Open access peer-reviewed chapter

# Quantum Walks

By Takuya Machida

Submitted: October 28th 2015Reviewed: February 12th 2016Published: August 24th 2016

DOI: 10.5772/62481

## Abstract

Quantum walks are quantum counterparts of random walks. While probability distributions of random walks diffusively spread out as the walkers are updating, quantum walks have ballistic behavior. Some of the ballistic behaviors have been revealed in long-time limit theorems and their probability distributions are all far away from the Gaussian distributions, which are known as limit distributions of random walks. In this chapter, we are going to be seeing limit distributions for a standard quantum walk on the line and two kinds of time-dependent quantum walk on the line.

### Keywords

• Quantum walk
• limit theorem
• Fourier analysis
• time-dependent walk
• line

## 1. Introduction

Quantum walks (QWs) are mathematical models on graphs whose systems repeatedly update according to time-evolution rules. They have been in an emerging field which describes the quantum world. Experts in mathematics, physics, and information theory have been interested in them and to study QWs, and a lot of fascinating properties of QWs have been discovered. Historically, QWs were independently introduced in science from several view points; mathematics in 1988 [1], physics in 1993 [2], and computer science in 1996 [3]. After a while, they began to get attention around 2000. Since QWs can be considered as quantum counterparts of random walks in mathematics, they are also called quantum random walks. The dynamics of QWs are similar to those of random walks in mathematical terms. But, whereas a random walker moves on a graph at random, a quantum walker spreads out as a wave on a graph. Although random walks are stochastic processes, QWs are different. They are unitary processes because the systems of quantum walkers get updated with unitary operators. In quantum physics, the update rules of QWs are interpreted as discretized models of Dirac equations. High dimensional Dirac equations are hard to solve even in numerics due to their complexities, and then we expect QWs to become alternative systems to solve the equations on computer. QWs also play an important role in quantum computers because they are quantum algorithms themselves. Indeed, some quantum algorithms based on QWs show quadratic speed-up, compared to the corresponding classical algorithms [4]. Such algorithms imply that quantum computers could give rise to excellent performance.

In this chapter, we are going to be seeing mathematical aspects of the QWs, which will be described as limit theorems. We first observe a standard QW on the line in Sec.2 Then we shift our focus to time-dependent QWs on the line in Secs. 3 and 4.

## 2. A quantum walk on the line

We start off with the description of a standard QW on the line. The system of the QW is defined in a tensor space of two Hilbert spaces. One is a Hilbert space ℋpspanned by an orthogonal normalized basis {|x:x}, the other is a Hilbert space ℋcspanned by an orthogonal normalized basis {|0,{|1}. We can consider |0and |1in the Hilbert space ℋcas the down-spin and the up-spin states of a quantum particle, respectively. The Hilbert space ℋpis called a position space and the Hilbert space ℋcis called a coin space. A QW on the line at time t (=0, 1, 2 …) is expressed in the tensor Hilbert space,

|Ψt=x|x|ψtxpc.E1

Customarily, |ψt(x)cis called a coin state or an amplitude at position xat time t. Given an initial state |Ψ0, the walker is repeatedly updating,

|Ψt+1=SC|Ψt,E2

with

C=x|xx|U,E3
S=x|x1x||00|+|x+1x||11|,E4

where Uis a unitary operator. In this chapter, we employ a form of the operator

U=cosθ|00|+sinθ|01|+sinθ|10|cosθ|11|(θ[0,2π)),E5

and an initial state

|Ψ0=|0α|0+β|1,E6

assuming the complex numbers αand βsatisfy the condition |α|2 + |β|2 = 1. The reason that we assigned unitarity to the operator Uand assumed the constraint |α|2 + |β|2 = 1 is that we define the probability that the quantum walker is observed at position xat time t,

Xt=x=Ψt|xx|j=01|jj|Ψt,E7

where the random variable Xtis regarded as the position of the walker at time t. Thanks to the assignment and the constraint, the right side of Eq. (7) certainly becomes a probability distribution. Figure 1 shows the probability distribution ℙ(Xt = x) at time 500 when the walker starts updating with α=1/2and β=i/2. In these pictures, only positive values of the probability are plotted. While random walkers normally show diffusive behavior, it is known that the quantum walker acts ballistic as time goes up. We see for certain the ballistic behavior of the probability distribution ℙ(Xt = x) in Figure 2 when we take α=1/2and β=i/2. As shown in Figure 1, the probability distribution ℙ(Xt = x) holds two sharp peaks and where they occur strongly depends on the value of the parameter θ. We get more detailed information about that from Figure 3.

The definition of the standard QW on the line has been done. So, what are we curious about? One of the major studies on QWs is to know how their probability distributions behave after they have updated a lot of times. For the probability distribution of the QW defined in this section, one can assert a limit theorem which tells us an approximate behavior of the probability distribution ℙ(Xt = x) after time tgoes enough up. First, we see a limit theorem (Theorem 1) when the value of the parameter θ, which is embedded in the operator U, is picked in the interval [0, π). Then we will extend it for θ ∈ [0, 2π) (Theorem 2) which is easily proved by making the most of Theorem 1.

Theorem 1Assume that θ ∈ [0, π) and θ ≠ 0, π/2. For a real number x, we have

limtXttx=xsinθπ1y2cos2θy2×1|α|2|β|2+sinθαβ¯+α¯βcosθyI|cosθ|,|cosθ|ydy,E8

where IA(x) is an indicator function such that

IAx=1xA0otherwise.E9

This limit theorem, to be exact which was a theorem for a general unitary operator U ∈ U(2), was proven for the first time by a combinatorial method in 2002 [5]. It was also possible to obtain by Fourier analysis introduced by Grimmett et al. [6]. Here we use the second method to derive the limit theorem. Let |ψ^tkbe the Fourier transform of the quantum walker in the form

|ψ^tk=xeikx|ψtx(k[π,π)).E10

Oppositely, we can obtain the coin states by inverse Fourier transformation,

|ψtx=12πππeikx|ψ^tkdk.E11

Equation (2) gives the evolution of the Fourier transform,

|ψ^t+1k=RkU|ψ^tk,E12

with

Rk=eik|00|+eik|11|.E13

The iteration by Eq. (12) connects the system at time tto the initial state,

|ψ^tk=RkUt|ψ^0k.E14

To prove the theorem, we concentrate on a convergence

limtEXt/tr=xrfxdxr=0,1,2,,E15

with

fx=sinθπ1x2cos2θx2×1|α|2|β|2+sinθαβ¯+α¯βcosθxI|cosθ|,|cosθ|x.E16

It is known from probability theory that the convergence guarantees Eq. (8). Before we compute the limit, let us depict the r-th moment EXtr=x=xrXt=xin Fourier picture. Since the initial state is given by the form of Eq. (6), the Fourier transform at time tis rewritten in a finite sum

|ψ^tk=x=eikx|ψtx=x=tteikx|ψtx.E17

Noting that

drdkr|ψ^tk=x=ttixreikx|ψtx,E18

we have

ψ^tk|irdrdkr|ψ^tk=(x=tteikxψtx|)y=ttyreiky|ψty=x=tty=ttxreikxyψtx|ψty.E19

Integrating Eq. (19) over the interval [−ππ), one can generate the r-th moment of the random variable Xt,

ππψ^tk|irdrdkr|ψ^tkdk=ππx=tty=ttyreikxyψtx|ψtydk=2πx=ttxrψtx|ψtx=2πx=xrψtx|ψtx=2πx=xrXt=x=2πEXtr,E20

from which the r-th moment results in a representation in Fourier picture,

EXtr=12πππψ^tk|irdrdkr|ψ^tkdk.E21

Here, let λj(k) (j = 1, 2) be the eigenvalues of the matrix R(k)U, and |vj(k)be the normalized eigenvector associated to the eigenvalue λj(k). Then the initial state of the Fourier transform is decomposed by the normalized eigenvectors,

|ψ^0k=v1k|ψ^0k|v1k+v2k|ψ^0k|v2k.E22

The Fourier transform at time tis, therefore, expressed with the eigenvalues and the eigenvectors,

|ψ^tk=RkUt|ψ^0k=j=12λjktvjk|ψ^0k|vjk,E23

which also gives a description of the derivative

drdkr|ψ^tk=trj=12λjktrλjkrvjk|ψ^0k|vjk+Otr1,E24

where tr=tt1t2××tr+1=j=tr+1tjand λj′(k) = (d/dk)λj(k). Recalling the initial state in Eq. (6), we now have the Fourier transform at time 0 in the form

|ψ^0k=α|0+β|1.E25

Equations (23) and (24) change the integral picture in Eq. (21),

EXtr=12πππψ^tk|irdrdkr|ψ^tkdk=trj=1212πππiλjkλjkr|vjk|ψ^0k|2dk+Otr1.E26

Dividing Eq. (26) by tr, we reach a representation

EXtrtr=trtrj=1212πππiλjkλjkr|vjk|ψ^0k|2dk+Otr1tr,E27

and obtain a convergence as t → ∞,

limtEXttr=limtEXtrtr=j=1212πππiλjkλjkr|vjk|ψ^0k|2dk.E28

Now what we need more has made sense. It is the eigensystem of the operator R(k)U. To make a computation about it, we give a standard basis to the Hilbert space ℋc,

|0=10,|1=01,E29

from which a matrix representation follows,

RkU=eikcosθeiksinθeiksinθeikcosθ.E30

The matrix contains two different eigenvalues

λjk=1j1c2sin2k+icsinkj=1,2,E31

in which cand sare short for cosθand sinθ, respectively. Differentiating Eq. (31) with respect to the variable k, we get

λjk=1jiccosk1c2sin2k1j1c2sin2k+icsink,E32

from which the function j′(k)/λj(k) is computed,

iλjkλjk=1jccosk1c2sin2k.E33

The normalized eigenvector associated to the eigenvalue λj(k) has a form

|vjk=1Njkseikccosk1j1c2sin2k,E34

with its normalized factor

Njk=21c2sin2k1c2sin2k+1jccosk.E35

Back in Eq. (28), we put j′(k)/λj(k) = x (j = 1, 2) and then obtain another integral form

limtEXttr=j=1212πππiλjkλjkrvjk|ψ^0k2dk=ccxr|c|sπc1x2c2x21|α|2|β|2+sαβ¯+α¯βcxdx=|c|cxrsπ1x2c2x21|α|2|β|2+sαβ¯+α¯βcxdx=xrsπ1x2c2x2×1|α|2|β|2+sαβ¯+α¯βcxI|c|,|c|xdx.E36

Equation (36) allows us to hold Eq. (8).

Now that we have obtained a limit theorem for the QW whose operator was defined by

U=cosθ|00|+sinθ|01|+sinθ|10|cosθ|11|(θ[0,π)),E37

Theorem 1 can be extended for the parameter θ ∈ [0, 2π).

Theorem 2Assume that θ ∈ [0, 2π) and θ ≠ 0, π/2, π, 3π/2. For a real number x, we have

limtXttx=xsinθπ1y2cos2θy2×1|α|2|β|2+sinθαβ¯+α¯βcosθyI|cosθ|,|cosθ|ydy.E38

Since we already had the limit theorem for the parameter θ ∈ [0, π) as Theorem 1, it is enough to prove Theorem 2 for θ ∈ [π, 2π) (θ ≠ π, 3π/2). Do you think we have to carry out the same calculation for such a parameter again? We do not actually have to do that and can avoid the same math by applying a small skill to the operator U. Let us slightly change the form of the operator U, which is described by U(θ) below,

Uθ:=U=cosθ|00|+sinθ|01|+sinθ|10|cosθ|11|=cosθπ|00|+sinθπ|01|+sinθπ|10|cosθπ|11|=Uθπ.E39

The negative sign in front of the operator U(θ − π) in Eq. (39) does not affect the probability distribution ℙ(Xt = x), which means the probability distribution given by the operator U(θ) is completely same as that given by the operator U(θ − π). Moreover, as long as the parameter θpicks a value in the interval [π, 2π), the variable θ − πstays in the interval [0, π). Since Theorem 1 works on the QW operated by U(θ − π) (θ ∈ [π, 2π)), one can assert a limit theorem for the parameter θ ∈ [π, 2π),

limtXttx=xsinθππ1y2cos2θπy21|α|2|β|2+sinθπαβ¯+α¯βcosθπyI|cosθπ|,|cosθπ|ydy=xsinθπ1y2cos2θy2×1|α|2|β|2+sinθαβ¯+α¯βcosθyI|cosθ|,|cosθ|ydyE40

We should remark that Eq. (40) holds under the condition θ − π ≠ 0, π/2, that is, θ ≠ π, 3π/2. As a consequence of Theorem 1 and Eq. (40), Theorem 2 comes up.

Figure 4 shows an example of the limit density function.

ddxlimtXttx=sinθπ1x2cos2θx2×1|α|2|β|2+sinθαβ¯+α¯βcosθxI|cosθ|,|cosθ|xdy.E41

We see that the limit density function reproduces the features of the probability distribution shown in Figure 1.

## 3. Two-period time-dependent QW

In this section, we see a time-dependent QW whose coin-flip operator depends on time. The evolution of the QW is given by two unitary operators,

U1=cosθ1|00|+sinθ1|01|+sinθ1|10|cosθ1|11|=c1|00|+s1|01|+s1|10|c1|11|,E42
U2=cosθ2|00|+sinθ2|01|+sinθ2|10|cosθ2|11|=c2|00|+s2|01|+s2|10|c2|11|,E43

with θj ∈ [0, 2π) (j = 1, 2), and cosθj(resp. sinθj) has been briefly written as cj(resp. sj). The total system at time tevolves to the next state at time t + 1 according to the time evolution rule

|Ψt+1=SC1|Ψtt=0,2,4,SC2|Ψtt=1,3,5,,E44

where

Cj=x|xx|Ujj=1,2.E45

It is plane that the operators C1 and C2 are alternately casted on the QW, which means that the unitary operator 2-periodically changes in time-line. If the parameters θ1 and θ2 take the same value, then the QW becomes the standard walk defined in Sec.

Now, we are looking at examples of the probability distribution when the walker starts off with |Ψ0=|01/2|0+i/2|1. Figure 5 draws the probability distribution at time 500 and two sharp peaks are observed in each picture. In the pictures, only positive values of probability are plotted. As shown in Figure 6, the probability distribution is spreading in proportion to time t. We also see how it depends on the parameters θ1 and θ2 in Figure 7.

The features, which we have seen in Figures 5 to 7 are caught by a limit theorem.

Theorem 3Assume that θ1θ2 ≠ 0, π/2, π, 3π/2. For a real number x, we have

limtXttx=x1ξθ1θ22π1y2ξθ1θ22y2×1|α|2|β|2+sinθ1αβ¯+α¯βcosθ1yIξθ1θ2,ξθ1θ2ydy,E46

with ξ(θ1θ2) = min{|cosθ1|, |cosθ2|}.

This limit theorem was proved by Fourier analysis in 2010 [7]. First, we find the time evolution of the Fourier transform |ψ^tk=xeikx|ψtx(k[π,π)),

|ψ^t+1k=RkU1|ψ^tkt=0,2,4,RkU2|ψ^tkt=1,3,5,,E47

which comes from Eq. (44). We should recall R(k) = eik|0⟩ ⟨0| + e− ik|1⟩ ⟨1|. From the recurrence, the transform at each time gets a connection to its initial state,

|ψ^2tk=RkU2RkU1t|ψ^0k,E48
|ψ^2t+1k=RkU1RkU2RkU1t|ψ^0k.E49

The eigensystem of the matrix R(k)U2R(k)U1 reforms Eqs. (48) and (49),

|ψ^2tk=j=12λjktvjk|ψ^0k|vjk,E50
|ψ^2t+1k=RkU1j=12λjktvjk|ψ^0k|vjk.E51

Arranging the r-th derivatives (r = 0, 1, 2, …) of the Fourier transform on the scale of time t,

drdkr|ψ^2tk=trj=12λjktrλjkrvjk|ψ^0k|vjk+Otr1,E52
drdkr|ψ^2t+1k=trRkU1j=12λjktrλjkrvjk|ψ^0k|vjk+Otr1,E53

we obtain the representations of the r-th moment of the random variable Xt,

EX2tr=trj=1212πππiλjkλjkrvjk|ψ^0k2dk+Otr1,E54
EX2t+1r=trj=1212πππiλjkλjkrvjk|ψ^0k2dk+Otr1.E55

Dividing these equations by time 2tor 2t + 1 followed by taking a limit makes the same expression,

limtEX2tr2tr=j=1212πππiλjk2λjkrvjk|ψ^0k2dk,E56
limtEX2t+1r2t+1r=limtEX2t+1r2tr2t2t+1r=j=1212πππiλjk2λjkr|vjk|ψ^0k|2dk,E57

which are combined as

limtEX2t2tr=limtEX2t+12t+1r=j=1212πππiλjk2λjkrvjk|ψ^0k2dk.E58

As a result, we have

limtEXttr=j=1212πππiλjk2λjkrvjk|ψ^0k2dk.E59

The Hilbert space ℋcspanned by Eq. (29) gives a matrix representation to the operator R(k)U2R(k)U1,

RkU2RkU1=c1c2e2ik+s1s2s1c2e2ikc1s2s1c2e2ik+c1s2c1c2e2ik+s1s2,E60

and one can find its eigenvalues

λjk=c1c2cos2k+s1s21ji1c1c2cos2k+s1s22j=1,2.E61

The normalized eigenvector associated to the eigenvalue λj(k) takes a form

|vjk=1Njks1c2e2ikc1s2ic1c2sin2k1j1c1c2cos2k+s1s22,E62

with

Njk=1c12s12c22s224c1c2s1s2cos2k+2c12c22sin22k+1j2c1c2sin2k1c1c2cos2k+s1s22.E63

Here we compute

iλjk2λjk=1jc1c2sin2k1c1c2cos2k+s1s22,E64

from Eq. (61). Putting j′(k)/2λj(k) = x (j = 1, 2) gives rise to another expression of Eq. (59),

limtEXttr=xr1ξθ1θ22π1x2ξθ1θ22x2×1|α|2|β|2+sinθ1αβ¯+α¯βcosθ1xIξθ1θ2,ξθ1θ2xdx.E65

For the same reason as the proof for Theorem 1, this convergence promises Theorem 3. As mentioned earlier, if the parameters θ1 and θ2 take the same value θ, then the 2-period time-dependent QW is the standard walk shown in Sec.2. In that case, Theorem 3 is in agreement with Theorem 2. Indeed, inserting a value, which is supposed to be θnow, to both θ1 and θ2 in Theorem 2 produces the limit probability distribution below,

limtXttx=xsinθπ1y2cos2θy2×1|α|2|β|2+sinθαβ¯+α¯βcosθyI|cosθ|,|cosθ|ydy.E66

Given an initial state with α=1/2,β=i/2, Figure 8 draws the limit density function

ddxlimtXttx=1ξθ1θ22π1x2ξθ1θ22x2Iξθ1θ2,ξθ1θ2x.E67

We confirm that the function has two singularities at the points ± ξ(θ1θ2) which correspond to two sharp peaks in Figure 5.

## 4. Three-period time-dependent QW

The standard QW in Sec.2 and the two-period time-dependent QW in Sec.3 have the same type of limit density function. In the final section, we see a three-period time-dependent QW and its limit density function. As a result, a different type of limit density function will be discovered. With a unitary operator U ∈ U(2), the system of three-period time-dependent QW is periodically updating,

|Ψt+1=SC|Ψtt=0,3,5,SC|Ψtt=1,4,6,S|Ψtt=2,5,7,,E68

where

C=x|xx|U,E69
S=x|x1x||00|+|x+1x||11|.E70

The 3-period time-dependent QW was studied by Grünbaum and Machida [8] when the unitary operator Uwas of the form

U=cosθ|00|+sinθ|01|+sinθ|10|cosθ|11|=c|00|+s|01|+s|10|c|11|,E71

with θ ∈ [0, 2π). Note that we have abbreviated cosθand sinθto cand sin Eq. (71), respectively. Let us view the probability distribution ℙ(Xt = x) when the initial state is given by |Ψ0=|01/2|0+i/2|1. The operator in Eq. (71) can give rise to probability distributions which have four sharp peaks, as shown in Figure 9. These four peaks can also be observed at relatively small time in Figure 10. Seeing Figure 11, we guess some values of the parameter θwhen the number of sharp peaks is three. They are π/3, 2π/3, 4π/3, and 5π/3, and these values can be exactly estimated by a limit theorem which will be introduced later.

We find a long-time limit theorem in the paper [8] and it asserts the convergence of a random variable rescaled by time t.

Theorem 4Assume that θ ≠ 0, π/2, π, 3π/2. For a real number x, we have

limtXttx=x[1ναβyfyI14c231+8c23y+1+να,β;yfyI1+8c23,14c23y]dy,E72

where

fx=|s||s|x+Dx2π1x2W+xWxDx,E73
ναβx=1c1+8c29c3|α|2|β|2+3s1+6c2αβ¯x+sc|s|1+8c2cs|α|2|β|21+2c2αβ¯Dx,E74
Dx=1+8c29c2x2,E75
W+x=14c2+312c2x2+2|s|xDx,E76
Wx=1+8c231+2c2x22|s|xDx,E77

and ℜ (z) denotes the real part of the complex number z.

This limit theorem can be derived by Fourier analysis as well. For the Fourier transform |Ψ^tk=xeikx|ψtx(k[π,π)), Eq. (68) produces a time evolution of the Fourier transform,

|ψ^3tk=RkRkU2t|Ψ^0k,|ψ^3t+1k=RkURkRkU2t|Ψ^0k,|ψ^3t+2k=RkU2RkRkU2t|Ψ^0k.E78

Given an orthogonal normalized basis such as Eq. (29), the matrix

RkRkU2=c2e3ik+s2eikcse3ikcseikcse3ik+cseikc2e3ik+s2eikE79

has two eigenvalues

λjk=c2cos3k+s2cosk1ji1c2cos3k+s2cosk2j=1,2.E80

A possible expression of the normalized eigenvector associated to the eigenvalue λj(k) is

|vjk=1Njk2cse2iksinkc2sin3k+s2sink+1j1c2cos3k+s2cosk2,E81

where Nj(k) is the normalization factor

Njk=21c2cos3k+s2cosk2+1jc2sin3k+s2sink1c2cos3k+s2cosk2.E82

With a decomposition |Ψ^3tk=j=12λjtkvjk|Ψ^0k|vjk, we get representations in the eigenspace,

|ψ^3tk=j=12λjktvjk|ψ^0k|vjk,E83
|ψ^3t+1k=RkUj=12λjktvjk|ψ^0k|vjk,E84
|ψ^3t+2k=RkU2j=12λjktvjk|ψ^0k|vjk,E85

and compute their derivatives

drdkr|ψ^3tk=trj=12λjktrλjkrvjk|ψ^0k|vjk+Otr1,E86
drdkr|ψ^3t+1k=trRkUj=12λjktrλjkrvjk|ψ^0k|vjk+Otr1,E87
drdkr|ψ^3t+2k=trRkU2j=12λjktrλjkrvjk|ψ^0k|vjk+Otr1.E88

The moments EX3tr,EX3t+1r, and EX3t+2rturn out to be of the form

trj=1212πππiλjkλjkrvjk|ψ^0k2dk+Otr1,E89

and we see

limtEX3t3tr=limtEX3t+13t+1r=limtEX3t+23t+2r=j=1212πππiλjk3λjkrvjk|ψ^0k2dk,E90

which is put together as

limtEXttr=j=1212πππiλjk3λjkrvjk|ψ^0k2dk,E91

where

iλjk3λjk=1j3c2sin3k+s2sink31c2cos3k+s2cosk2.E92

Setting iλjk/3λjk=xj=1,2in Eq. (91) leads us to an integral expression of the limit,

limtEXttr=xr[1ναβxfxI14c231+8c23x+1+να,β;xfxI1+8c23,14c23x]dx,E93

which guarantees Theorem 4. Figure 12 shows the limit density function (d/dx)ℙ(Xt/t ≤ x) when α=1/2,β=i/2, and we view the features of Figure 9 in the limit density function. The density function contains singular points at ±14c2/3,±1+8c2/3. When θ = π/4, they are found at ±1/3=±0.333,±5/3=±0.745in Figure 12-(a), and when θ = 2π/5, at ±51/6=±0.206,±45/3=±0.442in Figure 12-(b).

## Acknowledgments

The author is supported by JSPS Grant-in-Aid for Young Scientists (B) (No. 16K17648).

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Takuya Machida (August 24th 2016). Quantum Walks, Research Advances in Quantum Dynamics, Paul Bracken, IntechOpen, DOI: 10.5772/62481. Available from:

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