent quantum loop gas states with adjustably

weighted loop conﬁgurations. Combining such a PQC with the variational quantum eigensolver,

we obtain the accurate quantum circuit representation for the toric code model in an external

magnetic ﬁeld with any ﬁeld strength, where the system is not exactly solvable. The topological

quantum phase transition in this system is further observed in the optimized circuits by measuring

the magnetization and topological entanglement entropy.

Introduction.— The topologically ordered state, a new

category of exotic quantum states of matter, has continu-

ously attracted research interest in recent decades [1–3].

Many novel properties that emerge from it include the

degenerate ground state with long-range quantum en-

tanglement and the anyonic-type excitations, which are

also deemed essential for the development of quantum

computing [4, 5]. However, because of the existence of

the long-range quantum entanglement, it is challenging

to investigate the topologically ordered state. Except

for rare elaborately constructed exactly solvable models,

such as Kitaev’s toric code model [4] and Wen’s plaque-

tte model [6], it is diﬃcult to verify whether a topologi-

cally ordered phase exists in a certain microscopic quan-

tum many-body Hamiltonian and to identify its nature,

a famous example being the spin-1/2 antiferromagnetic

Heisenberg model on the kagome lattice [7–10].

On the other hand, rapidly developed synthetic con-

trollable quantum platforms, both in circuit-based quan-

tum computers [11, 12] and analog quantum simula-

tors [13–15], oﬀer a new way to explore these long-range-

entangled states. These controllable quantum systems

have an intrinsic quantum entangled nature and thus are

expected to be able to handle another quantum entangled

system relatively easier. Recently, this expectation has

been partially realized. One can use the optimal quan-

tum resource to realize the ground state of the toric code

model and accurately probe its properties [16], demon-

strating a huge potential for utilizing quantum computers

to study topologically ordered states.

Many algorithms have been proposed to eﬃciently use

the quantum resource for solving a generic non-exactly

solvable quantum many-body problem. Among them,

the most promising one, which is considered to be realiz-

able on current noisy-intermediate scale quantum (NISQ)

devices [17, 18], is the variational quantum eigensolver

(VQE) [19–21]. The VQE adopts a parametrized quan-

tum circuit (PQC) as the ansatz state. This PQC ansatz

is evaluated on a quantum computer, while its parame-

ters are optimized by a classical optimizer. Despite con-

siderable eﬀorts, the scalable realization of the VQE on

NISQ devices is still under extensive research. This is be-

cause proper PQC ansatze, which can faithfully describe

the ground state of a given quantum many-body Hamil-

tonian and yet can be performed accurately enough on

real devices, are still missing, especially when the sys-

tem is close to a phase transition point (for example, see

Ref. [22]).

In this paper, we address this issue by proposing

a PQC, which has adequate expressibility towards the

ground state of a non-exactly solvable correlated Hamil-

tonian and at the same time can be faithfully evalu-

ated on NISQ devices. We encode a quantum loop gas

state with adjustably weighted loop conﬁgurations into a

quantum circuit with the optimal circuit depth. The ad-

justable weights are controlled by circuit parameters. Us-

ing such a single PQC, we precisely reproduce the ground

state with an energy accuracy better than the order of

10−2for the toric code model in an external magnetic

ﬁeld, thus in both topologically ordered and ferromag-

netically ordered phases, by VQE calculations.

Toric code model in a magnetic ﬁeld.— The toric code

model [4] is deﬁned by qubits located on the bond centers

of a Lx×Lysquare lattice with Nbonds, and is given

by the following Hamiltonian:

HTC =−X

As−X

Bp,(1)

where As=Qi∈sσz

iand Bp=Qi∈pσx

i. Here, σz

i(σx

i) is

the Pauli Z(X) operator of qubit ilocated on the i-th

arXiv:2210.14662v2 [quant-ph] 27 Jan 2023

(a)

(b)

AsB

Ψ(θ)=|+++···

||

|

FIG. 1. (a) Schematic ﬁgure of the toric code model in

an external magnetic ﬁeld Bapplied along the zdirection

on the 6 ×5 lattice under OBCs, where qubits (indicated by

circles) are located on each bond of the lattice. Here, the

blue cross connecting four blue circles indicates an Asterm

and the green square connecting four green circles denotes a

Bpterm. Note that Asterms are also present at the corners

and edges of the lattice under OBCs, thus involving only two

and three qubits, respectively. The loops (indicated by red)

are formed by local qubit states |1ii, while qubits not along

any loops are |0ii. (b) Schematic ﬁgure of the ansatz state

|Ψ(θ)igenerated by a parametrized loop gas circuit. In these

two ﬁgures, the shade intensity of red for loops represents a

weight of the corresponding loops.

bond center, and s(p) sums over all the vertices (pla-

quettes) of the underlying square lattice [see Fig. 1(a)].

The Hamiltonian in Eq. (1) is exactly solvable, and its

ground state is a topological quantum spin liquid char-

acterized by a Z2topological order [4, 23]. For the toric

code model with open boundary conditions (OBCs) along

the two prime directions, its unique ground state can be

constructed as [16]

|Ψ0i=

p=1 1

√2

Ip+1

√2Bp|00 ···0i,(2)

where Ip=Qi∈pIiis the identity operator on the p-th

plaquette (Iibeing the identity operator at qubit i), Np

is the total number of the plaquettes, and |00 ···0i=

|0i1|0i2···|0iNrepresents a product of the local states

|0iiwith σz

i|0ii=|0iiand σz

i|1ii=−|1ii. Equation (2)

can be interpreted as generating an equally weighted su-

perposition of all possible basis conﬁgurations which con-

tain only closed loops formed by local states |1ii, i.e., a

quantum loop gas (LG).

An external magnetic ﬁeld along the zdirection can

drive the toric code model away from the exactly solvable

point and leads to a topological quantum phase transi-

tion [24, 25]. The corresponding Hamiltonian [also see

Fig. 1(a)] is described by

HTCM(x) = (1 −x)HTC −x

i=1

σz

i.(3)

When x= 0, HTCM(x) returns to the toric code case,

while it has the exact ground state |00 ···0iwhen x= 1.

Extensive quantum Mount Carlo studies have suggested

that HTCM(x) goes through a second-order quantum

phase transition at xc∼0.25 from the toric code state

to the ferromagnetically ordered state in the thermody-

namic limit [25].

It is quite insightful to understand the ground state of

HTCM(x) using the LG picture and consider the mag-

netic ﬁeld inducing tension to the loops [24]. In the

presence of the magnetic ﬁeld, the loop with a more ex-

tended perimeter costs more energy, and its weight in

the ground state should be lightened to minimize the en-

ergy, as schematically illustrated in Fig. 1(a). Therefore,

an LG with adjustable weights for each loop pattern can

be regarded as an appropriate description of the ground

state of HTCM(x), especially when the charge excitation

is gapped out and thus closed loop conﬁgurations remain

without having open strings. In the rest of this paper,

we will show how to represent this as a PQC with an

optimal circuit structure.

Parametrized loop gas circuit.— Before introducing the

PQC for a weight-adjustable LG, we brieﬂy recall the

method to construct the ground state |Ψ0iof the pure

toric code model HTC with an optimal quantum cir-

cuit [16, 26]. The core steps to generate |Ψ0iare to

apply a serial of (Ip+Bp)/√2 operators, which can be

realized by applying a Hadamard gate Hto a representa-

tive qubit associated with a plaquette and then applying

CNOT gates controlled by the representative qubit to the

other qubits in this plaquette. The circuit depth in this

construction grows linearly with Ly, meeting the lower

bound of the information spread in the quantum cir-

cuit to construct a globally entangled quantum state [27].

Furthermore, the circuit constructed by this method has

been realized on the current NISQ devices with high ac-

curacy [16].

Applying a Hadamard gate to |0iiis nothing but creat-

ing the equally weighted superposition between |0ii(i.e.,

vacuum) and |1ii(a precursor of the loop). In order to

construct an LG with diﬀerently weighted loop patterns,

we need to create an imbalanced superposition between

|0iiand |1ii, which is rather straightforward once we no-

tice the relation

H|0ii=Ry(π/2)|0ii,(4)

where

Ry(θ) = cos θ

2−sin θ

sin θ

2cos θ

2(5)

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Parametrizedquantumcircuitforweight-adjustablequantumloopgasRong-YangSun,1,2TomonoriShirakawa,1,2andSeijiYunoki1,2,3,41ComputationalMaterialsScienceResearchTeam,RIKENCenterforComputationalScience(R-CCS),Kobe,Hyogo,650-0047,Japan2QuantumComputationalScienceResearchTeam,RIKENCenterforQuantumComputing(...

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Parametrized quantum circuit for weight-adjustable quantum loop gas Rong-Yang Sun1 2Tomonori Shirakawa1 2and Seiji Yunoki1 2 3 4 1Computational Materials Science Research Team

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