Studying chirality imbalance with quantum algorithms Alexander M. Czajka1 2Zhong-Bo Kang1 2 3 4yYuxuan Tee1zand Fanyi Zhao1 2 3x 1Department of Physics and Astronomy University of California Los Angeles CA 90095 USA

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Studying chirality imbalance with quantum algorithms

Alexander M. Czajka,1, 2, ∗Zhong-Bo Kang,1, 2, 3, 4, †Yuxuan Tee,1, ‡and Fanyi Zhao1, 2, 3, §

1Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA

2Mani L. Bhaumik Institute for Theoretical Physics,

University of California, Los Angeles, CA 90095, USA

3Center for Quantum Science and Engineering, University of California, Los Angeles, CA 90095, USA

4Center for Frontiers in Nuclear Science, Stony Brook University, Stony Brook, NY 11794, USA

To describe the chiral magnetic eﬀect, the chiral chemical potential µ5is introduced to imitate

the impact of topological charge changing transitions in the quark-gluon plasma under the inﬂuence

of an external magnetic ﬁeld. We employ the (1+ 1) dimensional Nambu-Jona-Lasinio (NJL) model

to study the chiral phase structure and chirality charge density of strongly interacting matter with

ﬁnite chiral chemical potential µ5in a quantum simulator. By performing the Quantum imaginary

time evolution (QITE) algorithm, we simulate the (1 + 1) dimensional NJL model on the lattice

at various temperature Tand chemical potentials µ, µ5and ﬁnd that the quantum simulations

are in good agreement with analytical calculations as well as exact diagonalization of the lattice

Hamiltonian.

I. INTRODUCTION

In quantum chromodynamics (QCD), several major

challenges have gained considerable attention, includ-

ing how the vacuum structures of QCD are aﬀected in

extreme environments [1]. QCD research in hot and

dense conditions is of great importance, not only from

a purely theoretical perspective, but also for its numer-

ous applications to the studies of the quark matter in

the ultradense compact stars [2–8], and the Quark-Gluon

Plasma (QGP) which is abundantly produced in rela-

tivistic collisions of heavy ions [9,10]. Studying how

non-perturbative features of QCD are aﬀected by ther-

mal excitations at high temperatures Tand by baryon-

rich matter at ﬁnite chemical potentials µ[11] is highly

interesting.

Besides the eﬀects of ﬁnite Tand µ, the inﬂuence of

a strong magnetic ﬁeld Bis an exciting topic relevant to

phenomenology in relativistic heavy-ion collisions, where

strong magnetic ﬁelds are generated in non-central col-

lisions [12–17]. Many studies have been conducted on

the eﬀect of magnetic ﬁelds on the QCD vacuum [18–24],

and it has been determined that magnetic ﬁelds Bact as

a catalyst of dynamical chiral symmetry breaking [25–

27]. In the presence of a magnetic ﬁeld, a ﬁnite cur-

rent is induced along the direction of the ﬁeld lines due

to the anomalous production of an imbalance between

right- and left-handed quarks, namely that the number

of right-handed quarks NR1is not equal to the number

of left-handed quarks NL. This eﬀect is known as the

Chiral Magnetic Eﬀect (CME) [28–30].

The axial anomaly and topological objects in QCD are

∗aczajka74@physics.ucla.edu

†zkang@ucla.edu

‡yxtee0824@gmail.com

§fanyizhao@physics.ucla.edu

1More precisely, NRthe number of right-handed quarks minus the

number of left-handed antiquarks, with NLdeﬁned analogously.

the fundamental physics of the CME. At low or zero tem-

peratures, the change of non-trivial topological structure

is related to instanton [31,32] with the quantum tunnel-

ing eﬀect. However, at ﬁnite temperatures, the transition

is caused by sphalarons [33–35] and the chiral asymmetry

shows up. Unbalanced left- and right-handed quarks can

produce observable eﬀects that can be used to investigate

topological P- and CP-odd excitations [36–42]. Thus the

CME is a phenomenologically and experimentally inter-

esting eﬀect of the strong magnetic ﬁeld in heavy-ion col-

lisions. In [43], an observable sensitive to local P- and

CP-violation has been proposed for experiments. Mea-

surements of charge correlations were made by STAR at

RHIC [14,44–48], where conclusive evidence of charge az-

imuthal correlations was observed, which could be a pos-

sible result from CME with local P- and CP-odd eﬀects.

Furthermore, consistent experimental data was provided

by ALICE [49–53] and CMS [54,55] at the LHC, where

the azimuthal correlator was measured to search for the

CME in heavy-ion collisions.

By introducing a ﬁnite chiral chemical potential µ5

that imitates the eﬀects of the topological charge chang-

ing transitions, one can study the QCD phase dia-

gram [56] as well as the thermal behavior of the total

chirality charge N5=NR−NLunder the inﬂuence of

an external magnetic ﬁeld at ﬁnite temperature Tand

baryon chemical potential µ. At suﬃciently high tem-

peratures/densities, the strongly-interacting matter goes

through a deconﬁnement phase transition from hadronic

matter to quark-gluon plasma, and it is possible that

a chirality charge is produced in the phase transition

as a result of the ﬂip of fermion helicity in the interac-

tion with the gauge ﬁeld. Moreover, it has been demon-

strated that immediately after a heavy-ion collision, the

chirality charge comes to and stays at an equilibrium

value [7,57,58]. In light of these considerations, it is

evident that exploring the chiral imbalance in the QCD

phase diagrams is crucial for the description of heavy-ion

collisions.

To study the chiral magnetic eﬀect and the QCD chiral

arXiv:2210.03062v1 [hep-ph] 6 Oct 2022

phase transition, the Nambu-Jona-Lasinio (NJL) model

[59,60] has been playing an important role for many

years [5,61–70]. As an eﬀective model for QCD, the

NJL model is amenable to analytical calculations at ﬁnite

temperature Tand chemical potentials µand µ5.

In recent years, lattice QCD simulations have signif-

icantly improved our understanding of the QCD phase

diagram at zero or small chemical potentials µ[71–76].

However, as a result of the sign problem [77], at ﬁnite

µ, Monte-Carlo simulation is unable to be directly ap-

plied because the fermion determinant becomes complex,

and its phase ﬂuctuations prohibit its interpretation as a

probability density [71]. As a result, the sign problem is

a fundamental impediment to comprehending the phase

structure of nuclear matter. This is, however, not a ﬂaw

in QCD theory, but rather in the attempt to mimic quan-

tum statistics via the functional integral using a classical

Monte Carlo approach.

Fortunately, as indicated in [56,78], the statistical

properties of a quantum computer can be used to ob-

viate the necessity for a Monte Carlo study by modeling

the lattice system on a quantum computer. And there

have been abundant developments in applying quantum

computing to solving physics problems. In recent years,

it has been shown that using the current generation of

Noisy Intermediate-Scale Quantum (NISQ) technology

that quantum computers can solve complex problems

like simulating thermal properties [79–89], evaluating

ground states and real-time dynamics [78,90–104], mod-

eling many-body systems and relativistic eﬀects [105–

145], etc. Though digital quantum simulations on ther-

mal physical systems were researched earlier on, ﬁnite-

temperature physics is less well-known and still has to

be improved on quantum computers [146]. Several al-

gorithms for imaginary time evolution on quantum com-

puters, both with and without ansatz dependency, have

been introduced in recent years. In particular, the Quan-

tum Imaginary Time Evolution (QITE) algorithm applies

a unitary operation to simulate imaginary time evolution

and has been performed to simulate energy and mag-

netism in the Transverse Field Ising Model (TFIM) [147],

the chiral condensate in NJL model [148] and so on.

This study, along with previous studies, demonstrates

that NISQ quantum computers can provide consistent

and correct answers to physical problems that cannot be

solved eﬃciently or eﬀectively using classical computing

algorithms, indicating promising future applications of

quantum computing in non-perturbative QCD and be-

yond.

The remainder of this paper is organized as follows:

In Sec. II, we provide a brief description of the (1 + 1)

dimensional NJL model and the QITE algorithm used

for the quantum simulation. In Sec. III, we show the

analytic calculations of chiral condensate and chirality

charge density at ﬁnite temperature, baryon and chiral

chemical potentials. We then present and discuss our

numerical results from the quantum simulation in com-

parison with analytical computations and exact diago-

nalization results in Sec. IV. Finally, our conclusions are

summarized in Sec. V.

II. BACKGROUND

In this section, we ﬁrst brieﬂy introduce the (1 +

1)-dimensional Nambu-Jona-Lasinio (NJL) model, and

present the lattice discretization of the NJL Hamiltonian.

Next, we provide a brief introduction of the QITE algo-

rithm used for the quantum simulation.

A. The NJL model in (1 + 1) dimensions

The NJL model was deﬁned in [59,60] with the La-

grangian density

LNJL =¯

ψ(i/

∂−m)ψ+g(¯

ψψ)2+ ( ¯

ψiγ5ψ)2,(1)

where mand grepresent the bare quark mass and cou-

pling constant, respectively, and /

∂≡γµ∂µ. The explicit

representation of the (1 +1)-dimensional Cliﬀord algebra

{γµ, γν}= 2ηµν used in this work is

γ0=Z, γ1=−iY, γ5=γ0γ1=−X(2)

where the Pauli gates are

X=



0 1

1 0

, Y =



0−i

i0

, Z =



1 0

0−1

(3)

A simpliﬁed version of the NJL Model, the Gross-Neveu

(GN) model [149], is given by

L=¯

ψ(i/

∂−m)ψ+g(¯

ψψ)2.(4)

To study the chiral phase transition and chirality imbal-

ance in the GN model, we introduce additional terms re-

lated to non-zero chemical potential µand chiral chemical

potential µ5, which mimics the chiral imbalance between

right- and left-chirality quarks coupled with the chiral-

ity charge density operator n5=¯

ψγ0γ5ψ. Therefore, the

modiﬁed Lagrangian is

L=¯

ψ(i/

∂−m)ψ+g(¯

ψψ)2+µ¯

ψγ0ψ+µ5¯

ψγ0γ5ψ . (5)

In our previous work [148], we have studied the behavior

of the chiral condensate h¯

ψψiat µ5= 0 with ﬁnite and

non-zero temperature Tand chemical potential µ. The

Hamiltonian H=iψ†∂0ψ− L corresponding to Eq. (5)

is given by

H=¯

ψ(iγ1∂1+m)ψ−g(¯

ψψ)2−µ¯

ψγ0ψ

−µ5¯

ψγ0γ5ψ . (6)

For clariﬁcation, when we mention “NJL model” in this

work, we refer to the Hamiltonian given in Eq. (6).

As in our previous work [148], we ﬁrst use a stag-

gered fermion ﬁeld χ2n, χ2n+1 to discretize the Dirac

fermion ﬁeld ψ(x). With the lattice spacing aand

n= 0,··· , N/2−1 where Nis an even integer, one

has [150–155]

ψ(x) = 1

√a



χ2n

χ2n+1

.(7)

Therefore, one obtains the following discrete approxima-

tions of the various operators appearing in the Hamil-

tonian H=Rdx Hwhere periodic boundary conditions

are considered,

Zdx ¯

ψiγ1∂1ψ=a

N/2−1

n=0

ψ†

niγ5∂1ψn

=−i

2aN−2

n=0 χ†

nχn+1 −χ†

n+1χn

+χ†

N−1χ0−χ†

0χN−1,(8)

Zdx ¯

ψψ =a

N/2−1

n=0

ψ†

nγ0ψn=

N−1

n=0

(−1)nχ†

nχn,

(9)

Zdx(¯

ψψ)2=a

N/2−1

n=0

(ψ†

nγ0ψn)2

N/2−1

n=0 χ†

2nχ2n−χ†

2n+1χ2n+12

=−2

N/2−1

n=0 χ†

2nχ2nχ†

2n+1χ2n+1

N−1

n=0 χ†

nχn2,(10)

Zdx ¯

ψγ0ψ=a

N/2−1

n=0

ψ†

nψn=

N−1

n=0

χ†

nχn,(11)

Zdx ¯

ψγ0γ5ψ=a

N/2−1

n=0

ψ†

nγ5ψn

=−

N/2−1

n=0 χ†

2nχ2n+1 +χ†

2n+1χ2n.

(12)

Subsequently, the Hamiltonian in Eq. (6) becomes

H=Zdx¯

ψ(m+iγ1∂1−µγ0−µ5γ0γ5)ψ−g(¯

ψψ)2

N−1

n=0

(−1)nχ†

nχn−i

2aN−2

n=0 χ†

nχn+1 −χ†

n+1χn

+χ†

N−1χ0−χ†

0χN−1−µ

N−1

n=0

χ†

nχn

+µ5

N/2−1

n=0 χ†

2nχ2n+1 +χ†

2n+1χ2n−g

N−1

n=0 χ†

nχn2

+2g

N/2−1

n=0 χ†

2nχ2nχ†

2n+1χ2n+12.(13)

In order to implement the Hamiltonian to a quantum cir-

cuit, we write down the spin representation of the Hamil-

tonian using the Jordan-Wigner transformation [156],

χn=Xn−iYn

n−1

µ=0

(−iZµ),(14)

where Xn, Ynand Znare the Pauli-X, Y and Zma-

trices acting on the n-th lattice site. In such spin rep-

resentation, the discrete approximations of the relevant

operators are then given by

Zdx ¯

ψiγ1∂1ψ=

N−2

n=0

4a(XnXn+1 +YnYn+1)

+(−1)N/2

4a(XN−1X0+YN−1Y0)

N−2

i=1

Zi,(15)

Zdx ¯

ψψ =

N−1

n=0

(−1)nZn

2,(16)

Zdx(¯

ψψ)2=−1

N/2−1

n=0

(1+Z2n)(1+Z2n+1)

N−1

n=0

(1+Zn),(17)

Zdx ¯

ψγ0ψ=

N−1

n=0

2,(18)

Zdx ¯

ψγ0γ5ψ=1

N/2−1

n=0

(X2nY2n+1 −Y2nX2n+1).(19)

In Eqs. (15) and (17), we have imposed periodic bound-

ary conditions. With the relations in Eqs. (15)–(19), we

decompose the total (1 + 1)-dimensional NJL Hamilto-

nian into 6 pieces, writing H=

j=1

Hjwith

H1=

N/2−1

n=0

4a(X2nX2n+1 +Y2nY2n+1),(20)

H2=

N/2−1

n=1

4a(X2n−1X2n+Y2n−1Y2n) (21)

+(−1)N/2

4a(XN−1X0+YN−1Y0)

N−2

i=1

Zi,

H3=m

N−1

n=0

(−1)nZn,(22)

H4=g

2aN/2−1

n=0

(1+Z2n)(1+Z2n+1)−

N−1

n=0

(1+Zn)

(23)

H5=−µ

N−1

n=0

Zn,(24)

H6=−µ5

N/2−1

n=0

(X2nY2n+1 −Y2nX2n+1).(25)

Finally, with the decomposition of the Hamiltonian

shown in Eqs. (20)–(25), we are able to perform the

Suzuki-Trotter decomposition [157,158] to study the ef-

fects of the chiral chemical potential µ5on the ﬁnite tem-

perature properties of the chiral condensate h¯

ψψiand chi-

rality charge density n5of the (1 + 1)-dimensional NJL

model on a quantum simulator.

B. Quantum imaginary time evolution algorithm

In this section, we introduce the quantum imaginary

time evolution (QITE) algorithm [159], which is use for

evaluating the temperature dependence of the NJL model

for various values of the baryochemical potential µand

chiral chemical potential µ5. As pointed out in [159],

compared with other techniques for quantum thermal

averaging procedures [160–163], the QITE algorithm is

advantageous in generating thermal averages of observ-

ables without any ancillae or deep circuits. Moreover, the

QITE algorithm is more resource-eﬃcient and requires ex-

ponentially less space and time in each iteration than its

classical equivalents.

Generally, for a given Hamiltonian H, one can approx-

imate the (Euclidean) evolution operator e−βH by apply-

ing the Suzuki-Trotter decomposition [157,158]

e−βH =e−∆βH N+O(∆β2),(26)

where ∆βis a chosen imaginary time step and N=β/∆β

is the number of iterations needed to reach imaginary

time β= 1/T with temperature T. However, since the

evolution operator e−∆βH is not unitary, it cannot be

implemented as a sequence of unitary quantum gates. In

order to compute the Euclidean time evolution of a state

|Ψion a quantum computer, one needs to approximate

the action of the operator e−∆βH by some unitary oper-

ator. Fortunately, the QITE algorithm provides a proce-

dure for doing this.

In the QITE algorithm, to approximate the Euclidean

time evolution of |Ψi, a Hermitian operator Ais intro-

duced such that the eﬀect of the non-unitary operator

e−∆βH on a quantum state |Ψiis replicated by the uni-

tary operator e−i∆βA, namely2

pc(∆β)e−∆βH |Ψi ≈ e−i∆βA |Ψi,(27)

where the normalization c(∆β) = hΨ|e−2∆βH |Ψi.

When ∆βis very small, one is able to expand Eq. (27)

up to O(∆β), truncating after the ﬁrst nontrivial term.

Then at imaginary time β, the change of the quantum

states under the operators e−∆βH and e−i∆βA per small

imaginary time interval ∆βcan be represented by

|∆ΨH(β)i=1

∆β 1

pc(∆β)e−∆βH |Ψ(β)i−|Ψ(β)i!,

(28)

|∆ΨA(β)i=1

∆βe−i∆βA |Ψ(β)i−|Ψ(β)i,(29)

As proposed in [159], to determine the Hermitian opera-

tor A, we ﬁrst parameterize it in terms of Pauli matrices

as below

A(a) = X

aµˆσµ.(30)

Here ˆσµ=Qlσµl,l is a Pauli string and the subscript

µof aµlabels the various Pauli strings. To evaluate the

Hermitian operator A, we need to minimize the objective

function F(a) deﬁned by

F(a) =|||∆ΨH(β)i−|∆AΨ(β)i||2(31)

=|| |∆ΨH(β)i ||2+X

µ,ν

aνaµhΨ(β)|ˆσ†

νˆσµ|Ψ(β)i

+iX

aµ

pc(∆β)hΨ(β)|Hˆσµ−ˆσ†

µH|Ψ(β)i.

The ﬁrst term || |∆ΨH(β)i ||2is irrelevant to aµ. Thus,

we take the derivative with respect to aµand set it equal

to zero, yielding the linear equation (S+ST)a=b,

where the matrix Sand vector bare deﬁned by

Sµν =hΨ(β)|ˆσ†

νˆσµ|Ψ(β)i,(32)

bµ=−i

pc(∆β)hΨ(β)|Hˆσµ+ ˆσ†

µH|Ψ(β)i.(33)

From this equation, we are able to solve for aµand

evolve an initial quantum state under the unitary opera-

tor e−i∆βA to any imaginary time βby Trotterization,

|Ψ(β)i=e−i∆βAN|Ψ(0)i+O(∆β).(34)

In the remainder of this section, we compare the perfor-

mance of the QITE algorithm to the Variational Quantum

2Recall that quantum states are represented by rays {α|Ψi:α∈

C}in a Hilbert state, not by the vectors themselves, since the

normalization/phase of the state vectors are nonphysical.

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StudyingchiralityimbalancewithquantumalgorithmsAlexanderM.Czajka,1,2,Zhong-BoKang,1,2,3,4,yYuxuanTee,1,zandFanyiZhao1,2,3,x1DepartmentofPhysicsandAstronomy,UniversityofCalifornia,LosAngeles,CA90095,USA2ManiL.BhaumikInstituteforTheoreticalPhysics,UniversityofCalifornia,LosAngeles,CA90095,USA3Centerf...

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Studying chirality imbalance with quantum algorithms Alexander M. Czajka1 2Zhong-Bo Kang1 2 3 4yYuxuan Tee1zand Fanyi Zhao1 2 3x 1Department of Physics and Astronomy University of California Los Angeles CA 90095 USA

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