_{1}

^{*}

The present paper is focused on non-uniform quantum coins for the quantum random walk search algorithm. This is an alternative to the modification of the shift operator, which divides the search space into two parts. This method changes the quantum coins, while the shift operator remains unchanged and sustains the hypercube topology. The results discussed in this paper are obtained by both theoretical calculations and numerical simulations.

The search algorithms for unstructured databases are widely used in statistical data processing for searching the maximum or minimum element or an element corresponding to specific criteria. Effective search algorithms can provide a solution for one of the Non-Deterministic Polynomial Time Complete (NPTC) problems, from which a solution can be found to any NPTC problem by an algorithm with polynomial complexity. These are the reasons for the great interest in the quantum search algorithms and their experimental implementation.

The first quantum search algorithm for unstructured databases is created by Grover [

There are already many classical random walk algorithms that perform much better in their tasks than deterministic algorithms. Two classes of such algorithms are Las Vegas algorithms (which always end with a correct result when used for a finite time) and Monte Carlo algorithms (which depend on random input and might produce an incorrect result). Las Vegas algorithms are widely used in fields like artificial intelligence [

Another type of quantum algorithms are the ones based on the quantum random walk; they are analogous to the classical random walk. There are two types of those algorithms: continuous time evolution random walk algorithms (CTRWA) and discrete time random walk algorithms (DTRWA). The CTRWA were first introduced by Farhi and Gutmann [

Grover’s search, CTRWSA and DTRWSA differ conceptually in terms of working principle. This is the reason for their different advantages and disadvantages. Grover’s search algorithm and DTRWSA can be modified to find a solution with probability close to one. Long has shown that Grover’s search can be modified so that the probability of successfully finding a solution with it to be exactly equal to one [

The present paper is organized as follows. In Section 2, the discrete quantum random walk is reviewed, and the quantum random walk on a line is shown as an example. In Section 3, the quantum random walk on a hyper- cube and the quantum random walk search on a hypercube are reviewed. In Section 4, a new alternative way is demonstrated for the method shown in [

The classic random walk on a line starts at an initial state

The quantum random walk algorithm is the quantum analogue of the classic random walk algorithm. H^{C} is the Hilbert space of the quantum coin (coin space) and H^{S} is the Hilbert space of the nodes of the structure of the graph. Again, each step of the algorithm (which is described by the operator U) has two parts. First is the coin toss. The coin flip is defined by the unitary operator of the coin C_{0}, which acts in the coin space H^{C}. The coin operator acts upon the Hilbert space

An example for a shift operator is S_{L} corresponding to a quantum random walk on a line [

where x is the position of the particle on the line. The values of d (0 or 1) correspond to left and right directions. For the coin, a Hadamard matrix can be used:

Summarily, a DQRW step on line can be written as

The hypercube is a graph with _{C} is:

where d is the direction of the motion and

The Grover coin G is frequently chosen for a coin for quantum random walks on a hypercube. This coin is invariant to all permutations of the n edge directions, so it sustains the permutation symmetries of the hypercube.

where I is identity operator,

To make a quantum random walk search algorithm, a quantum oracle should be applied that marks the wanted element by applying a coin upon it. The oracle does this by using the function_{0} or C_{1},

Summarily, the operator of the coin becomes:

where C_{1} can be almost any unitary operator but most often it is taken

The quantum circuit of the random walk search algorithm is shown in

1) Initializing the starting state of the coin and node register in an equal weight superposition. This can be done by applying Hadamard gate on each qubit of the state

2) Applying quantum random walk search iteration

The steps of the quantum random walk search iteration are:

a) Applying a quantum oracle;

b) Applying an appropriate coin depending on the state of the control register;

c) Applying the quantum oracle;

d) Applying the shift operator.

Due to the symmetry of the hypercube, its nodes can always be re-labeled in such way that the marked node

The shift operator in this collapsed random walk basis

The quantum random walk on a line strongly depends on the position. In the basis

where

Potocek et al. have shown in [

In this chapter it will be demonstrated that the same result can also be obtained by using appropriate coins.

The generalized Householder reflection

where _{1}) with a minus sign is used. It can be viewed as _{0}), the Grover’s coin is used which is also a Householder reflection when

In [_{0} is again an equal superposition vector, but the phase is random. In this paper, only Householder reflection with phase equal to π will be discussed, when

Here we will view the case when C_{0} is a standard Grover coin, as it is in the original SKW search algorithm:

where

For the marking coin C_{1}, an arbitrary Householder reflection is taken with a phase π:

here a_{j} is real and

The coin and the random walk step are unchanged, as in the standard SKW search algorithm:

The perturbed random walk coin is:

This form is too general, so in the next subsection some examples for coins will be discussed.

The steps of this implementation of QRWS are the same as in the SKW search algorithm. The quantum circuit of this algorithm is almost the same as in SKW and is shown in

Some examples of coins suitable for a random walk search are proposed in this section. These examples are probably not only the useful ones but also can be performed relatively easily in experiments. A Householder reflection can easily be done with an N-pod system.

For simplicity, a hypercube with dimension 2K, instead of a hypercube with dimension N will be reviewed.

From here on, y_{i} will denote arbitrary values, and y_{i} may or may not be equal to y_{j} at_{i} at any i. The number of y_{i} as altogether is n − 1.

One case of asymmetrical coins is when

In the first searched subspace, the coin marks the element marked by the oracle. In the second searched subspace, the marking coin effectively marks one of the adjacent nodes. The matrix chosen to be used in the marking coin defines which of the nodes is marked. The division of the searched space in two requires an additional qubit (the number of states of the register prior to the division is 2K) in order to perform the search and to have a probability of finding a solution above 80%.

Here are two examples of such coins, depending on the way of doubling the number of states by adding a qubit:

The first type of such coin is when

This result can easily be explained when_{1}, C_{0}, _{0} marks all edges connected with the states with a plus sign. The coin _{1} marks all edges except the last one with a plus sign, and the last one?with a minus sign (

The number of steps needed depends on the exact values of x and y_{i}. The quantum circuit needed for those types of coins is shown in

The simulations are made by two qubit coins because of the absence of enough computational power to make simulations for coins with more qubits.

Values

With two-qubit coins,

It has been obtained by numerical simulations that a higher probability of finding the searched element is achieved when

Another type of such coin is the case when

When

With all other cases having this structure of the coin, when

Numerical simulations show that at least when the size of the node register N = 16, there are also other coins with different shape of the vector

Examples for such coins are when

and when

An example for such coins is when the formula (28) is used with the quantum circuit shown in

Another example of search coin is when the formula (26) is used with the quantum circuit shown in

The number of steps needed for the coins showed in this section is obtained by numerical simulations and has not been found empirically yet.

A discrete quantum random walk search algorithm optimized for hypercube is discussed. A new alternative DTRWS method for dividing the searched space in two is presented. The searched space is divided effectively into two by using asymmetric coins, which distribute the probability of shifting into neighboring nodes non-un- iformly.

The advantage of this method is that it preserves the topology of the hypercube and does not divide it by modifying the shift operator. The coins are obtained by using Householder reflection, which can be easily performed in experiments by using N-pod systems.