Quantum circuit for measuring an operators generalized expectation values and its applications to non-Hermitian winding numbers Ze-Hao Huang Äýj1 2Peng He UO1 2Li-Jun Lang Î3and Shi-Liang Zhu 13 4y

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Quantum circuit for measuring an operator’s generalized expectation values and its

applications to non-Hermitian winding numbers

Ze-Hao Huang (黄泽豪),1, 2 Peng He (何鹏),1, 2 Li-Jun Lang (郎利君),3, ∗and Shi-Liang Zhu (朱诗亮)3, 4, †

1National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China

2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

3Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,

School of Physics and Telecommunication Engineering,

South China Normal University, Guangzhou 510006, China

4Guangdong-Hong Kong Joint Laboratory of Quantum Matter,

Frontier Research Institute for Physics, South China Normal University, Guangzhou 510006, China

(Dated: May 11, 2023)

We propose a general quantum circuit based on the swap test for measuring the quantity hψ1|A|ψ2i

of an arbitrary operator Awith respect to two quantum states |ψ1,2i. This quantity is frequently

encountered in many ﬁelds of physics, and we dub it the generalized expectation as a two-state

generalization of the conventional expectation. We apply the circuit, in the ﬁeld of non-Hermitian

physics, to the measurement of generalized expectations with respect to left and right eigenstates of

a given non-Hermitian Hamiltonian. To eﬃciently prepare the left and right eigenstates as the input

to the general circuit, we also develop a quantum circuit via eﬀectively rotating the Hamiltonian pair

(H, −H†) in the complex plane. As applications, we demonstrate the validity of these circuits in the

prototypical Su-Schrieﬀer-Heeger model with nonreciprocal hopping by measuring the Bloch and

non-Bloch spin textures and the corresponding winding numbers under periodic and open boundary

conditions (PBCs and OBCs), respectively. The numerical simulation shows that non-Hermitian spin

textures building up these winding numbers can be well captured with high ﬁdelity, and the distinct

topological phase transitions between PBCs and OBCs are clearly characterized. We may expect

that other non-Hermitian topological invariants composed of non-Hermitian spin textures, such as

non-Hermitian Chern numbers, and even signiﬁcant generalized expectations in other branches of

physics would also be measured by our general circuit, providing a diﬀerent perspective to study

novel properties in non-Hermitian as well as other physics realized in qubit systems.

I. INTRODUCTION

Since the theoretical prediction and experimental ob-

servation of unique features in non-Hermitian systems

[1,2], such as the parity-time-reversal symmetry break-

ing [3–8], the breakdown of conventional bulk-boundary

correspondence [9–26], and the exceptional points (EPs)

[2,27,28], the non-Hermitian physics has been attract-

ing increasing attention. Many non-Hermitian phenom-

ena have been explored in various quantum platforms,

including quantum optics [11,29,30], quantum spin sys-

tems [31–33], ultracold atoms [7,8,34–40], and so on.

However, none of these studies has discussed the direct

measurement of the non-Hermitian generalization of the

expectation value, hψL|A|ψRi, with Abeing an arbitrary

operator and |ψL,Ribeing a pair of left and right eigen-

states, dubbed dual eigenstates, of a given non-Hermitian

Hamiltonian, which is deeply involved in many deﬁni-

tions of non-Hermitian quantities as the straightforward

generalization of the Hermitian counterparts [41]. Devel-

oping a method of measuring quantities of this form is

urgent and may be a prerequisite for studying in a uni-

versal manner the exotic non-Hermitian phenomena in

experiments.

∗ljlang@scnu.edu.cn

†slzhu@nju.edu.cn

One of the most interesting quantities involving

hψL|A|ψRiis the non-Hermitian topological invariant

[1], such as the non-Hermitian winding number [19]. For

example, the Su-Schrieﬀer-Heeger (SSH) model [42] with

nonreciprocal hopping is a prototypical non-Hermitian

topological model; the diﬀerence between it and its Her-

mitian counterpart is reﬂected by the winding number

deﬁned with the dual eigenstates [41]. Existing works try

to establish relations between this non-Hermitian wind-

ing number and the experimentally measurable expecta-

tion values to ﬁgure it out indirectly. It was shown that

the winding number can be calculated by the dynamic

winding numbers, deﬁned by the integral of the long-

time average of measurable expectation values [43]. On

the other hand, the authors of Ref. [33] reparametrize the

non-Hermitian Hamiltonian and use the measurable ex-

pectation values to ﬁt the parameters; the winding num-

ber is reconstructed by the parameters. The limitation

of these works is the lack of generality for measuring the

quantity hψL|A|ψRi, and the relations they found may

just be available in special cases.

Furthermore, a quantity such as hψL|A|ψRiis not only

restricted within the non-Hermitian physics if the dual

eigenstates are relaxed to two arbitrary quantum states

|ψ1,2i, i.e., hψ1|A|ψ2i, which we dub the generalized ex-

pectation of Ain the following as a two-state generaliza-

tion of the conventional expectation. Quantities in the

form of a generalized expectation are ubiquitous in quan-

arXiv:2210.12732v2 [quant-ph] 10 May 2023

tum physics, endowed with distinct meanings in diﬀerent

ﬁelds of physics [44], such as the overlap integral of two

states, matrix elements of an operator, scattering ampli-

tudes, and the Green’s function or Feynman propagator

in quantum ﬁeld theory. Since direct measurement of

them is outside of the conventional formalism, e.g., the

projective (von Neumann) measurement [45], some in-

direct methods have been proposed, such as the swap

test [46], weak measurement [47], and quantum circuits

for simulating the correlation function [48–50]. However,

these proposals are still only valid for speciﬁc situations;

that is, |ψ1,2icannot be arbitrary. A general scheme for

measuring generalized expectations remains to be settled

even in broader ﬁelds of quantum physics.

To deal with the measurement of hψ1|A|ψ2i, we pro-

pose a general circuit based on the swap test [46] in Fig.

1to directly capture the real and imaginary parts of the

generalized expectation. Meanwhile, to apply the gen-

eral circuit to hψL|A|ψRiin non-Hermitian systems, a

quantum circuit (Fig. 2) for eﬃciently preparing the dual

eigenstates of a given non-Hermitian Hamiltonian as the

input of the general circuit is also developed with the aid

of the dilation method [31]. By numerically simulating

these circuits in the nonreciprocal SSH model, we suc-

cessfully obtain the Bloch and non-Bloch spin textures

and the corresponding winding numbers under periodic

and open boundary conditions (PBCs and OBCs), re-

spectively, which demonstrates the validity of our cir-

cuits.

The paper is organized as follows. The general quan-

tum circuit for measuring generalized expectations is pro-

posed in Sec. II. Specially for non-Hermitian systems, the

quantum circuit for preparing the dual eigenstates of a

non-Hermitian Hamiltonian is developed in Sec. III. In

Sec. IV, we apply these circuits to the non-reciprocal

SSH model and numerically simulate the measurement

of Bloch and non-Bloch spin textures and the winding

numbers. Section Vprovides a conclusion.

II. A GENERAL MEASURMENT CIRCUIT FOR

GENERALIZED EXPECTATIONS

Our aim is to measure the quantity, hψ1|A|ψ2i, of an

arbitrary operator Awith respect to two quantum states

|ψ1iand |ψ2i, dubbed a generalized expectation of A,

which reduces to the conventional expectation when the

two states are identical, i.e., |ψ1i=|ψ2i. Because any

operator can be decomposed into Hermitian operators,

A=A+A†

2+iA−A†

2i, we just need to propose a

quantum circuit to measure the generalized expectation

of a Hermitian operator, i.e., hψ1|O|ψ2i, where Orepre-

sents an experimentally accessible, Hermitian operator.

Figure 1shows the quantum circuit for measuring

hψ1|O|ψ2i, which is the main result of this paper. Sup-

posing that |ψ1,2iare obtained, this circuit consists of

systems A and B, each of which is represented by n

qubits, and an ancilla qubit. Firstly, the two states |ψ1,2i

system A: |ψ1⟩

system B: |ψ2⟩

ancilla: |0⟩

|Ψ0⟩

|Ψ1⟩ |Ψ2⟩

O′

σx,y

Input Operation Readout

FIG. 1. The general quantum circuit for measuring the gen-

eralized expectation hψ1|O|ψ2iscaled by hψ1|O0|ψ2i. Two

quantum states |ψ1,2iare assumed to be prepared by other

methods as input in systems A and B, respectively, each of

which consists of nqubits. The ancilla qubit is initialized to

|0i. The successive operations include a Hadamard gate (de-

noted by H) on the ancilla and a controlled-swap (Fredkin)

gate with the ancilla as the control qubit. Oand O0denote

the experimentally accessible Hermitian operators. σx,y are

two operators of Pauli matrices. The details of the readout

process are given in Appendix B.

are put into systems A and B, respectively, and the an-

cilla qubit is initialized to |0i, yielding a product state

|Ψ0i=|ψ1i⊗|ψ2i⊗|0i(1)

as an initial state.

Secondly, by applying a Hadamard gate to the ancilla,

and then a controlled-swap (Fredkin) gate with the an-

cilla being the control qubit, we obtain successively

|Ψ1i=|ψ1i⊗|ψ2i ⊗ 1

√2(|0i+|1i),(2)

|Ψ2i=1

√2(|ψ1i⊗|ψ2i⊗|0i+|ψ2i⊗|ψ1i⊗|1i).(3)

Finally, the operator Oand an ancillary Hermitian

operator O0are introduced to systems A and B, respec-

tively. Because of the Hermiticity of Oand O0, we obtain

the following relations (see Appendix Afor the detailed

derivation):

hΨ2|O⊗O0⊗σx|Ψ2i

hΨ2|O0⊗O0⊗σx|Ψ2i= Re hψ1|O|ψ2i

hψ1|O0|ψ2i,

hΨ2|O⊗O0⊗σy|Ψ2i

hΨ2|O0⊗O0⊗σx|Ψ2i= Im hψ1|O|ψ2i

hψ1|O0|ψ2i,

(4)

where σx,y are two operators of Pauli matrices. Assum-

ing that Oand O0are experimentally accessible, from

Eq. (4) the generalized expectation hψ1|O|ψ2iscaled by

hψ1|O0|ψ2i 6= 0 can be ﬁgured out by measuring the tra-

ditional expectations [left-hand side of Eq. (4)] in ex-

periment. Based on current experiment scales [51], this

general measurement quantum circuit in Fig. 1can be

applied to the qubit systems with states and operators

being composed of up to ∼50 qubits of each. See Ap-

pendix Bfor details of the measurement.

For convenience, O0can be set as an identity opera-

tor, and the scaling factor hψ1|O0|ψ2ireduces to hψ1|ψ2i,

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Quantumcircuitformeasuringanoperator'sgeneralizedexpectationvaluesanditsapplicationstonon-HermitianwindingnumbersZe-HaoHuang(Äýj),1,2PengHe(UO),1,2Li-JunLang(Î)),3,andShi-LiangZhu(1×®)3,4,y1NationalLaboratoryofSolidStateMicrostructuresandSchoolofPhysics,NanjingUniversity,Nanjing210093,China2Collab...

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Quantum circuit for measuring an operators generalized expectation values and its applications to non-Hermitian winding numbers Ze-Hao Huang Äýj1 2Peng He UO1 2Li-Jun Lang Î3and Shi-Liang Zhu 13 4y

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