QUANTUM WALKS ON SIMPLEXES AND MULTIPLE PERFECT STATE TRANSFER HIROSHI MIKI

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QUANTUM WALKS ON SIMPLEXES AND MULTIPLE

PERFECT STATE TRANSFER

HIROSHI MIKI

Department of Mathematical Sciences, Faculty of Science and

Engineering, Doshisha University, Kyotanabe City, Kyoto, Japan

SATOSHI TSUJIMOTO

Graduate School of Informatics, Kyoto University, Sakyo-Ku, Kyoto,

606 8501, Japan

DA ZHAO

Graduate School of Informatics, Kyoto University, Sakyo-Ku, Kyoto,

606 8501, Japan

Abstract. In this paper, we study quantum walks on the exten-

sion of association schemes. Various state transfers can be achieved

on these graphs, such as multiple state transfer among extreme

points of a simplex, fractional revival on subsimplexes. Since only

few examples of multiple (perfect) state transfer are known, we

aim to make some additions in this collection.

1. Introduction

Given a graph Γ = (X, E), which might be weighted or oriented,

the continuous-time quantum walk on this graph is given by U(t) =

exp(−itA), where Ais a Hermitian matrix associated to the graph Γ.

This notion was raised by Farhi and Guttman [11] to develop quantum

E-mail addresses:hmiki@mail.doshisha.ac.jp, tsujimoto.satoshi.5s@kyoto-u.jp,

zhao.da.77r@st.kyoto-u.ac.jp.

2020 Mathematics Subject Classiﬁcation. 05C50, 15A16, 81P45.

Key words and phrases. quantum walk, perfect state transfer, association

scheme, extension.

arXiv:2210.13106v2 [quant-ph] 27 Jul 2023

2QUANTUM WALKS ON SIMPLEXES AND MULTIPLE PERFECT STATE TRANSFER

algorithms. Bose [4] investigated quantum walk on a path for the study

of quantum information transmission.

Let |ex⟩be the characteristic vector for the vertex x∈X, and let

V=⊕x∈XC|ex⟩be the space on which U(t) acts. Suppose one starts

with a state, say |eu⟩, after evolution by time t, one reaches

U(t)|eu⟩=X

x∈S

cx|ex⟩,(1)

where Sis a subset of Xand Px∈S|cx|2= 1. We say fractional revival

(FR) occurs on Sat time t. Several special cases have been focused in

the research.

•Generation of maximal entanglement (GME).

If S={v, w}and |cv|=|cw|=1

√2for x∈S, then the resulting

state at time tis a maximally entangled state of evand ew.

•Perfect state transfer (PST).

If S={v}, in other words there exists a real constant γsuch

that U(t)|eu⟩=eiγ|ev⟩, then we say that there exists perfect

state transfer from uto vat time t.

•Pretty good state transfer (PGST).

If there exists a real sequence {tk}and a constant γ∈Rsuch

that limk→∞ U(tk)|eu⟩=eiγ|ev⟩, then we say that there exists

pretty good state transfer from uto v.

•Multiple (perfect/pretty good) state transfer (MST) and uni-

versal state transfer (UST).

Let Cbe a subset of X. If for every u, v ∈C, there exists per-

fect/pretty good state transfer from uto v, then we say that

there exists multiple perfect state transfer/multiple pretty good

state transfer in C. In particular, if C=X, then we say that

there exists universal state transfer in the graph Γ.

•Zero transfer (ZT).

If there exists v∈Xsuch that for every t∈R,⟨ev|U(t)|eu⟩=

0, then we say that there is zero transfer between uand v. In

other words, the state on ucan never be seen on vat any time.

After Bose, the study turns to general graphs as perfect state transfer

between antipodal points of an unweighted path only occurs when the

length is 2 or 3. Christandl et al. [7] showed that perfect state transfer

can be achieved between antipodal points of a hypercube of arbitrary

dimension, which gives the perfect state transfer between the two ends

of a weighted path through projection. As perfect state transfers are

hard to obtain, the notation of pretty good state transfer was intro-

duced by Godsil [13]. It was shown by Kay [16] that in an unoriented

QUANTUM WALKS ON SIMPLEXES AND MULTIPLE PERFECT STATE TRANSFER3

graph, if one ﬁxes the initial state on u, then one can only achieve per-

fect state transfer between two vertices uand v, but not a third vertex.

Recently Chaves et al. [6] discussed ‘Why and how to add direction to

a quantum walk’. They also studied the phenomenon of zero transfer.

The ambitious universal state transfer was constructed by Cameron et

al. [5] in oriented graphs, which is impossible for unoriented graphs.

Graphs that carry universal perfect state transfer and universal pretty

good state transfer are partially constructed and characterized in [5;8].

However, these graphs are dense in the sense that almost every pair

of vertices are adjacent. Multiple state transfer, as a relaxation of

universal state transfer, was proposed by Godsil and Lato in [14].

In this paper, we focus on multiple perfect state transfer (MPST).

Since very few examples are known and the distance of transfer in

known examples is small, we aim to add new models that will give

multiple perfect state transfer with arbitrary length. In order to achieve

that end, we employ the techniques of commutative association schemes.

More speciﬁcally, from the small graph where MPST can be observed,

we will show that we can obtain the model that provide MPST by con-

sidering the symmetric tensor product of the underlying association

schemes. Furthermore, the multivariate Krawtchouk polynomials play

an important in explicit calculations. For example, in [13] Godsil and

Lato showed that the directed 3-gons carry multiple perfect state trans-

fer. Since the directed 3-gon is an adjacency graph of a commutative

association scheme, we obtain multiple perfect state transfer on trian-

gle simplex of arbitrary length by taking symmetric tensor product of

the association scheme.

The paper is organized as follows.

In Section 2, we introduce the deﬁnition and basic properties of asso-

ciation schemes. The extension of association schemes is emphasized.

In Section 3, we analyze quantum walks on extension of association

schemes. In Section 4, we apply the analysis in Section 3to several

base association schemes, which gives various examples of state trans-

fer, such as perfect state transfer among extreme points of a simplex,

multiple perfect state transfer among extreme points of a simplex, and

fractional revival on subsimplexes of a simplex.

2. Association scheme

In this section, we recall the deﬁnition and basic properties of com-

mutative association scheme. We will emphasize the extension of asso-

ciation scheme, which will be used in later construction of state trans-

fers.

4QUANTUM WALKS ON SIMPLEXES AND MULTIPLE PERFECT STATE TRANSFER

2.1. Commutative association scheme. A commutative associa-

tion scheme X= (X, {Ri}d

i=0) on Xof class dconsists of the rela-

tions Ri(0 ≤i≤d) whose adjacency matrices Ai(0 ≤i≤d) satisfy

the following conditions.

(1) A0=I;

(2) A0+A1+··· +Ad=J;

(3) A⊤

i=Ai′for some i′∈ {0,1, . . . , d};

(4) AiAj=AjAi=Pd

k=0 pk

i,j Ak.

The Bose-Mesner algebra A=⟨A0, A1, . . . , Ad⟩of a commutative as-

sociation scheme X= (X, {Ri}d

i=0) has another basis A=⟨E0, E1, . . . , Ed⟩

composed of primitive idempotents. They satisfy the following proper-

ties.

(1) E0=1

|X|J;

(2) E0+E1+··· +Ed=I;

(3) E⊤

i=Ei=Eˆ

ifor some ˆ

i∈ {0,1, . . . , d};

(4) Ei◦Ej=1

|X|Pd

k=0 qk

i,j Ek

Example 1. Let A0=1

1and A1=1

1. The primitive idem-

potents are given by E0=1

21 1

1 1and E1=1

21−1

−1 1. Then

A0E0=E0,A0E1=E1,A1E0=E0, and A1E1=−E1. In other

words AjEλ= (−1)j·λEλfor j, λ ∈ {0,1}. This association scheme is

called the trivial association scheme X2of size 2. Indeed this associ-

ation scheme is the underlying association scheme of a path of length

Example 2. Let Z=





...







be the circulant matrix of size

n. Let Ak=Zkfor k= 0,1, . . . , n −1and let Eℓ= (ζℓ(j−k)

√n)0≤j,k≤n−1,

where ζis the n-th root of unity. Then AkEℓ=ζkℓEℓfor k, ℓ ∈

{0,1, . . . , n −1}. This association scheme is called the association

scheme Znof directed n-gon.

Let X= (X, {Ri}d

i=0) be an association scheme on Xof class d. Let

Ai(0 ≤i≤d) be the adjacency matrices of X, and let Ei(0 ≤i≤d)

be the primitive idempotents of X. The transition matrices, the ﬁrst

eigenmatrix P= (Pi,j )0≤i,j≤dand the second eigenmatrix Q= (Qi,j )0≤i,j≤d,

QUANTUM WALKS ON SIMPLEXES AND MULTIPLE PERFECT STATE TRANSFER5

between the two bases are given by

Ai=

j=0

Pj,iEj, Ei=1

|X|

j=0

Qj,iAj(2)

In particular we have the valencies ki=P0,i and the multiplicities mi=

Q0,i. The two eigenmatrix are related by P Q =|X|Iand Pi,j

kj=Qj,i

mi.

We denote the cosine matrix of these values by C= (ci,j )0≤i,j≤d=

Pi,j

kj0≤i,j≤d. The transition between these two bases is very useful in

our computation. The reader is referred to [1] for a thorough discussion

of association scheme.

2.2. Extension of association schemes. In this subsection, we ex-

hibit two operations to construct new association schemes from old

ones. The ﬁrst operation is based on tensor product and the second

operation is based on group action. Combining these two operations,

we obtain the so-called extension of association schemes, also known

as the symmetric tensor product of association schemes [10].

Suppose X= (X, {Ri}d

i=0) is an association scheme on Xof class d

and Y= (Y, {Si}e

i=0) is an association scheme on Yof class e. Let

Ai(0 ≤i≤d) and Bj(0 ≤j≤e) be their adjacency matrices respec-

tively. Then

Ai⊗Bj(0 ≤i≤d, 0≤j≤e)

gives an association scheme on X×Yof class (d+1)(e+1)−1, denoted

by X⊗Y. In particular one can take tensor product of an association

scheme Xwith itself, which gives X⊗2=X⊗X. By composition and

associativity, we can deﬁne X⊗N=X⊗(N−1) ⊗X=X⊗ ··· ⊗ X. The

Bose-Mesner algebra of X⊗Nis denoted by A⊗N.

Let G≤Aut(A) be a subgroup of the automorphisms of the Bose-

Mesner algebra Aof a commutative association scheme X. Then the G-

invariant matrices in Aform a Bose-Mesner algebra of a fusion scheme

(also called subscheme) of X.

Let us consider the group action of the symmetric group SNon the

Bose-Mesner algebra of X⊗Ngiven by

g(Ai1⊗ ··· ⊗ AiN) = Aig(1) ⊗ ··· ⊗ Aig(N)

for every g∈SN. Here Ai1, Ai2, . . . , AiNare taken from A0, A1, . . . , Ad.

It can be easily veriﬁed that these actions give automorphisms of A⊗N.

The fusion scheme, denoted by Sym(X, N), is called the extension of

X, also known as the N-th symmetric tensor product of X. A typical

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QUANTUMWALKSONSIMPLEXESANDMULTIPLEPERFECTSTATETRANSFERHIROSHIMIKIDepartmentofMathematicalSciences,FacultyofScienceandEngineering,DoshishaUniversity,KyotanabeCity,Kyoto,JapanSATOSHITSUJIMOTOGraduateSchoolofInformatics,KyotoUniversity,Sakyo-Ku,Kyoto,6068501,JapanDAZHAOGraduateSchoolofInformatics,Kyoto...

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