Parity-time symmetric holographic principle Xingrui Song1and Kater Murch1 1Department of Physics Washington University St. Louis Missouri 63130 USA

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Parity-time symmetric holographic principle

Xingrui Song1and Kater Murch1

1Department of Physics, Washington University, St. Louis, Missouri 63130, USA

(Dated: October 5, 2022)

Originating from the Hamiltonian of a single qubit system, the phenomenon of the avoided level

crossing is ubiquitous in multiple branches of physics, including the Landau-Zener transition in

atomic, molecular and optical physics, the band structure of condensed matter physics and the

dispersion relation of relativistic quantum physics. We revisit this fundamental phenomenon in the

simple example of a spinless relativistic quantum particle traveling in (1+1)-dimensional space-time

and establish its relation to a spin-1/2 system evolving under a PT -symmetric Hamiltonian. This

relation allows us to simulate 1-dimensional eigenvalue problems with a single qubit. Generalizing

this relation to the eigenenergy problem of a bulk system with Nspatial dimensions reveals that its

eigenvalue problem can be mapped onto the time evolution of the edge state with (N−1) spatial

dimensions governed by a non-Hermitian Hamiltonian. In other words, the bulk eigenenergy state

is encoded in the edge state as a hologram, which can be decoded by the propagation of the edge

state in the temporal dimension. We argue that the evolution will be PT -symmetric as long as the

bulk system admits parity symmetry. Our work ﬁnds the application of PT -symmetric and non-

Hermitian physics in quantum simulation and provides insights into the fundamental symmetries.

In the 1980s, Richard Feynman envisioned the advan-

tage of using quantum mechanical systems to simulate

quantum physics [1]. As most formulations of quantum

mechanics consider the systems to be governed by Hermi-

tian Hamiltonians, the community established the theory

of quantum computation based on combinations of uni-

tary gate operations [2–5]. However, after decades of

eﬀort toward implementing quantum computers, we re-

alize that even the most highly controlled quantum sys-

tems are open quantum systems [6–8]. These open quan-

tum systems are non-unitary, suﬀering from the residual

coupling with the environment causing dissipation and

decoherence. In contrast to the stereotype that such non-

unitary eﬀects of evolution are always harmful, dissipa-

tion is now considered an important resource for quan-

tum technologies, with extensive applications in quan-

tum control, sensing, and simulation [9–17]. In this let-

ter, we extend Feynman’s logic by taking advantage of

open quantum systems as an eﬃcient resource for quan-

tum simulation. In particular, we show how the open

quantum system evolution described by a non-Hermitian

Hamiltonian [15, 18–30] allows one to map the eigen-

value problem of a 1-dimensional Hermitian system onto

the time evolution of a qubit. Extending this reason-

ing suggests that an N-dimensional Hermitian bulk sys-

tem can be mapped onto the non-Hermitian time evolu-

tion of a (N−1)-dimensional edge system. In parallel

with the research highlighting the bulk-boundary corre-

spondence of non-Hermitian systems [31–35], our work

ﬁnds the new application of these concepts to reduce the

quantum resources required for encoding the spatial de-

grees of freedom in quantum simulation for Hermitian

systems. We present two speciﬁc examples that illus-

trate the power of this concept: First, we propose an ex-

perimental scheme exploiting this relationship to perform

quantum simulation of a scattering problem with reduced

quantum resources. Second, we examine the system of a

Kitaev chain and show how the non-Hermitian evolution

can model the physics of Majorana zero modes in one

of the simplest topologically nontrivial models [36–40].

After demonstrating our method with two examples, we

discuss the relation between Parity symmetry and Parity-

Time (PT )-symmetry and generalize the result to show

how PT -symmetric evolution of the edge state generates

the solution of the eigenvalue problem of a bulk Hermi-

tian system.

The eigenvalue problem and PT -symmetric evolu-

tion—We start by considering the simple case of a mass-

less Fermion moving in one dimension, which has a linear

dispersion relation as shown in Fig. 1(a). The right and

left moving particles form a two-state system {|Li,|Ri},

which can be mapped onto a qubit. In this gapless limit,

the Dirac equation is uncoupled. Once the two compo-

nents are coupled, the particle acquires a mass resulting

in an avoided crossing (Fig. 1(b)). In this case, the parti-

cle at rest is given by an equal superposition of the states

|Liand |Ri. In general, the state of a traveling particle is

given by some linear combination of |Liand |Ri. Partic-

ularly, the state |±i =1

√2(|Li ± |Ri) represents the par-

ticle at rest with eigenenergy equal to ±m. This picture

implicitly incorporates the idea of separation of variables,

where we downgrade the momentum from an operator to

a parameter −i∂x→k. Hence we can write the corre-

sponding Schr¨odinger equation for the particle with the

time coordinate treated as the independent variable,

i∂tψL

ψR=−k m

m k ψL

ψR=H(k)ψL

ψR,(1)

where we have set ~=c= 1 for simplicity. This ap-

proach, however, is not the only option, and may not be

the most convenient approach under some circumstances.

Indeed, the Dirac equation does not prefer a speciﬁc time

arXiv:2210.01128v1 [quant-ph] 3 Oct 2022

FIG. 1. Dispersion relation, avoided crossings, and

qubits. (a) The linear dispersion relation of the left-moving

and right-moving massless Fermions. These two states are la-

beled by |Liand |Ri, respectively. They can be mapped onto

a qubit, represented north and south pole on a Bloch sphere.

(b) The dispersion relation of massive Dirac Fermion. Once

the coupling is introduced, the dispersion relation exhibits

an avoided crossing. The two states represent the particle at

rest with ±meigenenergy are represented on the Bloch sphere

along the ±Xaxis.

or spatial coordinate.

Instead, we consider replacing the energy, ωwith a

parameter and treat the spatial coordinate as the inde-

pendent variable,

i∂xψL

ψR=ω−m

m−ωψL

ψR=Heﬀ (ω)ψL

ψR.(2)

The matrix that appears here as an eﬀective Hamilto-

nian, Heﬀ , is clearly non-Hermitian, and embodies the

well-known concept of the PT -symmetry breaking tran-

sition [18, 20–27, 29, 30], as is shown in Fig. 2. Solving

the characteristic polynomial for the eigenvalues kof this

eﬀective Hamiltonian,

k=±pω2−m2(3)

we can see that if ωis a real number, the solution for

kcan alternatively be real, or imaginary. This em-

bodies the regions of respectively unbroken (Fig. 2(a))

and broken PT -symmetry (Fig. 2(c)), with an excep-

tional point (EP) occurring for |ω|=m(Fig. 2(b))

[15, 23, 27, 28, 41, 42]. Additionally, since Heﬀ is non-

Hermitian, its eigenvectors are generally non-orthogonal.

This establishes the principle that we employ to har-

ness quantum evolution in a resource-eﬃcient manner for

quantum simulation: we now let the laboratory time co-

ordinate represent the spatial coordinate of Eq. 2, such

FIG. 2. PT -symmetry. The parameter space is catego-

rized by the status of the PT -symmetry. In each case, the

horizontal dashed line represents the eigenenergy and the ar-

rows represent the eigenvectors of Heﬀ . (a) When |ω|> m,

the PT -symmetry is unbroken. The eigenmomenta of Heﬀ are

a pair of real numbers with opposite signs. (b) When |ω|=m,

the system is at the exceptional point with the eigenmomenta

coalescing to 0. (c) When |ω|< m, the PT -symmetry is bro-

ken. The eigenmomenta of Heﬀ are a pair of purely imaginary

numbers with opposite signs.

that the real-time evolution of a qubit captures the spa-

tial solution of a chosen problem. We refer to this prin-

ciple as the qubit hologram because the qubit encodes

the wavefront of the spatial solution. In general, we are

interested in the scattering or bound states of a given po-

tential V(x), which can now be obtained from the time

evolution under,

H(t) = E−V(t)−m

m−E+V(t),(4)

where we have replaced ωwith Eto emphasize it now

represents the eigenenergy of the problem.

Example 1: Scattering phase shifts—The problem of

solving the cross section of an elastic neutron scattering

oﬀ an atomic nucleus described by an optical potential

can be reduced to a 1D problem through partial wave

decomposition (Fig. 3(a)) [43], which gives a diﬀerential

equation for the radial wave function ulj (r) for the orbital

angular momentum quantum number land total angular

momentum number j,

u00

lj (r) + k2−2m

~2Vlj (r)−l(l+ 1)

r2ulj (r) = 0,(5)

where Vlj (r) is given by the global optical model CH89

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Parity-timesymmetricholographicprincipleXingruiSong1andKaterMurch11DepartmentofPhysics,WashingtonUniversity,St.Louis,Missouri63130,USA(Dated:October5,2022)OriginatingfromtheHamiltonianofasinglequbitsystem,thephenomenonoftheavoidedlevelcrossingisubiquitousinmultiplebranchesofphysics,includingtheLanda...

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Parity-time symmetric holographic principle Xingrui Song1and Kater Murch1 1Department of Physics Washington University St. Louis Missouri 63130 USA

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