Higher-Dimensional Quantum Walk in Terms of Quantum Bernoulli Noises.

As a discrete-time quantum walk model on the one-dimensional integer lattice Z , the quantum walk recently constructed by Wang and Ye [Caishi Wang and Xiaojuan Ye, Quantum walk in terms of quantum Bernoulli noises, Quantum Information Processing 15 (2016), 1897-1908] exhibits quite different features. In this paper, we extend this walk to a higher dimensional case. More precisely, for a general positive integer d ≥ 2 , by using quantum Bernoulli noises we introduce a model of discrete-time quantum walk on the d-dimensional integer lattice Z d , which we call the d-dimensional QBN walk. The d-dimensional QBN walk shares the same coin space with the quantum walk constructed by Wang and Ye, although it is a higher dimensional extension of the latter. Moreover we prove that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type, which is in sharp contrast with the case of the usual higher dimensional quantum walks. Some other results are also obtained.

1. Introduction

As quantum analogs of classical random walks, quantum walks [1] have found wide application in quantum information, quantum computing and many other fields [2,3]. In the past two decades, quantum walks with a finite number of internal degrees of freedom have been intensively studied and many deep results have been obtained (see [2,3,4,5,6] and references therein). For example, Konno [5] found that a one-dimensional quantum walk with two internal degrees of freedom usually has a limit probability distribution with scaling speed n, instead of , which is far from being Gaussian.

Quantum Bernoulli noises refer to the family of annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation (CAR) in equal time, and can provide an approach to the effects of environment on an open quantum system [7,8]. In 2016, by using quantum Bernoulli noises, Wang and Ye [9] introduced a discrete-time quantum walk model on the one-dimensional integer lattice , which we call the one-dimensional QBN walk below.

Belonging to the category of unitary quantum walks, the one-dimensional QBN walk, however, exhibits quite different features. It takes the space of square integrable Bernoulli functionals as its coin space, hence has infinitely many internal degrees of freedom since is infinite-dimensional. Moreover, for some special choices of the initial state, it has the same limit probability distribution as the classical random walk [9], which is in marked contrast to the case of the usual unitary quantum walks (e.g., the Hadamard walk [5]). From a physical point of view [10], this behavior of the one-dimensional QBN walk might help understand the effects of decoherence in quantum walks.

Recent years have seen much attention paid to quantum walks on higher dimensional integer lattices. Mackay et al. [11] extended the Hadamard walk to a higher dimensional case and examined the time dependence of the standard deviation. Segawa and Konno [12] considered a quantum walk driven by many coins and found that the number of coins can have an important effect on the behavior of the walk. More recently, Komatsu and Konno [4] investigated stationary amplitudes of quantum walks on the higher-dimensional integer lattice. There are other works about quantum walks on higher dimensional integer lattices (see e.g., [13,14,15]).

In this paper, we would like to extend the one-dimensional QBN walk to a higher dimensional case. More precisely, for a general positive integer , we will use quantum Bernoulli noises to introduce a model of discrete-time quantum walk on the d-dimensional integer lattice . Our main work is as follows.

For each , by using quantum Bernoulli noises, we construct self-adjoint operators , , which act on the space of square integrable Bernoulli functionals. We prove that , are pairwise orthogonal and moreover their sum is unitary.

By taking the operators , , as coin operators, we establish a model of discrete-time quantum walk on , which we call the d-dimensional QBN walk. Of this walk, we obtain a unitary representation in the function space and a characterization in the tensor space .

Under some mild conditions, we obtain a link between amplitudes of the d-dimensional QBN walk and those of the one-dimensional QBN walk. And based on this link, we find that, for a range of choices of its initial state, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type.

As is seen, the coin space of the d-dimensional QBN walk is just the space of square integrable Bernoulli functionals, which is infinite-dimensional. Thus the d-dimensional QBN walk has infinitely many internal degrees of freedom. It should be also mentioned that the d-dimensional QBN walk shares the same coin space with the one-dimensional QBN walk, although it is a higher dimensional extension of the latter.

This paper consists of five sections. In Section 2, we briefly recall some necessary notions and facts about quantum Bernoulli noises. Our main work then lies in Section 3 and Section 4. Here, among others, we prove several supporting theorems, define our quantum walk model and examine its fundamental properties. Finally in Section 5, we make some conclusion remarks.

2. Preliminaries

In this section, we briefly recall some necessary notions and facts about quantum Bernoulli noises. We refer to [7] for details about quantum Bernoulli noises.

Throughout this paper, always denotes the set of all integers, while means the set of all nonnegative integers. We denote by the finite power set of , namely where means the cardinality of . Unless otherwise stated, letters like j, k and n stand for nonnegative integers, namely elements of .

Let be the set of all mappings , and the sequence of canonical projections on given by

Let be the -field on generated by the sequence , and a given sequence of positive numbers with the property that for all . Then there exists a unique probability measure on such that for , () with when and with . Thus one has a probability measure space , which is referred to as the Bernoulli space and random variables on it are known as Bernoulli functionals.

Let be the sequence of Bernoulli functionals generated by sequence , namely where . Clearly is an independent sequence of random variables on the probability measure space . Let be the space of square integrable complex-valued Bernoulli functionals, namely We denote by the usual inner product of the space , and by the corresponding norm. It is known that Z has the chaotic representation property. Thus form an orthonormal basis (ONB) of , which is known as the canonical ONB of . Here and

Clearly is infinite-dimensional as a complex Hilbert space.

([7]) For each where

The operators and are usually known as the annihilation and creation operators acting on Bernoulli functionals, respectively. And the family is referred to as quantum Bernoulli noises. The next lemma shows that quantum Bernoulli noises satisfy the canonical anti-commutation relations (CAR) in equal-time.

([7]) Let k, and where I is the identity operator on

For a nonnegative integer , one can define, respectively, two self-adjoint operators and on in the following manner where I is the identity operator on . It then follows from Lemma 2 that the operators , , , form a commutative family, namely

([9]) For all

In view of the commutativity of family , we can naturally introduce the following symbols and , the identity operator on . Similarly we can define for any . It can be verified that , , also form a commutative family of self-adjoint operators on . Additionally, it can be shown that whenever , with .

3. Definition and Fundamental Properties

In this section, we prove some supporting theorems, present the definition of our quantum walk and examine its fundamental properties.

In what follows, we always assume that is a given positive integer and . We denote by the d-fold cartesian product of , and by the d-fold tensor product space of . In addition, we assume that is a fixed unitary isomorphism. Such a unitary isomorphism exists because is infinite-dimensional and separable.

3.1. Coin Operators

This subsection constructs our coin operators, which will play a fundamental role in defining our quantum walk.

Recall that , for . In what follows, for notational convenience we rewrite , . And for , we use the symbol to mean the tensor product of , , ⋯, . Clearly, is a bounded operator on for each .

For where, as indicated above,

Let

For each , it follows from the fact of being self-adjoint for all that is a self-adjoint operator on , which, together with the fact of being unitary, implies that the operator defined by (15), namely , is self-adjoint as an operator on .

Let , with . Then there is some such that , where and are the jth components of and , respectively. By Lemma 3, , which implies that . Thus, we have

This completes the proof of property (i).

Next, we verify property (ii). In fact, for each , it follows from Lemma 3 that is a unitary operator on . Thus, by the property of operator tensor product, we know that is a unitary operator on , which, together with fact that is a unitary isomorphism, implies that is a unitary operator on . □

3.2. Definition and Unitary Representation

In this subsection, we present the definition of our quantum walk and find out its unitary representation. As usual, we set to be the d-fold cartesian product of , and we denote by the space of square summable functions defined on and valued in , namely where means the norm in . As is known, is a separable Hilbert space with the inner product given by where is the inner product in . By convention, elements of are usually known as vectors. A vector is called a unit vector if , where stands for the norm in . Note that each unit vector makes a probability distribution on .

Let and the function all belong to

By using Theorem 1, we have which together with the invariance gives which, together with the fact that is a unitary operator on , implies that

Therefore and . Similarly, we can show that and . □

Based on Theorem 2, we can now present the definition of our quantum walk on as follows.

The d-dimensional QBN walk is a discrete-time quantum walk on the d-dimensional integer latice

The walk takes

The time evolution of the walk is governed by equation where

In that case, the function

It is well known that . This just means that describes the position of the d-dimensional QBN walk, while describes its internal degrees of freedom. As usual, is called the coin space of the walk. Clearly, the d-dimensional QBN walk has infinitely many internal degrees of freedom because its coin space is infinite-dimensional.

For each and where

For each , denote by the function given by which, by Theorem 2, belongs to . Thus, we can define an operator on in the following manner

It is easy to see that is linear. And moreover, by Theorem 2, we know that is even an isometry, which means that has an adjoint .

Let U, . Then, by general properties of the adjoint of an operator, we have which, together with the fact of being self-adjoint, gives where is the function given by

It then follows from the arbitrariness of that , namely

This shows that (22) holds. A direct calculation yields that where means the identity operator on . Therefore, is a unitary operator satisfying (21) and (22). □

Applying Theorem 3 to Definition 2, we come to the next theorem, which shows that the d-dimensional QBN walk belongs to the category of unitary quantum walks.

Let where

3.3. Characterization in Tensor Space

As is seen, the d-dimensional QBN walk is formulated in the function space . In the present subsection, we reformulate it in the tensor space , which is isomorphic to in the sense of unitary isomorphism.

Let be the canonical unitary isomorphism. Then, satisfies that where is the function defined by , . As is indicated in Theorem 4, unitary operator sequence plays an important role in describing the d-dimensional QBN walk.

In the following, for each , we denote by the counterpart of in tensor space , namely Then is a sequence of unitary operators on . Thus, from a physical point of view, we naturally come to the next observation.

The d-dimensional QBN walk can be viewed as a unitary evolution determined by the unitary operator sequence

We now consider the structure of unitary operators , . Let and . Then, for , by letting , , we have where the series on the righthand side converges in the norm of . By Theorem 3, we have which implies that

Thus, as vectors in , we have which, together with and , yields which together with (27) implies that

Therefore, by the arbitrariness of choosing and , we come to the next result, which actually offers a characterization of the d-dimensional QBN walk in tensor space.

Let where

4. Limit Probability Distribution

In the present section, we focus on exploring limit probability distribution of the d-dimensional QBN walk. To be convenient, we additionally denote by the space of square summable functions defined on and valued in .

([9]) For each and where

4.1. Amplitude Formula

For vectors , , ⋯, , it can be verified that the function defined by belongs to . Moreover, this function even becomes a unit vector in whenever , , ⋯, are unit vectors in .

As is shown above, , are unitary operators on . Thus, for all unit vector , vectors obviously make a sequence of unit vectors in .

Let

Then, for all where with

By Lemma 4, for all and , is a unit vector in . Now, for each nonnegative integer , we define a function as

Then, as indicated above, , , are unit vectors in , and in particular which implies that .

On the other hand, for all and , by using (33) and Lemma 4, we find that which, together with the notation and (see Section 3.1 for details), gives where as specified in Section 3. Thus, by taking tensor product, we get where . Taking the action of operator on both sides and then using (15) yields ,, which together with (34) and Theorem 3 implies that ,. Thus which, together with the fact and Theorem 4, implies that , which together with (34) gives (32). This compete the proof. □

According to [

As an immediate consequence of Theorem 6, we have the following useful corollary, which offers a formula for calculating the probability to find out the walker at a position in .

Let

Then, for all where with

4.2. Limit Probability Distribution

For , we write , where and are the creation and annihilation operators on , see Section 2 for details. By Lemma 2, , make a commutative sequence of self-adjoint operators on . Moreover, by the CAR in equal time, one has where I denotes the identity operator on . In the following, we write and

It can be verified that form a commutative family of self-adjoint unitary operators on .

For , we write . Additionally, for and , we define a functional on in the following manner where and means the cardinality of as a set.

Let

Suppose that where  □

For each , by using the method of Fourier transform for vector-valued functions, we can get an expression of of the following form

Let and . Then, with the notation , we have

On the other hand, for each , it follows easily that

Thus which implies that

Note that vectors and are orthogonal for , with . Thus, by (44) and (45), we have which together with the definition of functional gives (43). □

A vector

Every basis vector in the canonical OBN

Let . Then, for all , we have . On the other hand, we can verify that make an orthonormal system in . Thus, for any and , we have which implies that has the ABD property. □

Let

Suppose that

Then it holds that

Let . Then, by Theorem 7, we have

On the other hand, since has the ABD property, there exist constant and such that which implies that which, together with , yields that

Therefore

This completes the proof. □

The next result establishes a limit theorem for the d-dimensional QBN walk, which shows that for a range of choices of its initial state the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type.

Let the initial state where where namely

For each , we can define a function in the following manner:

Clearly, , , ⋯, are unit vectors in , and moreover they admit the following relations

Thus, by Corollary 1, we have where with for all index .

Now, let us consider the characteristic function of random vector . By definition, we have where . Using (54) gives . For each , by using Theorem 8, we find

Therefore which implies that converges in law to the d-dimensional standard Gaussian distribution. □

5. Conclusions Remarks

As is well known, the Hadamard walk is a one-dimensional quantum walk, whose coin space is a two-dimensional space (typically ). In 2002, by extending the Hadamard walk to a higher dimensional case, Mackay et al. [11] actually introduced a d-dimensional quantum walk for a general . However, their d-dimensional quantum walk takes a -dimensional space as its coin space, hence has a finite number of internal degrees of freedom. In other words, as a higher dimensional extension of the Hadamard walk, the d-dimensional quantum walk introduced by Mackay et al. [11] does not share the same coin space with the Hadamard walk.

As is seen, in this paper we introduce a d-dimensional quantum walk in terms of quantum Bernoulli noises, which is called the d-dimensional QBN walk. The coin space of the d-dimensional QBN walk is the space of square integrable Bernoulli functionals, which is infinite-dimensional. Thus the d-dimensional QBN walk has infinitely many internal degrees of freedom. Moreover, the d-dimensional QBN walk shares the same coin space with the one-dimensional QBN walk (namely the one recently introduced in [9]), although it is a higher dimensional extension of the latter.

It should be noted that the existence of a unitary isomorphism plays a key role in constructing the d-dimensional QBN walk. For a finite dimensional space, say , there exists no unitary isomorphism from to unless . This just means that our approach in this paper differs from that used by Mackay et al. in [11].

Decoherence is one of important topics in the study of quantum walks. Physically, decoherence means a deviation from pure quantum behavior. If a quantum walk shows some classical asymptotic behavior, then it contains an amount of decoherence. Kendon and Tregenna [16] showed for the first time that decoherence can be useful in quantum walks. Brun et al. [17] investigated quantum walks with decoherent coins. Chisaki et al. [18] analyzed a class of quantum walks with position measurements and found that those walks have limit probability distributions of Gauss type under some situations, which means that quantum walks with position measurements can produce decoherence. There are other works addressing decoherence in quantum walks (see [10] and references therein). As is seen, as a model of higher-dimensional quantum walk constructed in terms of quantum Bernoulli noises, the d-dimensional QBN walk has a limit probability distribution of d-dimensional standard Gauss type for some choices of its initial state, which together with the work of [9] implies that quantum Bernoulli noises can provide an alternative way to produce decoherence in quantum walks.