PHYSICAL REVIEW A 84, 012324 (2011)

Limit theorems for quantum walks associated with Hadamard matrices

Clement Ampadu*

31 Carrolton Road, Boston, MA 02132, USA(Received 24 March 2011; published 20 July 2011)

We study a one-parameter family of discrete-time quantum walk models on Z and Z2 associated with theHadamard walk. Weak convergence in the long-time limit of all moments of the walker’s pseudovelocity onZ and Z2 is proved. Symmetrization on Z and Z2 is theoretically investigated, leading to the resolution of theKonno-Namiki-Soshi conjecture in the special case of symmetrization of the unbiased Hadamard walk on Z. Anecessary condition for the existence of a phenomenon known as localization is given.

DOI: 10.1103/PhysRevA.84.012324 PACS number(s): 03.67.Ac

I. INTRODUCTION

The Hadamard walk plays a key role in studies of thequantum walk, thus the generalization of the Hadamard walkis one of the many fascinating challenges. As is well known,the simplest and well studied example of a quantum walk,Ref. [1] for example, is the Hadamard whose unitary matrix isdefined by

H = 1√2

[1 1

1 −1

]. (1.1)

The dynamics of this walk corresponds to the symmetricrandom walk in the classical case. A generalization of (1.1)that leads to symmetric random walks and have been studiedby a number of authors, including those in Refs. [2–4], is thefollowing:

H (η,φ,ψ) = eiη

2

[ei(φ+ψ) e−i(φ−ψ)

ei(φ−ψ) e−i(φ+ψ)

]. (1.2)

In this paper we study certain generalizations of theHadamard walk in one and two dimensions and clarifytheoretically the properties of localization and symmetrization.The paper begins in Sec. II, where the stochastic process of theone- and two-dimensional quantum walk is introduced; therethe generator of the process in the generalized Hadamard walkin both the one- and two-dimensional models is given. Becausethe process of building the stochastic model is intimatelyrelated to the work of the authors in Ref. [5], we follow theconvention of putting a superscript T on the left-hand sidethroughout to denote the transpose of a vector or matrix. InSec. III we prove convergence in the long-time limit of thewalker’s pseudovelocity in one dimension, and do the samefor the x and y components of the walker’s pseudovelocityin two dimensions. Since a common method of proof inthe analysis of the long-time behavior of quantum walkswhose time evolution is given by unitary transformations isto diagonalize the time-evolution matrix, we use the methodgiven by the authors in Ref. [5] which is essentially due to thosein Ref. [2], which have been successfully used by a numberof authors including those in Refs. [6] and [7]. Due to theunitary nature of the time-evolution matrix, the convergencetheorems are in a sense weak [8]. We also give in Sec. IIIanalytic expressions for the probability distribution in both the

*drampadu@hotmail.com

one- and two-dimensional models. This first requires express-ing the dynamics of the Hadamard walk considered in thispaper in terms of difference equations [9–11]. In Sec. IV westudy localization in the generalized Hadamard walk model. Ineither the one- or two-dimensional models we wish to answerthe following question:

If, say, a quantum walker which could be a quantumparticle exists only at one site initially on the line in the one-dimensional model or in the xy plane in the two-dimensionalmodel, will the quantum walker remain trapped with highprobability near the initial position?

The answer to this question has been investigated by variousauthors in varying contexts under the pseudonym “localiza-tion.” For example, Inui et al. [12] studied a generalizedHadamard walk in one dimension with three inner statesand concluded that the quantum walker (quantum particle)is trapped near the origin with high probability. Watabeet al. [5], on the other hand, were able to control localizationaround the origin for a one-parameter family of discrete-timequantum walk models on the square lattice, which includedthe Grover walk, which is related to the Grover’s algorithmin computer science, as a special case. For the Grover walkitself in two dimensions, Inui et al. [13] were able to showlocalization analytically. Liu and Pentulante [14] were able tooffer a theoretical explanation for localization in the case ofdiscrete quantum random walks on a linear lattice with twoentangled coins. Following the localization studies by theseand many other authors in the literature, in this section weoffer a theoretical criterion for localization with respect to thegeneralized Hadamard walk models considered in this paper.If the theoretical criterion holds, then we should observe incomputer simulations oscillatory behavior in the distributionof the pseudovelocity in both the one- and two-dimensionalmodels. In particular, as the time step increases, a spikein the oscillatory behavior should be observed. However,convergence of any moments given by either Theorem 3.5(two-dimensional model) or Theorem 3.2 (one-dimensionalmodel) in this paper implies that, if we smear out the oscillatorybehavior, the averaged values of the distribution will becaptured by the derived probability density function in bothcases. Due to the expected dynamics as seen by previousauthors, we first give the criterion for localization anywhereon the line or in the plane by expressing the probability oflocalization in terms of the intensity of the Dirac δfunction,by making use of the stationary probability distribution inboth dimensions. These criteria are given in Theorems 4.1

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CLEMENT AMPADU PHYSICAL REVIEW A 84, 012324 (2011)

and 4.2, respectively. Due to the similar nature in the proofof Theorem 4.2 to that of Theorem 4.1, we omit it, butremark that it involves invoking the mean value property asit pertains to double integrals [15]. In Sec. V, symmetry ofthe distribution in both the one- and two-dimensional modelsis considered theoretically, and the necessary and sufficientconditions for symmetrization are given. Along the way wesettle the Konno-Namiki-Soshi conjecture as it relates tothe symmetrization of the unbiased Hadamard walk in onedimension. Section VI is devoted to the conclusions, and therewe leave the reader with an open problem.

II. THE QUANTUM WALK MODELS

A. The one-dimensional model

Let R = {x : x ∈ Z}, whereZ denotes the set of all integersz = {. . . , − 2, − 1,0,1,2, . . .}. Corresponding to the fact thatthere are two nearest-neighbor sites for each site x ∈ R, weassign a two-component wave function

ζ (x,t) =(

φ1(x,t)

φ2(x,t)

)to a quantum walker, each component of which is a complexfunction of location x ∈ R and discrete time t = 0,1,2, . . .. Aquantum coin will be given by a 2 × 2 unitary matrix, U =(Umn)2

m,n=1, and a spatial shift operator on R in the wave-number space x ′ ∈ [−π,π ) by the matrix

V (x ′) =(

eix ′0

0 eix ′

),

where i = √−1. We assume that at the initial time t = 0 thewalker is located at the origin with a two-component quiditT � = ( d1 d2 ) ∈ C2, where C denotes the set of complexnumbers, and

∑2i=1 |di |2 = 1.

Definition 2.2 (Wave function of the walker): Let M(x ′) ≡V (x ′)U . The wave function of the walker at time t isgiven by ζ̂ (x ′,t) = [M(x ′)]t�, t = 0,1,2, . . . in the x ′space.Time evolution in Z is obtained by performing the Fouriertransformation

ζ (x,t) =∫ π

−π

dx ′

2πeix ′x ζ̂ (x ′,t),

where the inverse Fourier transform is given by ζ̂ (x ′,t) =∑x∈Z ζ (x,t)e−ix ′x .Definition 2.3 (Stochastic process of one-dimensional

quantum walk). Let Xt be the x coordinate of the positionof the quantum walker at time t. The probability that wefind the walker at x ∈ R at time t is given by P (x,t) ≡Prob(Xt = x) = T ζ̄ (x,t)ζ (x,t), where the bar denotes com-plex conjugation. The moment of Xt is given by

⟨Xα

t

⟩ =∑x∈Z

xαP (x,t) =∫ π

−π

dx ′

2π

T ¯̂ζ (x ′,t)(

i∂

∂x ′

)α�

ζ (x ′,t)

for α = 0,1,2, . . ..

B. The two-dimensional model

Let M = {(x,y) : x,y ∈ Z}, where Z denotes the set of allintegers Z = {. . . , − 2, − 1,0,1,2, . . .}. Corresponding to the

fact that there are four nearest-neighbor sites for each site(x,y) ∈ M , we assign a four-component wave function

�(x,y,t) =

⎛⎜⎜⎜⎝

ς1(x,y,t)

ς2(x,y,t)

ς3(x,y,t)

ς4(x,y,t)

⎞⎟⎟⎟⎠

to a quantum walker, each component of which is a com-plex function of location (x,y) ∈ M and discrete time t =0,1,2, . . .. A quantum coin will be given by a 4 × 4 unitarymatrix, B = (Bst )4

s,t=1, and a spatial shift operator on M inthe wave-number space (m′,n′) ∈ [−π,π ) × [−π,π ) by thematrix

T (m′,n′) =

⎛⎜⎜⎜⎝

eim′0 0 0

0 e−im′0 0

0 0 ein′0

0 0 0 e−in′

⎞⎟⎟⎟⎠ ,

where i = √−1. We assume that at the initial time t = 0the walker is located at the origin with a four-componentquidit T θ = ( k1 k2 k3 k4 ) ∈ C4, where C denotes the setof complex numbers, and

∑4i=1 |ki |2 = 1.

Definition 2.4 (Wave function of the walker): LetQ(m′,n′) ≡ T (m′,n′)B. The wave function of the walker attime t is given by �̂(m′,n′,t) = [Q(m′,n′)]t θ , t = 0,1,2, . . .

in the (m′,n′) space. Time evolution in M is obtained byperforming the Fourier transformation

�(x,y,t) =∫ π

−π

dm′

2π

∫ π

−π

dn′

2πei(–m′x+n′y) �

�(m′,n′,t),

where the inverse Fourier transform is given by

�̂(m′,n′,t) =∑

(x,y)∈M

�(x,y,t)e−i(m′x+n′y).

Definition 2.5 (Stochastic process of two-dimensionalquantum walk). Let Xt be the x coordinate and let Yt bethe y coordinate of the position of the quantum walker attime t. The probability that we find the walker at (x,y) ∈ M

at time t is given by P (x,y,t) ≡ Prob[(Xt,Yt ) = (x,y)] =T �̄(x,y,t)�(x,y,t), where the bar denotes complex conju-gation. The joint moment of Xt and Yt is given by⟨

Xαt Y

βt

⟩ ≡∑

(x,y)∈M

xαyβP (x,y,t)

=∫ π

−π

dm′

2π

∫ π

−π

dn′

2π

T ¯̂�(m′,n′,t)

×(

i∂

∂m′

)α (i

∂

∂n′

)β

�̂(m′,n′,t)

for α,β = 0,1,2, . . ..

C. Generalizations of the Hadamard walk

In this paper we consider the following generalization ofthe Hadamard walk for the one-dimensional model,

H ∗(p,q) =[√

p√

q√q −√

p

], q = 1 − p, (2.3.3)

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where p ∈ (0, 1). In the two-dimensional model we willconsider the following:

H ∗∗(p,q) =

⎡⎢⎢⎢⎣

p√

pq√

pq q√pq −p q −√

pq√pq q −p −√

pq

q −√pq −√

pq p

⎤⎥⎥⎥⎦ ,

q = 1 − p, (2.3.4)

where p ∈ (0,1). When p = 12 , H ∗(p,q) reduces to the

Hadamard matrix in Eq. 2.3.1, while H ∗∗(p,q) reduces toa two-dimensional generalization of (2.3.1). We close thissection by noting that the generator of the process in theone-dimensional model is

M(x ′) =[ √

p eix ′ √qeix ′

√qe−ix ′ −√

pe−ix ′

](2.3.5)

and in the two-dimensional model it is

Q(m′,n′)

=

⎡⎢⎢⎢⎣

peim′ √pqeim′ √

pqeim′qeim′

√pqe−im′ −pe−im′

qe−im′ −√pqe−im′

√pqein′

qein′ −pein′ √pqein′

qe−in′ −√pqe−in′ −√

pqe−in′pe−in′

⎤⎥⎥⎥⎦ .

(2.3.6)

In both (2.3.5) and (2.3.6), q = 1 − p and p ∈ (0,1).

III. LIMIT DISTRIBUTION OF LONG-TIME BEHAVIOR

A. Weak convergence in the one-dimensional model

We first diagonalize the time-evolution matrix in the one-dimensional model which is given by the matrix in Eq. (2.3.5).The eigenvalues of the matrix in Eq. (2.3.5) can be shown tobe μ1 = eiσ (x ′) and μ2 = −e−iσ (x ′), where σ (x ′) is determinedby the equation

sin σ (x ′) = √p sin(x ′). (3.1.1)

The eigenvectors corresponding to the eigenvalues μj , forj = 1,2, are given by the following column vectors:

�hj (x ′) = Nj

⎛⎜⎝

√pq+√

qμj eix′

q

1

⎞⎟⎠ , (3.1.2)

where Nj is an appropriate normalization factor. Define the2 × 2 unitary matrix by P (x ′) = (�h1 �h2), where �hj (x ′) is theeigenvector corresponding to μj . Put

D =(

μ1 0

0 μ2

),

and then it follows that [P (x ′)]−1M(x ′) P (x ′) = D, and thusM(x ′) is diagonalizable. Since P (x ′) is unitary, it followsthat P (x ′) is invertible and that [P (x ′)]−1 = T P (x ′). Nowrecall that the wave function of the walker at time t inthe one-dimensional model is given by ζ̂ (x ′,t) = [M(x ′)]t�,and it follows from [P (x ′)]−1M(x ′) P (x ′) = D that wecan write [M(x ′)]t = [M(x ′)D{M(x ′)}−1]t . By induction

on t we can show that [M(x ′)]t = [M(x ′)D{M(x ′)}−1]t =M(x ′)Dt [M(x ′)]−1. So the wave function of the walker attime t in the one-dimensional model can be written asζ̂ (x ′,t) = M(x ′) Dt [M(x ′)]−1�. Using sigma notation we canshow that the matrix ζ̂ (x ′,t) = M(x ′) Dt [M(x ′)]−1� can bewritten as

ζ̂ (x ′,t) =2∑

j=1

μtj�hjcj (x ′),

where cj (x ′) ≡ T⇀

hj �. It follows from

ζ̂ (x ′,t) =2∑

j=1

μtj�hjcj (x ′)

that we have(i

∂

∂x ′

)α�

ζ (x ′,t) =(

i∂

∂x ′

)α[eiσ (x ′)t ⇀

h1c1 + (−1)t e−iσ (x ′)t ⇀

h2c2]

for α = 1,2, . . .. By induction on α, we can show that(i

∂

∂x ′

)α�

ζ (x ′,t) = (−1)α−1

[∂

∂x ′ σ (x ′)]α

× [(−1)t⇀

h2c2eiσ (x ′)t − ⇀

h1c1eiσ (x ′)t ]tα

for α = 1,2, . . .. Since P (x ′) is unitary and the eigenvectors⇀

hj have been normalized, it follows from the unitary of

P (x ′) that the vectors⇀

hj are orthonormal since in particularT P (x ′)P (x ′) = I2, where I2 is the 2 × 2 identity matrix, andthus

T hm(x ′)hm′(x ′) ={

1, if m = m′,0, if m �= m′.

Now,

T�

ζ (x ′,t)(

i∂

∂x ′

)α

ζ̂ (x ′,t)

= [(−1)t T �h2c̄2e−iσ (x ′)t − T

⇀

h1c̄1e−iσ (x ′)t ](−1)α−1

×[

∂

∂x ′ σ (x ′)]α

[(−1)t �h2c2eiσ (x ′)t − �h1c1e

iσ (x ′)t ]tα.

(3.1.3)

Since

T hm(x ′) hm′(x ′) ={

1, if m = m′,0, if m �= m′,

and noting that |cj |2 = cj c̄j , it follows that we write theexpression in Eq. (3.1.3) as

T ζ̂ (x ′,t)(

i∂

∂x ′

)α

ζ̂ (x ′,t)

=[

∂

∂x ′ σ (x ′)]α

[|c2|2 + |c1|2](−1)α−1tα. (3.1.4)

Definition 3.1 (Pseudovelocity of quantum walker). For theone-dimensional model this is given as Vt = Xt

t, t = 1,2, . . ..

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CLEMENT AMPADU PHYSICAL REVIEW A 84, 012324 (2011)

Recall that⟨Xα

t

⟩ =∫ π

−π

dx ′

2π

T ¯̂ζ (x ′,t)(

i∂

∂x ′

)α�

ζ (x ′,t).

However, Eq. (3.1.4) above implies we have⟨Xα

t

⟩ =∫ π

−π

dx ′

2π{|c2|2 + |c1|2}(−1)α−1

[∂

∂x ′ σ (x ′)]α

tα.

(3.1.5)

From Eq. (3.1.1) we have σ (x ′) = arcsin[√

p sin(x ′)], andthus

∂

∂x ′ σ (x ′) =√

p cos(x ′)√1 − p sin2 x ′ . (3.1.6)

From (3.1.5) and (3.1.6) we arrive at the main result, theconvergence theorem in the long-time limit of the moment ofthe walker’s pseudovelocity Xt

t.

Theorem 3.2.

limt→∞

⟨(Xt

t

)α⟩

=∫ π

−π

dx ′

2π{|c1|2 + |c2|2}(−1)α−1

[ √p cos(x ′)√

1 − p sin2(x ′)

]α

.

B. Analytic expression for the probability distribution in theone-dimensional model

The dynamics of the one-dimensional model consideredherein can be expressed in terms of difference equations givingthe system of linear equations

φ1(x,t) = eik[√

pφ1(x − 1,t − 1) + √qφ2(x − 1,t − 1)],

(3.2.1)

φ2(x,t) = eik[√

qφ1(x + 1,t − 1) − √pφ2(x + 1,t − 1)],

(3.2.2)

for a given

ζ (x,0) =(

d1

d2

)δx,0,

where k ∈ R,

δx,0 ={

1, if x = 0,

0, otherwise.

φ1 and φ2 are the components of ζ (x,t), and q = 1 − p. Nowapplying the Fourier transform ζ̂ (x ′,t) = ∑

x∈Z φ(x,t) −ix ′x to(3.2.1) and (3.2.2), we obtain

�

ζ 1(x ′,t) = eik[√

pe−ix ′ζ̂1(x ′,t − 1) + √

qe−ix ′ �

ζ 2(x ′,t − 1)],

(3.2.3)�

ζ 2(x ′,t) = eik[√

qeix ′ζ̂1(x ′,t − 1) − √

peix ′ζ̂2(x ′,t − 1)].

(3.2.4)

In matrix form the system consisting of Eqs. (3.2.3) and

(3.2.4) can be written as�

ζ (x ′,t) = eikS(x ′)ζ (x ′,t − 1), where

S(x ′) =(√

pe−ix ′ √qe−ix ′

√qeix −√

peix ′

).

Now applying the Fourier transform ζ̂ (x ′,t) =∑x∈Z φ(x,t)e−ix ′x to the initial condition φ(x,0), we

get the equivalent initial condition(ζ̂1(x ′,0)

ζ̂2(x ′,0)

)=

(φ1(0,0)

φ2(0,0)

).

By induction on t we can show that the wave function of thewalker can be written as ζ̂ (x ′,t) = eitkSt (x ′)ζ̂ (x ′,0). By wayof the relation eiθ = cos θ + i sin θ and properties of the Paulimatrices we can show that

S(x ′) =(√

pe−ix ′ √q e−ix ′

√q eix ′ −√

peix ′

)

admits the following decomposition S(x ′) = cos[θ (x ′)]I +i sin[θ (x ′)]⇀

c(x ′) · ⇀σ , where θ and

⇀

c are real functions of x ′,and the matrix vector ⇀

σ has Pauli components

⇀σ 1 =

[0 1

1 0

],

⇀σ 2 =

[0 −i

i 0

],

⇀σ 3 =

[1 0

0 −1

],

where I is the 2 × 2 identity matrix. The decomposi-tion S(x ′) = cos[θ (x ′)]I + i sin[θ (x ′)]⇀

c(x ′) · ⇀σ is exponen-

tial and implies we can write St (x ′) = cos[tθ (x ′)]I +i sin[tθ (x ′)]⇀

c(x ′) · ⇀σ .

Definition 3.3. The Chebyshev polynomial Tn(x) of thefirst kind is a polynomial in x of degree n, defined by therelation Tn(x) = cos(nθ ) when x = cos(θ ), and the Chebyshevpolynomial Un(x) of the second kind is a polynomial in x ofdegree n, defined by the relation Un(x) = [sin(n + 1)θ]/(sin θ )when x = cos(θ ) [16].

Using Definition 3.3 we can write St (x ′) =Tt (cos[θ (x ′)])I + Ut−1( cos [θ (x ′) ]) i sin [θ (x ′)]⇀

c(x ′) · ⇀σ . In

Mason et al. [16] it is shown that Un(x) = 2xUn−1(x) −Un−2(x) and 2Tn(x) = Un(x) − Un−2(x). From both of theserelations we can deduce Tn(x) = Un(x) − xUn−1(x), fromwhich it follows that we can write

St (x ′) = Ut (cos[θ (x ′)])I − Ut−1(cos[θ (x ′)])

×{cos[−θ (x ′)]I + i sin[−θ (x ′)]⇀

c(x ′) · ⇀σ }.

(3.2.5)

However, we notice that the expression within the curlybraces in Eq. (3.2.5) can be written using the exponential

notation as [eiθ(x ′)⇀c (x ′)·⇀σ ]−1 = [S(x ′)]−1. So we can writeEq. (3.2.5) as

St (x ′) = Ut (cos[θ (x ′)])I − Ut−1(cos[θ (x ′)])[S(x ′)]−1.

(3.2.6)

Following Fuss et al. [9] we can also determine thefunction cos[θ (x ′)] in terms of the components of S(x ′)by defining the inner product (A,B) = 1

2 Tr(AB), where Trdenotes the trace function, on the vector space of 2 × 2unitary matrices and hence obtain an inner product spacewith {I,σ1,σ2,σ3} as an orthonormal basis. The coefficientof the identity matrix on the right-hand side of S(x ′) =cos[θ (x ′)]I + i sin[θ (x ′)]⇀

c(x ′) · ⇀σ is cos[θ (x ′)]. To determine

the coefficient of the identity matrix on the left-hand side ofS(x ′) = cos[θ (x ′)]I + i sin[θ (x ′)]⇀

c(x ′) · ⇀σ , we take the inner

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LIMIT THEOREMS FOR QUANTUM WALKS ASSOCIATED . . . PHYSICAL REVIEW A 84, 012324 (2011)

product (I,S(x ′)) = 12 Tr[IS(x ′)], which gives −i

√p sin(x ′).

We saw earlier that the wave function of the walker at anarbitrary time t is given by ζ̂ (x ′,t) = eitkSt (x ′)ζ̂ (x ′,0), thus

between two times t0 and t1, ζ̂ (x ′,t) = eitkSt (x ′)ζ̂ (x ′,0) hasthe equivalent interpretation T (t1,t0) = ei(t1−t0)kS(t1−t0)(x ′). Sowe can write

T (t,0) = eitkSt (x ′)= eitk[Ut (cos[θ (x ′)])I − Ut−1(cos[θ (x ′)])[S(x ′)]−1]

= eitk[Ut (−i√

p sin(x ′))I − Ut−1(−i√

p sin(x ′))[S(x ′)]−1]. (3.2.7)

From ζ̂ (x ′,t) = eitkSt (x ′)ζ̂ (x ′,0), T (t1,t0) = ei (t1−t0)kS(t1−t0)(x ′), and Eq. (3.2.7) we can write

ζ̂ (x ′,t) = eitk�Ut (−i√

p sin(x ′))I − Ut−1(−i√

p sin(x ′))[S(x ′)]−1�ζ̂ (x ′,0). (3.2.8)

In order to write Eq. (3.2.8) in terms of ζ̂1(x ′,t)and ζ̂2(x ′,t)we first note that

[S(x ′)]−1 =[√

peix ′ √qe−ix ′

√qeix ′ −√

peix ′

], ζ̂ (x ′,t) =

(ζ̂1(x ′,t)ζ̂2(x ′,t)

),

ζ (x ′,0) =(

ζ̂1(x,0)

ζ̂2(x,0)

)=

(φ1(0,0)

φ2(0,0)

)=

(d1

d2

).

Thus we have

ζ̂1(x ′,t) = eitk[Ut (−i√

p sin(x ′))d1 − (√

peix ′d1 + √

qe−ix ′d2)

×Ut−1(−i√

p sin(x ′))], (3.2.9)

ζ̂2(x ′,t) = eitk[Ut (−i√

p sin(x ′))d2 + (√

peix ′d2 − √

qe−ix ′d1)

×Ut−1(−i√

p sin(x ′))]. (3.2.10)

Using the definition

ut (−i√

p : x) = 1

2π

∫ π

−π

Ut (−i√

p sin(x ′))eixx ′dx ′,

and applying the inverse Fourier transform to Eqs. (3.2.9) and(3.2.10), respectively, we have

φ1(x,t) = eitk[d1ut (−i√

p : x) − √pd1ut−1(−i

√p : x + 1)

−√qd2ut−1(−i

√p : x − 1)], (3.2.11)

φ2(x,t) = eitk[d2ut (−i√

p : x) + √pd2ut−1(−i

√p : x + 1)

−√qd1ut−1(−i

√p : x − 1)]. (3.2.12)

Here

ζ (x,t) =(

φ1(x,t)

φ2(x,t)

)∈ C2

gives the time evolution of the quantum random walk for dis-crete times t � 0 on a line x ∈ Z. We note that for a quantumparticle starting from the initial state ( d1

d2), Eqs. (3.2.11) and

(3.2.12) imply that the quantum walk probability density isgiven by |φ1(x,t)|2 + |φ2(x,t)|2. Write P (x,t) = |φ1(x,t)|2 +|φ2(x,t)|2, and then from Eqs. (3.2.11) and (3.2.12) we havethat

P (x,t) = |d1ut (−i√

p : x) − √pd1ut−1(−i

√p : x + 1)

−√qd2ut−1(−i

√p : x − 1)|2

× |d2ut (−i√

p : x) + √pd2ut−1(−i

√p : x + 1)

−√qd1ut−1(−i

√p : x − 1)|2. (3.2.13)

To write Eq. (3.2.13) in closed form we use the power seriesmethod of Ref. [9]. According to the authors we can write

Ut (y) =[t/2]∑m=0

(−1)m(

t − m

m

)(2y)t−2m, (3.2.14)

where (t − m

m

)= (t − m)!

m!(t − 2m)!

for 0 � m � t − m, and [t/2] is the floor value of t/2.If we put 2y = √

p(e−ix ′ − eix ′) and expand the powers

using the binomial theorem, it follows we can write

Ut (−i√

p sin(x ′))

=[t/2]∑m=0

(−1)m(

t − m

m

) (√p)t−2m

t−2m∑k=0

(t − 2m

k

)

× (−1)ke−ix ′[t−2(m+k)]. (3.2.15)

To put Eq. (3.2.15) in a suitable form, we change the indexof summation in the second sum by putting j = m + k; itfollows we can then write (3.2.15) as

Ut (−i√

p sin(x ′))

=[t/2]∑m=0

(−1)m(

t − m

m

) (√p)t−2m

t∑j=0

(t − 2m

j − m

)

× (−1)ke−ix ′(t−2j ), (3.2.16)

where the property ( q

r) = 0 is to be used when r < 0 or r > q.

Reordering the summation in Eq. (3.2.16) gives

Ut (−i√

p sin(x ′))

=t∑

j=0

[[t/2]∑m=0

(−1)m+k

(t − m

m

)

×(

t − 2m

j − m

) (√p)t−2m

]e−ix ′(t−2j ). (3.2.17)

Let P tt−2j (−i

√p) denote the expression in square brackets

in Eq. (3.2.17), then

Ut (−i√

p sin(x ′)) =t∑

j=0

P tt−2j (−i

√p)e−ix ′(t−2j ). (3.2.18)

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CLEMENT AMPADU PHYSICAL REVIEW A 84, 012324 (2011)

Recall that Ut (−i√

p sin(x ′)) has the inverse Fouriertransform

ut (−i√

p : x) = 1

2π

∫ π

−π

Ut (−i√

p sin(x ′))eixx ′dx ′.

However,by (3.2.18) we can write

ut (−i√

p : x) = 1

2π

∫ π

−π

t∑j=0

P tt−2j (−i

√p)eix ′(x−(t−2j ) dx ′.

(3.2.19)

We see from Eq. (3.2.19) that

ut (−i√

p : x) =t∑

j=0

P tt−2j (−i

√p),

provided that x = t − 2j , and ut (−i√

p : x) = 0, providedthat x �= t − 2j . Thus in terms of the Kronecker delta we canwrite

ut (−i√

p : x) =t∑

j=0

P tt−2j (−i

√p)δx,t−2j ,

where P tt−2j (−i

√p) is the expression inside the square

brackets in Eq. (3.2.17). The main result of this section isnow ready and we summarize as follows.

Theorem 3.4. The quantum probability density functionfor the generalized Hadamard walk in one dimension is

given by

P (x,t) = |d1ut (−i√

p : x) − √pd1ut−1(−i

√p : x + 1)

−√qd2ut−1(−i

√p : x − 1)|2 + |d2ut

× (−i√

p : x) + √pd2ut−1(−i

√p : x + 1)

−√qd1ut−1(−i

√p : x − 1)|2,

whereut (−i

√p : x) =

t∑j=0

P tt−2j (−i

√p)δx,t−2j

and P tt−2j (−i

√p) is the expression in square brackets in

Eq. (3.2.17).

C. Weak convergence in the two-dimensional model

We first note that H ∗∗(p,q) = H ∗(p,q) ⊗ H ∗(p,q), whereH ∗(p,q) is given by Eq. (2.3.3). From the matrix in Eq. (2.3.5)one can deduce that the eigenvalues and the eigenvectorsof H ∗∗(p,q) are the product of the eigenvalues and theeigenvectors of the matrices M(m′+n′

2 ) and M(m′−n′2 ), where

the matrices M(m′+n′2 ) and M(m′−n′

2 ) come from the matrixin Eq. (2.3.5) upon making the appropriate substitution, andthen using the method of Sec. III A the convergence theoremin the long-time limit of the joint moment of the walker’spseudovelocity Xt

tand Yt

tis given by the following.

Theorem 3.5:

limt→∞

⟨(Xt

t

)α (Yt

t

)β⟩

=∫ ∫

[−π,π )2

⎛⎝ −√

p cos(

m′+n′2

)2√

1 − p sin2(

m′+n′2

) −√

p cos(

m′−n′2

)2√

1 − p sin2(

m′−n′2

)⎞⎠

α ⎛⎝ −√

p cos(

m′+n′2

)2√

1 − p sin2(

m′+n′2

) +√

p cos(

m′−n′2

)2√

1 − p sin2(

m′−n′2

)⎞⎠

β

× ∣∣c1(m′,n′)∣∣2 dm′

2π

dn′

2π

+∫ ∫

[−π,π )2

⎛⎝ −√

p cos(

m′+n′2

)2√

1 − p sin2(

m′+n′2

) +√

p cos(

m′−n′2

)2√

1 − p sin2(

m′−n′2

)⎞⎠

α ⎛⎝ −√

p cos(

m′+n′2

)2√

1 − p sin2(

m′+n′2

) −√

p cos(

m′−n′2

)2√

1 − p sin2(

m′−n′2

)⎞⎠

β

× ∣∣c2(m′,n′)∣∣2 dm′

2π

dn′

2π

+∫ ∫

[−π,π )2

⎛⎝ −√

p cos(

m′−n′2

)2√

1 − p sin2(

m′−n′2

) +√

p cos(

m′+n′2

)2√

1 − p sin2(

m′+n′2

)⎞⎠

α ⎛⎝ √

p cos(

m′−n′2

)2√

1 − p sin2(

m′−n′2

) +√

p cos(

m′+n′2

)2√

1 − p sin2(

m′+n′2

)⎞⎠

β

× ∣∣c3(m′,n′)∣∣2 dm′

2π

dn′

2π

+∫ ∫

[−π,π )2

⎛⎝ √

p cos(

m′+n′2

)2√

1 − p sin2(

m′+n′2

) +√

p cos(

m′−n′2

)2√

1 − p sin2(

m′−n′2

)⎞⎠

α ⎛⎝ √

p cos(

m′+n′2

)2√

1 − p sin2(

m′+n′2

) −√

p cos(

m′−n′2

)2√

1 − p sin2(

m′−n′2

)⎞⎠

β

× ∣∣c4(m′,n′)∣∣2 dm′

2π

dn′

2π.

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LIMIT THEOREMS FOR QUANTUM WALKS ASSOCIATED . . . PHYSICAL REVIEW A 84, 012324 (2011)

D. Analytic expression for the probability distribution in thetwo-dimensional model

Note that the dynamics of the two-dimensional Hadamardmodel considered in this paper can be expressed in terms ofdifference equations giving the system of linear equations

φ1(x,y,t) = eik{pφ1(x − 1,y,t − 1) + √pqφ2

× (x − 1,y,t − 1) + √pqφ3(x − 1,y,t − 1)

+ qφ4(x − 1,y,t − 1)}, (3.4.1)

φ2(x,y,t) = eik{√pqφ1(x + 1,y,t − 1) − pφ2

× (x + 1,y,t − 1) + qφ3(x + 1,y,t − 1)

−√pqφ4(x + 1,y,t − 1)}, (3.4.2)

φ3(x,y,t) = eik{√pqφ1(x,y − 1,t − 1) + qφ2

× (x,y − 1,t − 1) − pφ3(x,y − 1,t − 1)

−√pqφ4(x,y − 1,t − 1)}, (3.4.3)

φ4(x,y,t) = eik{qφ1(x,y + 1,t − 1) − √pqφ2

× (x,y + 1,t − 1) − √pq φ3(x,y + 1,t − 1)

+pφ4(x,y + 1,t − 1)}, (3.4.4)

for a given

�(x,y,0) =

⎛⎜⎜⎜⎝

q1

q2

q3

q4

⎞⎟⎟⎟⎠∈ C4,

understood when x = y = 0, where C is the set of com-plex numbers and

∑4j=1 |qj |2 = 1, k ∈ R, q = 1 − p, and

φj for j = 1,2,3,4 are the components of �(x,y,t), andthen using the method of Sec. III B we get the probabilitydistribution for the generalized Hadamard walks as follows.

Theorem 3.6. The quantum walk probability density func-tion is given by

P (x,y,t) =4∑

i=1

|φi(x,y,t)|2,

where φ1,φ2,φ3,φ4 are given by

φ1(x,y,t) = eitk{q1ut (ip : (x,y)) − q1put−1(ip : (x + 1,y))

− q2√

pqut−1(ip : (x,y + 1)) − q3√

pqut−1

× (ip : (x,y − 1)) − q4qut−1(ip : (x − 1,y))},φ2(x,y,t) = eitk{q2ut (ip : (x,y)) − q1

√pqut−1

× (ip : (x + 1,y)) + q2put−1(ip : (x + 1,y))

− q3qut−1(ip : (x,y − 1)) + q4√

pqut−1

× (ip : (x,y − 1))},φ3(x,y,t) = eitk{q3ut (ip : (x,y)) − q1

√pqut−1

× (ip : (x + 1,y)) − q2qut−1(ip : (x,y + 1))

+ q3put−1(ip : (x + 1,y)) + q4√

pqut−1

× (ip : (x,y + 1))},φ4(x,y,t) = eitk{q4ut (ip : (x,y)) − q1qut−1(ip : (x + 1),y))

+ q2√

pqut−1(ip : (x + 1,y)) + q3√

pqut−1

× (ip : (x + 1,y)) − q4put−1(ip : (x + 1,y))},

and ut (ip : (x,y)) in the expressions for φ1,φ2,φ3,φ4 isgiven by

ut (ip : (x,y)) =⎧⎨⎩

t−k∑j=0

Q(p), if (x,y) = (−q,t − k − 2j ),

0, otherwise,,

where

Q(p) =[t/2]∑m=0

t−2m∑k=0

k∑q=0

[t

m

]pt−2m

(t − k − 2m

j − m

)

×(

t − 2m

k

)(k

q

)(−1)t−2m−v−q,

and [t

m

]= (−1)m

(t − m

m

)and (

t − m

m

)is the binomial coefficient as defined in Eq. (3.2.14).

IV. LOCALIZATION IN THE GENERALIZED HADAMARDMODEL: THEORETICAL INVESTIGATION

To give the criterion we first give the probability oflocalization on the line in the one-dimensional model.

Theorem 4.1. Let P (x) = limt→∞ P (x,t), and then theprobability of localization at x = x0 is given by

P (x0) =∫ ∞

−∞P (x)δ(x − x0) dx,

where P (x,t) is the quantum probability density function forthe generalized Hadamard walk in one dimension as given byTheorem 3.4. and δ(x) is the one-dimensional Dirac δ function.

Proof. We make use of the following characterization of theDirac δ function which is available in Ref. [17]. Let ε > 0 begiven. The Dirac δ function δ(x) can be defined as the limit ofa sequence of discontinuous functions δε(x), where

δε(x) ={

12ε

, |x| < ε,

0, |x| > ε.

From this definition we see that by shifting δε(x) to the rightx0 units we can define δ(x − x0) as the limit of a sequence ofdiscontinuous functions δε(x − x0) defined as

δε(x − x0) ={

12ε

, |x − x0| < ε,

0, |x − x0| > ε.

Each δε(x − x0) has a unit area under the curve andlimε→0 δε(x − x0) = 0 if x �= x0. However, on formally in-terchanging the limit process with integration, we obtain∫ ∞

−∞δε(x − x0) dx = lim

ε→0

∫ ∞

−∞δε(x − x0) dx = 1.

Since P (x) = limt→∞ P (x,t) is a continuous function byvirtue of the continuity of the quantum probability density

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CLEMENT AMPADU PHYSICAL REVIEW A 84, 012324 (2011)

function P (x,t), it follows upon using the mean value theoremfor integrals before passing to the limit that

P (x0) =∫ ∞

−∞P (x)δ(x − x0)dx,

and the proof is finished. In the two-dimensional model theprobability of localization is given by the following.

Theorem 4.2. Let P (x,y) = limt→∞ P (x,y,t), and then theprobability of localization at (x,y) = (x0,y0) is given by

P (x0,y0) =∫ ∞

−∞

∫ ∞

−∞P (x,y)δ(x − x0,y − y0)dx dy,

where P (x,y,t) is the quantum walk probability densityfunction for the generalized Hadamard walk in two dimensionsas given by Theorem 3.6, and δ(x,y) is the two-dimensionalDirac δ function.

Definition 4.3. We say localization has occurred in eitherthe one- or two-dimensional models if P (x0) or P (x0,y0) issufficiently large within the confines of the interval [0,1].

V. SYMMETRY OF THE DISTRIBUTION:THEORETICAL INVESTIGATION

In this section we give rigorous results on the symmetry ofthe distribution for both the one- and two-dimensional modelsconsidered in this paper. We follow closely the ideas in thepaper of Konno et al. [18].

A. Necessary and sufficient condition in theone-dimensional model

Recall that in the one-dimensional model that the Hadamardtransformation is given by

H ∗(p,q) =[√

p√

q√q −√

p

].

H ∗(p,q) acts on chirality states, which signifies the directionof motion of the particle. In the one-dimensional model theparticle can either move left or right in the direction of thex axis, depending on its chirality state. In particular, H ∗(p,q)acts on the chirality states |L〉 and |R〉, where L and R referto the right and left chirality states, respectively, and

|L〉 =[

1

0

]and |R〉 =

[0

1

].

In particular, H ∗(p,q) acts on the two chirality statesvia |L〉 �→ √

p |L〉 + √q|R〉, |R〉 �→ √

q|L〉 − √p|R〉, and

thus H ∗(p,q)|L〉 = √p|L〉 + √

q|R〉 and H ∗(p,q)|R〉 =√q|L〉 − √

p|R〉. Now define the matrices P and Q asfollows:

P = √p

[1 0

0 0

]+ √

q

[0 1

0 0

]and

Q = √q

[0 0

1 0

]+ √

p

[0 0

0 −1

],

with H ∗(p,q) = P + Q. Note that P represents the left-moving particle and Q represents the right-moving parti-cle. Now we define the dynamics of the Hadamard walk

in one dimension using P and Q. To do so we definethe following (2N + 1) × (2N + 1) matrix H̄ ∗

N : (C2)2N+1 →(C2)2N+1, where C is the set of complex numbers, P and Q

are as above, and

02×2 =[

0 0

0 0

],

H̄ ∗N =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

02×2 P 02×2 . . . . . . 02×2 Q

Q 02×2 P 02×2 . . . . . . 02×2

02×2 Q 02×2 P 02×2 . . . 02×2

.... . .

. . .. . .

. . .. . .

...02×2 . . . 02×2 Q 02×2 P 02×2

02×2 . . . . . . 02×2 Q 02×2 P

P 02×2 . . . . . . 02×2 Q 02×2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Now let

ζ tx (�) =

(φt

1,x (�)

φt2,x (�)

)= φt

1,x(�)|L〉 + φt2,x(�)|R〉 ∈ C2

be the two-component vector of amplitudes of the particlebeing at site x and at time t with the chirality being left (uppercomponent) and right (lower component), and let ζ t (�) =T

[ζ t−N (�),ζ t

−(N−1)(�), . . . ,ζ tN (�)] ∈ (C2)2N+1 be the quibit

states at time t , where T represents the transposed operator.Here the initial quibit state is given by

ζ 0(�) =[ N︷ ︸︸ ︷02×1, . . . 02×1 ,�,

N︷ ︸︸ ︷02×1, . . . 02×1

]∈ (C2)2N+1,

where

02×1 =[

0

0

], � =

[d1

d2

],

with

2∑i=1

|di |2 = 1.

The time evolution of the one-dimensional Hadamard walkconsidered in this paper is then given by [H̄ ∗

Nζ t (�)]x =Qζ t

x−1(�) + Pζ tx+1(�), where [ζ t (�)]x = ζ t

x(�). Since P

and Q satisfy PP ∗ = P ∗P + Q∗Q = I and PQ∗ = QP ∗ =Q∗P = P ∗Q = 02×2, where the asterisk represents the adjointoperator, it follows that the matrix H̄ ∗

N is also a unitary matrix.Now define the set of initial quibit states by

� ={

� =[

d1

d2

]∈ C2 :

2∑i=1

|di |2 = 1

},

and define the probability distribution of the generalizedHadamard walk in one dimension by P (X�

t = x) = |ζ tx(�)|2.

To study the symmetry of the distribution of H ∗( 12 , 1

2 ),Konno et al. [18] introduced the following classes of initial

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LIMIT THEOREMS FOR QUANTUM WALKS ASSOCIATED . . . PHYSICAL REVIEW A 84, 012324 (2011)

qubit states:

�⊥ ={� =

[d1

d2

]∈ � : |d1| = |d2|,d1d̄2 + d̄1d2 = 0

},

�s = {� ∈ � : P

(X�

t = x) = P

(X�

t = −x)

for any t ∈ Z+ and x ∈ Z},

�0 = {� ∈ � : E

(X�

t

) = 0, for any t ∈ Z+},

where Z+ represents the set of positive integers and Zrepresents the integers. It is noted that if d1d2 �= 0, thend̄1d2 + d1d̄2 = 0 implies that d1 and d2 are orthogonal. If� ∈ �s , then the distribution of X�

t is symmetric for anyt ∈ Z+ and the authors concluded the following theorem.

Theorem 5.1: For H ∗( 12 , 1

2 ) we have �⊥ = �s = �0. Inorder for the theorem to hold, the inclusion �⊥ ⊆ �s ⊆�0 ⊆ �⊥ needs to be proven. However, their proof of theinclusion �0 ⊆ �⊥ lacks rigor. To show this inclusion basedon their class of initial qubit states, we need to show thatif � ∈ �0, then � ∈ �⊥. However, if � ∈ �0, then it iseasily seen that E(X�

t ) = 0, but this does not necessarilyimply that � ∈ �⊥. In particular, the authors have shownfor t ∈ {1, . . . ,10} we have E(X�

t ) = −at (|d1|2 − |d2|2) −bt (d1d̄2 + d̄1d2) where a1 = 0, a2 = 0, a3 = 1

2 , a4 = 1, a5 =98 , a6 = 5

4 , a7 = 2716 , a8 = 17

8 , a9 = 293128 , a10 = 157

64 , b1 = 1,b2 = 1, b3 = 1, b4 = 3

2 , b5 = 2, b6 = 178 , b7 = 9

4 , b8 = 4316 ,

b9 = 258 , b10 = 421

128 , and conjectured that bt+1 = at + 1 forany t � 1. It follows at once, irrespective of the coefficientsat and bt , that we can establish the following: If � ∈ �0,then E(X�

t ) = 0 if and only if � ∈ �⊥. This then begs thefollowing question: If the Konno-Namiki-Soshi conjecture istrue and � /∈ �⊥, does there exist at and bt both not zero suchthat E(X�

t ) = 0? We prove the answer is true and use it toprovide the correct the class of initial qubit states for whichTheorem 5.1 holds and generalizes immediately if we replaceH ∗( 1

2 , 12 ) with H ∗(p,q).

To see the conjecture, one can use the principle of mathe-matical induction on t . First their conjecture can be reformu-lated as follows: Is b2 − a1 = b3 − a2 = · · · = bt − at−1 =bt+1 − at = 1 ∀t � 1? If t ∈ {1,2, . . . ,10}, the authors haveshown the answer is affirmative. It follows that the basis stepis automatic. Now we assume for t = k that b2 − a1 = b3 −a2 = · · · = bk − ak−1 = bk+1 − ak = 1, then the inductive hy-pothesis imply we also have the following b2 − a1 = b3 −a2 = · · · = bk − ak−1 = bk+1 − ak = bk+2 − ak+2 = 1, so theequality holds for t = k + 1, and the conjecture is resolved.Now we show that if � /∈ �⊥, then there exist at and bt both notzero such that E(X�

t ) = 0. Notice since the conjecture is true itfollows that we can write E(X�

t ) = L − bt+1L − btQ, whereL = |d1|2 − |d2|2 and Q = d1d̄2 + d̄1d2. Now if E(X�

t ) = 0,then this implies we have bt+1L + btQ = L. Puttingb

homogeneoust = Xt , and considering the related homogeneous

equation bt+1L + btQ = 0, it is easy to check that its solutionb

homogeneoust is given by

bhomogeneoust = (−1)t

(Q

L

)t

.

Now for bt+1L + btQ = L itself we put bconstantt = X,

and it easy to see that the associated solution is bconstantt =

LL+Q

, and thus bt = bhomogeneoust + bconstant

t = (−1)t (Q

L)t +

LL+Q

, which is the solution to bt+1L + btQ − L = 0. Since

the left-hand side of bt+1L + btQ − L = 0 is E(X�t ), we

have found bt and at = bt+1 − 1 both not zero such thatE(X�

t ) = 0.Now we give the correct class of initial qubit states for

which Theorem 5.1 is valid and generalizes if we replaceH ∗( 1

2 , 12 ) with H ∗(p,q). The class of initial qubit states are as

follows:

�s = {� ∈ � : P

(X�

t = x)

= P(X�

t = −x)

for any t ∈ Z+ and x ∈ Z},

�⊥ ={� =

[d1

d2

]∈ � : |d1| = |d2|,d1d̄2 + d̄1d2 = 0

},

�0 = �′0 ∪ �′′

0, where �0 = {� ∈ � : E

(X�

t

) = 0},

�′0 = {

� ∈ �⊥ : E(X�

t

) = 0},

�′′0 = {

� /∈ �⊥ : E(X�

t

) = 0}.

In order to prove Theorem 5.1, the following lemma isneeded, which is essentially Lemma 1 in Ref. [18] uponmaking the necessary change in notation, so we omit theproof here.

Lemma 5.2. Let

J =[

0 −1

1 0

].

For any x ∈ Z,t ∈ Z+, we have

ζ tx(�) =

⎧⎪⎪⎨⎪⎪⎩

(−1)t iJ ζ t−x(�), if � =

√12eiθ

[1

i

],

(−1)t (−i) Jζ t−x (�) , if � =

√12eiθ

[1

−i

],

where θ ∈ [0,2π ).Proof of Theorem 5.1. We need to show �⊥ ⊆ �s ⊆ �0

and �0 ⊆ �s ⊆ �⊥ for equality to hold. Equivalently wecan show the following inclusion: �⊥ ⊆ �s ⊆ �0 ⊆ �⊥. Wefirst show the inclusion �s ⊆ �0. Note that if � ∈ �s , thenP (X�

t = x) = P (X�t = −x) and so we can deduce that∑

x

xP(X�

t = x) −

∑−x

(−x) P(X�

t = −x) = 0,

that is, E(X�t ) = 0.

This implies that � ∈ �0 and the inclusion �s ⊆ �0 iscomplete. We now show the inclusion �⊥ ⊆ �s . First we notethat for θ1 �= θ2, where θ1,θ2 ∈ [0,2π ), it is easy to check that√

1

2eiθ1

[1

i

]and

√1

2eiθ2

[1

−i

]are in �⊥. Since the matrix A = [ V1 V2 ] has linearlyindependent columns, where

V1 = √peiθ1

[1

i

]and V2 =

√1

2eiθ2

[1

−i

]

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CLEMENT AMPADU PHYSICAL REVIEW A 84, 012324 (2011)

it follows that

�⊥ = span

{√1

2eiθ1

[1

i

],

√1

2eiθ2

[1

−i

]}.

So if � ∈ �⊥, then from Lemma 2.14 we can deduce thefollowing:

P(X�

t = x) = ∣∣ζ t

x (�)∣∣2 = ζ t

x (�) ζ tx (�)∗

= [(−1)n iJ ζ t

−x (�)] [

(−1)n i∗J ∗ζ t−x (�)∗

]= ζ t

−x (�) ζ t−x (�)∗

= ∣∣ζ t−x (�)

∣∣2 = P(X�

t = −x),

which implies that � ∈ �s and the inclusion �⊥ ⊆ �s

is complete. Now we show the inclusion �0 ⊂ �⊥. If� ∈ �0, then E(X�

t ) = 0, but �0 = �′0 ∪ �′′

0, where �′0 =

{� ∈ �⊥ : E(X�t ) = 0} and �′′

0 = {� /∈ �⊥ : E(X�t ) = 0} so

� ∈ �′0 ∪ �′′

0 and the definition of �′0 implies that � ∈ �⊥,

and the inclusion �0 ⊂ �⊥ is complete.Thus we immediately see that Theorem 5.1 generalizes if

we replace H ∗( 12 , 1

2 ) with H ∗(p,q). To see the generalizationwe first generalize Lemma 5.2 as follows.

Lemma 5.3. Let

J =[

0 −1

1 0

].

For any x ∈ Z, t ∈ Z+, we have

ζ tx(�) =

⎧⎪⎪⎨⎪⎪⎩

(−1)t iJ ζ t−x(�), if � = √

peiθ

[1

i

],

(−1)t (−i) Jζ t−x (�) , if � = √

peiθ

[1

−i

],

where θ ∈ [0,2π ).Proof. The proof is essentially the same as Lemma 5.2 or

Lemma 1 in Ref. [18], upon replacing the matrix P with

P = √p

[1 0

0 0

]+ √

q

[0 1

0 0

]and the matrix Q with

Q = √q

[0 0

1 0

]+ √

p

[0 0

0 −1

],

and making the necessary change in notation. Thus, to seethe generalized form of Theorem 5.1, we essentially useLemma 5.3 in the proof of Theorem 5.1, and the generalizedform of Theorem 5.1 follows, which gives the necessaryand sufficient condition for symmetry in the one-dimensionalmodel considered in this paper.

B. Necessary and sufficient condition in thetwo-dimensional model

Recall for the two-dimensional model considered in thispaper that the time evolution is given by the transformation

H ∗∗(p,q) =

⎡⎢⎢⎣

p√

pq√

pq q√pq −p q −√

pq√pq q −p −√

pq

q −√pq −√

pq p

⎤⎥⎥⎦ ,

q = 1 − p.

Recall that the direction of motion of the particle depends on itschirality state. In two dimensions we have four chirality states:left, right, up, or down. The evolution of the two-dimensionalHadamard walk proceeds as follows: At each time step, if theparticle has left chirality, it moves one step to the left, andif it has right chirality, it moves one step to the right, and ifthe chirality state is up or down, then it moves up or down,respectively. In particular, H ∗∗(p,q) acts on four chiralitystates |L〉, |R〉, |U 〉, and |D〉, where L, R, U, and D referto the left, right, up, and down chirality states, respectively, asfollows:

|L〉 �→ p|L〉 + √pq(|R〉 + |D〉) + q|U 〉,

|R〉 �→ √pq(|L〉 − |U 〉) − p|R〉 + q|D〉,

|D〉 �→ √pq(|L〉 − |U 〉) + q|R〉 − p|D〉,

|U 〉 �→ q|L〉 − √pq(|R〉 + |D〉) + p|U 〉.

Define

|L〉 =

⎡⎢⎢⎢⎣

1

0

0

0

⎤⎥⎥⎥⎦ , |R〉 =

⎡⎢⎢⎢⎣

0

1

0

0

⎤⎥⎥⎥⎦ ,

|D〉 =

⎡⎢⎢⎢⎣

0

0

1

0

⎤⎥⎥⎥⎦ , and |U 〉 =

⎡⎢⎢⎢⎣

0

0

0

1

⎤⎥⎥⎥⎦ ,

and then we have

H ∗∗(p,q)|L〉 �→ p|L〉 + √pq(|R〉 + |D〉) + q|U 〉,

H ∗∗(p,q)|R〉 �→ √pq(|L〉 − |U 〉) − p|R〉 + q|D〉,

H ∗∗(p,q)|D〉 �→ √pq(|L〉 − |U 〉) + q|R〉 − p|D〉,

H ∗∗(p,q)|U 〉 �→ q|L〉 − √pq(|R〉 + |D〉) + p|U 〉.

Now we introduce the following matrices:

PL =

⎡⎢⎢⎢⎣

p√

pq√

pq q

0 0 0 0

0 0 0 0

0 0 0 0

⎤⎥⎥⎥⎦ ,

PR =

⎡⎢⎢⎣

0 0 0 0√pq −p q −√

pq

0 0 0 0

0 0 0 0

⎤⎥⎥⎦ ,

PD =

⎡⎢⎢⎢⎣

0 0 0 0

0 0 0 0√pq q −p −√

pq

0 0 0 0

⎤⎥⎥⎥⎦ ,

PU =

⎡⎢⎢⎢⎣

0 0 0 0

0 0 0 0

0 0 0 0

q −√pq −√

pq p

⎤⎥⎥⎥⎦ ,

with H ∗∗(p,q) = PL + PR + PD + PU . Here PL, PR , PD ,PU represents the probability the particle moves left, right,

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LIMIT THEOREMS FOR QUANTUM WALKS ASSOCIATED . . . PHYSICAL REVIEW A 84, 012324 (2011)

down, and up, respectively. By using PL, PR , PD , and PU , wedefine the dynamics of the generalized Hadamard walk in twodimensions by the following (2N + 1)2 × (2N + 1)2 matrixH̄ ∗∗

N : (C4)4N2+4N+1 �→ (C4)4N2+4N+1 by H̄ ∗∗N = H̄ ∗

N ⊗ H̄ ∗N ,

where the (2N + 1) × (2N + 1) matrix H̄ ∗N : (C2)2N+1 �→

(C2)2N+1 is given by

H̄ ∗N =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

02×2 P 02×2 . . . . . . 02×2 Q

Q 02×2 P 02×2 . . . . . . 02×2

02×2 Q 02×2 P 02×2 . . . 02×2

.... . .

. . .. . .

. . .. . .

...

02×2 . . . 02×2 Q 02×2 P 02×2

02×2 . . . . . . 02×2 Q 02×2 P

P 02×2 . . . . . . 02×2 Q 02×2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

where

02×2 =[

0 0

0 0

],

P =[√

p√

q

0 0

],

and

Q =[

0 0√q −√

p

],

andC is the set of complex numbers. Note that the (2N + 1) ×(2N + 1) matrix H̄ ∗

N : (C2)2N+1 �→ (C2)2N+1 defined aboveis associated with the dynamics of the generalized Hadamardwalk in one dimension. In particular, recall that the relationshipbetween the transformations in the one- and two-dimensionalmodels is given by H ∗∗(p,q) = H ∗(p,q) ⊗ H ∗∗(p,q).Now let

�tx,y (θ ) =

⎡⎢⎢⎢⎣

ζ t1,x,y (θ )

ζ t2,x,y (θ )

ζ t3,x,y (θ )

ζ t4,x,y (θ )

⎤⎥⎥⎥⎦ = ζ t

1,x,y(θ )|L〉 + ζ t2,x,y(θ )|R〉

+ ζ t3,x,y(θ )|D〉 + ζ t

4,x,y(θ )|U 〉 ∈ C4

be the four-component vector of amplitudes of the particlebeing at site (x,y) at time t , where the first, second, third,and fourth components of the column vector correspondto the chirality states left, right, down, and up, respec-tively, and let �t (θ ) =T [�t

−N (θ ),�t−(N−1)(θ ), . . . ,�t

−N (θ )] ∈(C4)4N2+4N+1 be the qubit states at time t , where T means thetransposed operator. The initial qubit state is given by

�0 (θ ) =T

[ 2N2+2N︷ ︸︸ ︷04×1, . . . 0

4×1,θ,

2N2+2N︷ ︸︸ ︷04×1, . . . 04×1

]∈ (C4)4N2+4N+1,

where

04×1 =

⎡⎢⎢⎢⎣

0

0

0

0

⎤⎥⎥⎥⎦ and θ =

⎡⎢⎢⎢⎣

k1

k2

k3

k4

⎤⎥⎥⎥⎦ ,

with4∑

i=1

|ki |2 = 1.

The following equation defines the time evolution of thegeneralized two-dimensional Hadamard walk,

[H̄ ∗∗N �t (θ )]x,y = PR�t

x−1,y(θ ) + PL�tx+1,y(θ )

+PU�tx,y−1(θ ) + PD�t

x,y+1(θ ),

where [�t (θ )]x,y = �tx,y(θ ). Note that PR,PL,PU ,PD

satisfy PRP ∗R + PLP ∗

L + PUP ∗U + PDP ∗

D + P ∗RPR + P ∗

LPL +P ∗

UPU + P ∗DPD = I4×4, and also satisfy

PRP ∗L = PLP ∗

R = P ∗LPR = P ∗

RPL =

⎡⎢⎢⎢⎣

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

⎤⎥⎥⎥⎦

= PUP ∗D = PDP ∗

U = P ∗DPU = P ∗

UPD,

where the asterisk denotes the adjoint operator. The aboverelations imply that H̄ ∗∗

N is also a unitary matrix. Define theset of initial qubit states as follows:

� =

⎧⎪⎪⎪⎨⎪⎪⎪⎩θ =

⎡⎢⎢⎢⎣

k1

k2

k3

k4

⎤⎥⎥⎥⎦ ∈ C4 :

4∑i=1

|ki |2 = 1

⎫⎪⎪⎪⎬⎪⎪⎪⎭ .

Now we define the probability distribution of the general-ized Hadamard walk in two dimensions starting from theinitial qubit state θ ∈ � by P ((Xθ

t ,Yθt ) = (x,y)) = |�t

x,y(θ )|2.Nowwe introduce the following class of initial qubit states akinto the one-dimensional case.

�≈s = {

θ ∈ � :∣∣�t

−x,y (θ )∣∣2 = ∣∣�t

x,−y (θ )∣∣2 = ∣∣�t

−x,−y (θ )∣∣2

= ∣∣�tx,y (θ )

∣∣2 for any t ∈ Z+ and (x,y) ∈ Z2},

�≈⊥ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩θ =

⎡⎢⎢⎢⎣

k1

k2

k3

k4

⎤⎥⎥⎥⎦ ∈ � : |k1|2 = |k2|2 = |k3|2 = |k4|2 ,

∑i,ji �=j

ki k̄j = 0

⎫⎪⎪⎬⎪⎪⎭ ,

�≈0 = {

θ ∈ �≈⊥ : E

(Xθ

t Yθt

) = 0 for any t ∈ Z+}

∪ {θ /∈ �≈

⊥ : E(Xθ

t Yθt

) = 0 for any t ∈ Z+}.

Similar to the one-dimensional case it also follows that ifk1k2k3k4 �= 0, then ∑

i,ji �=j

ki k̄j = 0

implies that k1,k2,k3,k4 are orthogonal. For θ ∈ �≈s , the distri-

bution of the generalized Hadamard walk in two dimensions is

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CLEMENT AMPADU PHYSICAL REVIEW A 84, 012324 (2011)

also symmetric, and the following theorem below establishesthe necessary and sufficient condition.

Theorem 5.4. �≈s = �≈

⊥ = �≈0 . The proof of the above

theorem is similar in nature to the proof of generalized formof Theorem 5.1, which was obtained by replacing H ∗( 1

2 , 12 )

with H ∗(p,q), and involves invoking the following Lemma,whose proof is also similar to that of the generalized form ofLemma 5.3.

Lemma 5.5: Let

J≈ =

⎡⎢⎢⎢⎣

0 0 0 1

0 0 −1 0

0 −1 0 0

1 0 0 0

⎤⎥⎥⎥⎦ .

For any (x,y) ∈ Z2, t ∈ Z+, and γ ∈ [0, 2π ), we have

�tx,y (θ ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(−1)n iJ≈�t−x,−y (θ ) , if θ = pe2iγ

⎡⎢⎢⎢⎣

1

i

i

−1

⎤⎥⎥⎥⎦ ,

(−1)n (−i) J≈�t−x,−y (θ ) , if θ = pe2iγ

⎡⎢⎢⎢⎣

1

−i

−i

−1

⎤⎥⎥⎥⎦ ,

�tx,−y (θ ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(−1)n iJ≈�t−x,−y (θ ) , if θ = pe2iγ

⎡⎢⎢⎢⎣

1

i

i

−1

⎤⎥⎥⎥⎦ ,

(−1)n (−i) J≈�t−x,−y (θ ) , if θ = pe2iγ

⎡⎢⎢⎢⎣

1

−i

−i

−1

⎤⎥⎥⎥⎦ .

VI. CONCLUDING REMARKSWe have obtained weak convergence theorems in the long-

time limit of the walker’s pseudovelocity in both one and twodimensions for the Hadamard walk model considered in thispaper. Theoretical criteria is given for localization

and symmetrization in both the one- and two-dimensionalmodels. It is an interesting problem to study symmetrizationvia a variant of the quantum Pascal triangle as considered byRef. [18] in the case of the unbiased Hadamard walk in onedimension.

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