## 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 ℋ_{p} spanned by an orthogonal normalized basis _{c} spanned by an orthogonal normalized basis _{c} as the down-spin and the up-spin states of a quantum particle, respectively. The Hilbert space ℋ_{p} is called a position space and the Hilbert space ℋ_{c} is called a coin space. A QW on the line at time *t* (=0, 1, 2 …) is expressed in the tensor Hilbert space,

Customarily, *x* at time *t*. Given an initial state

with

where *U* is a unitary operator. In this chapter, we employ a form of the operator

and an initial state

assuming the complex numbers *α* and *β* satisfy the condition |*α*|^{2} + |*β*|^{2} = 1. The reason that we assigned unitarity to the operator *U* and assumed the constraint |*α*|^{2} + |*β*|^{2} = 1 is that we define the probability that the quantum walker is observed at position *x* at time *t*,

where the random variable *X*_{t} is 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(*X*_{t} = *x*) at time 500 when the walker starts updating with *X*_{t} = *x*) in **Figure 2** when we take **Figure 1**, the probability distribution ℙ(*X*_{t} = *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(*X*_{t} = *x*) after time *t* goes 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, 2*π*) (Theorem 2) which is easily proved by making the most of Theorem 1.

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

*where I*_{A}(*x*) *is an indicator function such that*

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

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

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

with

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

To prove the theorem, we concentrate on a convergence

with

It is known from probability theory that the convergence guarantees Eq. (8). Before we compute the limit, let us depict the *r*-th moment *t* is rewritten in a finite sum

Noting that

we have

Integrating Eq. (19) over the interval [−*π*, *π*), one can generate the *r*-th moment of the random variable *X*_{t},

(20) |

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

Here, let *λ*_{j}(*k*) (*j* = 1, 2) be the eigenvalues of the matrix *R*(*k*)*U*, and *λ*_{j}(*k*). Then the initial state of the Fourier transform is decomposed by the normalized eigenvectors,

The Fourier transform at time *t* is, therefore, expressed with the eigenvalues and the eigenvectors,

which also gives a description of the derivative

where *λ*_{j}′(*k*) = (*d*/*dk*)*λ*_{j}(*k*). Recalling the initial state in Eq. (6), we now have the Fourier transform at time 0 in the form

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

Dividing Eq. (26) by *t*^{r}, we reach a representation

and obtain a convergence as *t* → ∞,

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 ℋ_{c},

from which a matrix representation follows,

The matrix contains two different eigenvalues

in which *c* and *s* are short for *cosθ* and *sinθ*, respectively. Differentiating Eq. (31) with respect to the variable *k*, we get

from which the function *iλ*_{j}′(*k*)/*λ*_{j}(*k*) is computed,

The normalized eigenvector associated to the eigenvalue *λ*_{j}(*k*) has a form

with its normalized factor

Back in Eq. (28), we put *iλ*_{j}′(*k*)/*λ*_{j}(*k*) = *x* (*j* = 1, 2) and then obtain another integral form

(36) |

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

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

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

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

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*, 2*π*) (*θ* ≠ *π*, 3*π*/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,

(39) |

The negative sign in front of the operator *U*(*θ* − *π*) in Eq. (39) does not affect the probability distribution ℙ(*X*_{t} = *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 *θ* picks a value in the interval [*π*, 2*π*), the variable *θ* − *π* stays in the interval [0, *π*). Since Theorem 1 works on the QW operated by *U*(*θ* − *π*) (*θ* ∈ [*π*, 2*π*)), one can assert a limit theorem for the parameter *θ* ∈ [*π*, 2*π*),

(40) |

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

**Figure 4** shows an example of the limit density function.

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,

with *θ*_{j} ∈ [0, 2*π*) (*j* = 1, 2), and *cosθ*_{j} (resp. *sinθ*_{j}) has been briefly written as *c*_{j} (resp. *s*_{j}). The total system at time *t* evolves to the next state at time *t* + 1 according to the time evolution rule

where

It is plane that the operators *C*_{1} and *C*_{2} are alternately casted on the QW, which means that the unitary operator 2-periodically changes in time-line. If the parameters *θ*_{1} and *θ*_{2} 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 **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 *θ*_{1} and *θ*_{2} in **Figure 7**.

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

**Theorem 3** *Assume that θ*_{1}, *θ*_{2} ≠ 0, *π*/2, *π*, 3*π*/2*. For a real number x, we have*

*with ξ*(*θ*_{1}, *θ*_{2}) = *min*{|*cosθ*_{1}|, |*cosθ*_{2}|}.

This limit theorem was proved by Fourier analysis in 2010 [7]. First, we find the time evolution of the Fourier transform

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

The eigensystem of the matrix *R*(*k*)*U*_{2}*R*(*k*)*U*_{1} reforms Eqs. (48) and (49),

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

we obtain the representations of the *r*-th moment of the random variable *X*_{t},

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

which are combined as

As a result, we have

The Hilbert space ℋ_{c} spanned by Eq. (29) gives a matrix representation to the operator *R*(*k*)*U*_{2}*R*(*k*)*U*_{1},

and one can find its eigenvalues

The normalized eigenvector associated to the eigenvalue *λ*_{j}(*k*) takes a form

with

Here we compute

from Eq. (61). Putting *iλ*_{j}′(*k*)/2*λ*_{j}(*k*) = *x* (*j* = 1, 2) gives rise to another expression of Eq. (59),

For the same reason as the proof for Theorem 1, this convergence promises Theorem 3. As mentioned earlier, if the parameters *θ*_{1} and *θ*_{2} 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 *θ* now, to both *θ*_{1} and *θ*_{2} in Theorem 2 produces the limit probability distribution below,

Given an initial state with **Figure 8** draws the limit density function

We confirm that the function has two singularities at the points ± *ξ*(*θ*_{1}, *θ*_{2}) 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,

where

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

with *θ* ∈ [0, 2*π*). Note that we have abbreviated *cosθ* and *sinθ* to *c* and *s* in Eq. (71), respectively. Let us view the probability distribution ℙ(*X*_{t} = *x*) when the initial state is given by **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 *θ* when the number of sharp peaks is three. They are *π*/3, 2*π*/3, 4*π*/3, and 5*π*/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 4** *Assume that θ* ≠ 0, *π*/2, *π*, 3*π*/2*. For a real number x, we have*

where

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

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

has two eigenvalues

A possible expression of the normalized eigenvector associated to the eigenvalue *λ*_{j}(*k*) is

where *N*_{j}(*k*) is the normalization factor

With a decomposition

and compute their derivatives

The moments

and we see

which is put together as

where

Setting

which guarantees Theorem 4. **Figure 12** shows the limit density function (*d*/*dx*)ℙ(*X*_{t}/*t* ≤ *x*) when **Figure 9** in the limit density function. The density function contains singular points at *θ* = *π*/4, they are found at **Figure 12-(a)**, and when *θ* = 2*π*/5, at **Figure 12-(b)**.