Ito's formula for the discrete-time quantum walk in two dimensions

Following [Konno, arXiv:1112.4335], it is natural to ask: What is the Ito's formula for the discrete time quantum walk on a graph different than Z, the set of integers? In this paper we answer the question for the discrete time quantum walk on Z^2, the square lattice.


I. Introduction
Ito's formula which is related to the Ito's lemma is used in stochastic calculus to find the differential of a function of a particular type of stochastic process, and has a wide range of applications. According to the author in [1], the Ito's formula for the random walk has been investigated [2,3]. However, in the quantum walk case, the Ito's formulas are unknown. In [1] the author presents the Ito formula for the one-dimensional discrete-time quantum walk and gives some examples including Tanaka's formula by using the formula. Integrals for the quantum walk is also discussed.
In the present paper new results on the Ito's formula for the discrete-time quantum walk on the square lattice is given. This paper is organized as follows. In Section II we define the quantum walk on the square lattice, there the dynamics of the walk in the Fourier picture is recorded in Lemma 1. In Section III we present an Ito's formula for the discrete-time quantum walk on the square lattice as well as a Tanaka-type formula for the quantum walk. In [4] the author of the present paper computed some sojourn times of the Grover walk in two dimensions. The Tanaka-type formula may be useful in computing local time at the origin. Section IV is devoted to the conclusions, there two types of problems for further exploration is discussed. The first concerns integrals for the quantum walk whilst the second concerns another relation on the Ito's formula for the discrete-time quantum walk on the square lattice.

II. Definitions
Recall that the discrete-time quantum walk is the quantum analogue of the classical random walk with an additional degree of freedom call chirality. In the two-dimensional setting on the square lattice, the chirality takes values left, right, downward, and upward, and means the direction of motion of the walker. At each time step, the particle moves according to its chirality state. For example, if the chirality state is upward, then the particle moves one step up. Let us define , where C is the set of complex numbers. The unitarity of U gives 1 , where z denotes complex conjugation, and bc ad U matrix which is also unitary. In order to define the dynamics of the model we write We should note that represents the walker moves to the left, right, downward, and upward directions respectively at position ) , ( y x at each time step. Let the set of initial quibit states at the origin for the quantum walk be given by . We should note that the definition implies we can write The probability that the quantum walker is in position . From now on we consider the Fourier transform of , ( , we obtain the following Lemma 1: For any is also a power of n by the adjoint operation.

III. Ito-Type Formulas for the Discrete-Time Quantum Walk on the Square Lattice
. From now on we consider the quantum walk , and noting that , a direct computation gives the following . For any   We should remark that the expressions in part (A) of Proposition 2 is the Ito formulas for the discrete-time quantum walk on the square lattice.
is defined in a similar way, then from Proposition 2 we immediately obtain the following . For any    Next we present a Tanaka-type formula for the discrete-time quantum walk on the square lattice.
Making a similar substitution in the second expression in part (A) of Theorem 3, we get the Tanakatype formula for the discrete-time quantum walk on the square lattice as follows.

Corollary 4:
Similarly, we can show that the second term on the RHS of the first expression in part (b) of Theorem 3, can be written as