High-delity realization of the AKLT state on a NISQ-era quantum processor Tianqi Chen1Ruizhe Shen2yChing Hua Lee2zand Bo Yang1 3x 1School of Physical and Mathematical Sciences Nanyang Technological University Singapore 639798

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High-ﬁdelity realization of the AKLT state on a NISQ-era quantum processor

Tianqi Chen,1, ∗Ruizhe Shen,2, †Ching Hua Lee,2, ‡and Bo Yang1, 3, §

1School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 639798

2Department of Physics, National University of Singapore, Singapore 117542

3Institute of High Performance Computing, A∗STAR, Singapore 138632

The AKLT state is the ground state of an isotropic quantum Heisenberg spin-1 model. It exhibits

an excitation gap and an exponentially decaying correlation function, with fractionalized excitations

at its boundaries. So far, the one-dimensional AKLT model has only been experimentally realized

with trapped-ions as well as photonic systems. In this work, we successfully prepared the AKLT

state on a noisy intermediate-scale quantum (NISQ) era quantum device for the ﬁrst time. In

particular, we developed a non-deterministic algorithm on the IBM quantum processor, where the

non-unitary operator necessary for the AKLT state preparation is embedded in a unitary operator

with an additional ancilla qubit for each pair of auxiliary spin-1/2’s. Such a unitary operator is

eﬀectively represented by a parametrized circuit composed of single-qubit and nearest-neighbor CX

gates. Compared with the conventional operator decomposition method from Qiskit, our approach

results in a much shallower circuit depth with only nearest-neighbor gates, while maintaining a

ﬁdelity in excess of 99.99% with the original operator. By simultaneously post-selecting each ancilla

qubit such that it belongs to the subspace of spin-up |↑i, an AKLT state can be systematically

obtained by evolving from an initial trivial product state of singlets plus ancilla qubits in spin-up

on a quantum computer, and it is subsequently recorded by performing measurements on all the

other physical qubits. We show how the accuracy of our implementation can be further improved

on the IBM quantum processor with readout error mitigation.

I. INTRODUCTION

The current age has witnessed tremendous progress in

the quantum simulation of novel many-body phenom-

ena [1–6]. In particular, there has been intense recent

focus on using noisy intermediate-scale quantum (NISQ)-

era [7] quantum computers to assist in large-scale tasks

with the goal of eventual quantum supremacy [8,9].

Among them, programmable digital quantum computers

have so far been successfully used for the implementa-

tion and study of discrete time crystals (DTC) [10,11],

quantum chemistry problems with Hartree-Fock meth-

ods [12], fractional quantum Hall states [13,14], spin

chain dynamics [15,16], interacting topological lattice

models [17,18], many-body localization [19], lattice

gauge theory [20] and quantum spin liquid states [21].

These examples in general involve (but are not limited

to) three categories of usage of quantum computers for

condensed matter physics: time evolution, ground state

search and state preparation. Such eﬀorts are made

with the goal of overcoming major drawbacks in current

numerical approaches. These include the exponential

“curse” of exact diagonalization (ED) [22], the sign prob-

lem in Fermionic quantum Monte Carlo simulations [23],

and the rapid growth of entanglement in tensor network

states [24,25].

At the current juncture, there are still limitations and

challenges in using NISQ-era quantum computers for

∗tianqi.chen@ntu.edu.sg

†ruizhe20@u.nus.edu

‡phylch@nus.edu.sg

§yang.bo@ntu.edu.sg

large scale simulations. Some major issues include large

circuit depth, low gate ﬁdelity, and thermal noise from

the execution of the quantum circuit [26]. In response,

many classical algorithms and approaches based on ma-

trix product state (MPS) have been recently proposed

for state preparation [27–29]. Another challenge, which

would be present even for a perfect quantum computer,

is that state preparation is a fundamentally non-unitary

process which requires the implementation of non-unitary

operators. Progress has lately been made through imag-

inary time evolution approaches combined with varia-

tional algorithms [30,31], or through constructing a de-

terministic measurement operator [32,33]. However,

these techniques may not always be practical for NISQ-

era quantum processors such as the IBM Q system due

to short qubit coherence times. In general, various req-

uisite operators cannot be directly decomposed into the

fundamental unitary gates on NISQ-era quantum com-

puters, posing diﬃculties for existing schemes for state

preparation.

In this work, we propose an algorithm and demon-

strate the preparation of a particular type of quantum

many-body state, the so-called Aﬄeck, Kennedy, Lieb,

and Tasaki (AKLT) state [34,35], on NISQ-era quan-

tum computers. As a type of Valence-Bond-Solid (VBS)

state [35], it is the exact ground state of the spin-1 AKLT

model, which is the paradigm of a strongly correlated

symmetry protected topological (SPT) phase with a Hal-

dane gap [36] and fractional excitations at its bound-

aries [34,37,38]. SPT phases of matter received much at-

tention recently on quantum computers [17,39–44], and

the two-dimensional generalization of the AKLT model

on a trivalent lattice is proposed to be a universal re-

source [45,46] for measurement-based quantum compu-

arXiv:2210.13840v2 [quant-ph] 9 Feb 2023

tation [47–49]. So far, the 1D AKLT state has been ex-

perimentally realized and characterized on photonic im-

plementations [50] using cluster states [51] and in trapped

ions [52]. Recently, we notice that there have been much

eﬀorts to construct the VBS state, including in partic-

ular the AKLT state in 1D with measurement assisted

preparation [53], and in 2D with a post-selection algo-

rithm [54]. With the usage of tensor network states,

both 1D and 2D AKLT states can be prepared adia-

batically [55]. For our work, instead of performing the

variational searching of the AKLT state as the ground

state of the spin-1 AKLT model [56], we show that the

AKLT state can be obtained by evolving from a trivial

initial product state composing of a chain of singlets.

On a NISQ-era quantum computer (e.g., IBMQ), the

main challenge is the non-unitarity of the state prepa-

ration, and our new approach is based on augmenting

the non-unitary subspace with additional ancilla qubits,

such that an eﬀectively non-unitary operator can be real-

ized through measurement-based post-selection. This al-

lows us to implement non-unitary operators with unitary

gates, achieving the simultaneous non-unitary projection

on every site of an initial product state made up of a chain

of singlets. For an eﬃcient quantum circuit realization

of this unitary operator, another matrix product state

(MPS)-based algorithm on a classical computer is used

to transform the operator into a parametrized circuit

via variational optimization [17,57–60]. Most recently,

MPS-based algorithms have been applied for the investi-

gations of translational-invariant systems [61,62]. Com-

pared with other recent AKLT state preparation methods

[53–55], our approach only requires nearest-neighbor CX

gates, and the full circuit that prepares the AKLT state

is much shallower than that from Qiskit’s default isome-

try decomposition method [63]. Also, the evolution from

the initial state has only one step, and it does not require

any mid-circuit measurements on IBM Q [64].

This paper is organized as follows. First, in Sec. II,

we introduce the AKLT model and its ground state, i.e.

the AKLT state. Sec. III discusses the details of the

approach used in this work to prepar the AKLT state,

which includes transforming the projection operator into

a unitary one, a variational parametrized circuit for the

three-qubit operator, and post-selection of the results.

Sec. IV presents the characterization of AKLT states for

L= 2,3,4 and 5 on IBMQ devices, and discusses vari-

ous factors which could aﬀect the ﬁdelity of the prepared

state. Finally, we highlight the conclusion of this work

in Sec. V.

II. THE AKLT STATE

Below, we brieﬂy introduce the AKLT state. Consider a

1D spin chain with 2Lspin-1/2s, grouped into pairs of

adjacent spins as illustrated in FIG. 1(a). In general, each

pair of adjacent spin-1/2s either forms a spin-0 singlet

state (| ↑↓i − | ↓↑i)/√2, or one of the three symmetric

states

|+i=| ↑↑i (1)

|Oi= 1/√2 (|↑↓i +|↓↑i)

|−i =|↓↓i

which spans the spin-1 subspace. To construct the AKLT

state, we ﬁrst project onto the spin-1 subspace of each

pair of adjacent spin-1/2s [circled in FIG. 1(a)], such that

we obtain an eﬀective chain of Lspin-1s.

Before any constraints are applied, each pair of adja-

cent spin-1s can have a total spin of S= 0,1 or 2. The

AKLT state is the unique state satisfying the constraint

that every pair of adjacent spin-1s (i.e. the four consecu-

tive spin-1/2s in two adjacent circles) is allowed to have

a total spin of S= 0 or 1, but not 2. In terms of the

constituent spin-1/2s, this is equivalent to the constraint

that each spin-1/2 forms a (spin-0) singlet with another

spin-1/2 from an adjacent spin-1 pair, as illustrated in

FIG. 1(a). This would be the picture that our AKLT

state algorithm is based on - we shall ﬁrst prepare the

spin singlets, and next project spin-1/2 pairs connected

to adjacent singlets onto their total S= 1 subspace.

The above spin chain picture can be recasted as an

MPS representation of the AKLT state |ψi, for both pe-

riodic and open boundary conditions (PBCs and OBCs):

|ψiPBC =X

Tr [Aσ1Aσ2···AσL]|σ1σ2···σLi,(2)

|ψiOBC =X

σhbl

TAσ1Aσ2···AσLbr

Ai|σ1σ2···σLi,

(3)

where σi∈ {+, O, −} labels the i-th spin-1 basis state,

with corresponding MPS matrices Aσgiven by

A+= +r2

3τ+, A0=−r1

3τz, A−=−r2

3τ−,(4)

τzand τ±=τx±iτyspanning the set of Pauli ma-

trices [65,66]. Since (τ±)2= 0, this matrix representa-

tion keeps track of the AKLT constraint that two adja-

cent spin-1s do not add up to total spin S= 2. Under

PBCs, there has to be an equal number of |+iand |−i

in |σ1σ2···σLi, as enforced by the trace operator Tr.

Under OBCs, which is the more convenient scenario for

implementation on the quantum processor [see Fig. 4],

we will have to ﬁx the end spins – in the above, we

have chosen these boundary vectors to be bl

A=1 0T,

and br

A=0 1T, up to a normalization factor. This

means that both boundary spins are ﬁxed as spin up,

which is the same as the MPS representation described

in Ref. [67]. As is shown below in Fig. 2, this choice is

of convenience for our implementation, as all qubits are

initialized as spin up on the IBM Q system. For this

choice, there is necessarily one more |+icompared to the

FIG. 1. Structure of the AKLT state. (a) The AKLT

state with open boundary conditions (OBCs) with L= 5:

solid lines connect two spin-1/2 qubits, forming singlet states.

Each pair of spin-1/2s (circled) from two consecutive singlets

are projected via ˆ

Ponto their total spin S= 1 sector, via

Eq. (6). (b) With the help of ancilla qubits (brown spheres),

the non-unitary projectors ˆ

Pcan be embedded in unitary op-

erators ˆ

Uacting on three qubits (red dashed square): the two

spin-1/2s (blue spheres) plus the ancilla qubit. The solid black

line connecting two physical qubits forms a singlet. The ac-

tual physical embedding into IBM quantum processor qubits

is shown Sect. III D.

number of |−i. The explicit forms of |ψiPBC and |ψiOBC

are given in Appendix Cfor L= 2 and L= 3.

Although we shall directly prepare the AKLT state

through an MPS (Eqs. 2and 3) quantum circuit, we note

in passing that the AKLT state can also be obtained as

the unique zero ground state [34,35] of the following

projection operator:

PS=2 =

L−1

i=1 Si·Si+1 +1

3(Si·Si+1)2+2

3,(5)

which projects onto the total spin S= 2 sector in all

pairs of adjacent spin-1s. It can be derived [34] by con-

sidering the total spin operator (Si+Si+1)2with eigen-

values proportional to S(S+ 1), where Siand Si+1 are

the spin-1 operators of adjacent spin-1s.

III. PREPARING THE AKLT STATE ON A

QUANTUM COMPUTER

A. Implementing local projections within unitary

operators

The preparation of the AKLT state on a quantum cir-

cuit crucially requires non-unitary operators for project-

ing onto the spin-1 subspace of each adjacent spin-1/2

pair [FIG. 1(a)]. Inspired by the techniques introduced

in Refs. [29,68,69], we develop an approach for preparing

the AKLT state by embedding the projection operator on

each spin-1/2 pair into a 3-qubit unitary operator that

admits an additional ancilla qubit. By subsequently pro-

jecting the ancilla qubit onto a chosen state |↑i by post-

selection [see FIG. 1(b)], we can realize the non-unitary

S= 1 projection on the two spin-1/2s. This approach

allows us to prepare the AKLT state according to the

MPS formalism given in [70].

Explicitly, as shown in FIG. 1(a), a local spin-1 in

the bulk, which forms one “site” of an AKLT chain in

OBCs, is built from a pair of adjacent spin-1/2 through

the projection operator

P=|+ih+|+|OihO|+|−ih−| (6)

=





1 0 0 0

0 1/2 1/2 0

0 0 0 1





,

expressed in the spin-1/2 basis {| ↑↑i,| ↑↓i,| ↓↑i,| ↓↓i},

with |+i=|↑↑i,|Oi= 1/√2(|↑↓i +|↓↑i) and |−i =

|↓↓i deﬁned as before. Note that the spin-1/2s are the

physical degrees of freedom on a quantum processor, even

though they are often referred to as virtual spins in the

AKLT literature.

This spin-1 projector of Eq. (6) is non-unitary as

Pˆ

P†6=I, which is not possible to be directly imple-

mented on a quantum computer such as IBM Q. To re-

alize it in a quantum circuit, we embed it in a 3-qubit

unitary operator ˆ

Uwhich takes the form

U=ˆ

Pˆ

Qˆ

P(7)

in the product basis of the ancilla qubit and the two spin-

1/2 qubits. Here, we stipulate

Q=





0 0 0 0

0 1/2−1/2 0

0−1/2 1/2 0

0 0 0 0





,(8)

such that ˆ

Uis unitary, i.e. ˆ

U†ˆ

U=I. As both ˆ

Pand ˆ

are symmetric, it is easy to verify that ˆ

P2+ˆ

Q2=I4×4,

and ˆ

Qˆ

P+ˆ

Pˆ

Q= 04×4.

For this three-site subsystem consisting of two original

spin chain qubits and an ancilla qubit, we examine input

states of the form

|ψi=|↑i ⊗ |φi=1

0⊗ |φi,(9)

where |↑i represents an ancilla qubit in the spin-up state,

and |φirepresents the two adjacent qubits which are

paired as a singlet. Applying Eq. (7) to the above three-

qubit state, we have

U |ψi=ˆ

P |φi

Q|φi=|↑i ⊗ ˆ

P |φi+|↓i ⊗ ˆ

Q|φi.(10)

Therefore, it is clear that after projecting the output an-

cilla qubit onto |↑i, say via post-selection, the target state

|φiis indeed acted on by the nonunitary projector ˆ

Pi.e.

h↑| ˆ

U(|↑i ⊗ |φi) = ˆ

P|φi.(11)

The above technique contains only one single-step evo-

lution that does not require any mid-circuit measurement

for the preparation of the AKLT state [64], which pro-

vides a new approach towards embedding a non-unitary

projection operator ˆ

Pinto a unitary operator ˆ

Ufor fur-

ther decomposition into basis gates on a quantum com-

puter, which will be discussed shortly. We also remark

that our approach can be used to prepare the AKLT state

under both OBCs to PBCs, although the PBC case re-

quires a quantum device geometry such that a closed loop

of qubits exists, and are accompanied by appropriately

located branches functioning as ancilla qubits [see Fig. 4]

[71].

B. Quantum circuit implementation

We break up the preparation of the MPS-based AKLT

state into two steps, as sketched in FIG. 2. We ﬁrst

prepare the paired singlet states [two solid dots con-

nected with a solid line in Fig. 1(a)] as initial states

through the combination of Xgates, a Hadamard gates

and a CX gate (see notations in Qiskit [63]), which

corresponds to the operations to the left of the red

dashed line in FIG. 2. An Xgate is essentially a Pauli-

Xoperator, and a Hadamard gate Hmaps |↑i[|↓i] to

(|↑i +|↓i)/√2(|↑i − |↓i)/√2:

X=0 1

1 0, H =1

√21 1

1−1,(12)

while a CX gate is a two-qubit controlled-X gate which

performs a Pauli-Xoperation on the target qubit when-

ever the control is in state |↓i.

CX =





1000

0100

0001

0010





.(13)

which is expressed in the same basis as in Eq. (6).

Next, we perform the S= 1 projections ˆ

Pon spins-1/2

pairs from adjacent singlets, which is undertaken by the

unitary operation ˆ

Uof Eq. (7). The ancilla qubits asso-

ciated with the ﬁrst, second, third etc pairs are labeled

“q0” , “q3” and “q6” etc. in FIGs. 1(b) and 2.

FIG. 2shows the circuit structure for a small illus-

trative system with L= 3; we point out that the afore-

mentioned procedure can be extended to arbitrarily large

system sizes, as one can apply the unitary operator ˆ

Usi-

multaneously to all corresponding sites, i.e.

h0|

q2X H X X

h0|

q5X H X X

h0|

FIG. 2. Illustrative quantum circuit for the prepara-

tion of a 2L= 6-qubit AKLT state with OBC. The roles

of the 9 qubits q0to q8are given in Fig. 1(b), with Lancilla

qubits q0, q3, q6and the remain 2Lqubits representing the

spin-1/2 chain. Qubits q1and q8are boundary qubits while

qubits q2, q4and q5, q7pair up as singlets. The state prepa-

ration consists of two steps, as separated by the red dashed

line: First, the initial state consisting of L−1 = 2 singlet pairs

is initialized as shown to the left of the red line, where each

combination of CX, X and Hadamard gate (H) gates creates

a singlet. Next, to the right of the red line, the 3-qubit unitary

operation ˆ

Ufrom Eq. (7) is eﬀected, where every third qubit

q3kis an ancilla. To recover the non-unitary spin-1 projection

Pfrom Eq. (6), post-selection “h0|” operations are performed

on the ancilla qubits, as described by Eq. (10). The OBC

AKLT state shown in FIG. 1(a) is obtained through measure-

ments and post-selections on the physical qubits. With the

circuit geometry given in FIG. 1(b), the CX gates between

q2, q4and q5, q7act between nearest neighbor qubits when

embedded in a quantum processor (also see Fig. 4).

|ψiAKLT = L−1

k=0 h↑|3k!hˆ

U(0,1,2) ⊗ˆ

U(3,4,5) ⊗ ···|ψi0i

= L−1

k=0 h↑|3k!



L−1

j=0

U(3j, 3j+ 1,3j+ 2) |ψi0



L−1

j=0

Pj|φi0

(14)

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High-delityrealizationoftheAKLTstateonaNISQ-eraquantumprocessorTianqiChen,1,RuizheShen,2,yChingHuaLee,2,zandBoYang1,3,x1SchoolofPhysicalandMathematicalSciences,NanyangTechnologicalUniversity,Singapore6397982DepartmentofPhysics,NationalUniversityofSingapore,Singapore1175423InstituteofHighPerformanc...

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High-delity realization of the AKLT state on a NISQ-era quantum processor Tianqi Chen1Ruizhe Shen2yChing Hua Lee2zand Bo Yang1 3x 1School of Physical and Mathematical Sciences Nanyang Technological University Singapore 639798

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