Implementation of quantum walks on IBM quantum computers

Abstract

The development of universal quantum computers has achieved remarkable success in recent years, culminating with the quantum supremacy reported by Google. Now it is possible to implement short-depth quantum circuits with dozens of qubits and to obtain results with significant fidelity. Quantum walks are good candidates to be implemented on the available quantum computers. In this work, we implement discrete-time quantum walks with one and two interacting walkers on cycles, two-dimensional lattices, and complete graphs on IBM quantum computers. We are able to obtain meaningful results using the cycle, the two-dimensional lattice, and the complete graph with 16 nodes each, which require 4-qubit quantum circuits up to depth 100.

Appendices

A Appendix

Proposition 1

The decomposition of P given by Eq. (11) in terms of multi-controlled Toffoli gates is

$$\begin{aligned} P = X_{k-1}\,C_{k-1}(X_{k-2})C_{k-2,k-1}(X_{k-3})\cdots C_{1,\ldots ,k-1}\,(X_{0}), \end{aligned}$$

where k is the number of qubits.

Proof

From Eq. (11), we have

$$\begin{aligned} P|q\rangle ={\left\{ \begin{array}{ll} |q+1\rangle , &{} \text {if }q<N-1 \\ |0\rangle , &{}\text {if }q=N-1, \end{array}\right. } \end{aligned}$$

(34)

where $|q\rangle$ is a generic state of the computational basis in decimal notation. The binary representation of q is $(q_0\ldots q_{k-1})_2$. In the case $q=N-1$, the binary representation of the state is $|(1\ldots 1)_2\rangle$ and it is straightforward to verify that the circuit of P in Fig. 2 generates the desired output state $|(0\ldots 0)_2\rangle$. Suppose that $q<N-1$. The action of P on a generic qubit state is

$$\begin{aligned} P|q_0\cdots q_{k-1}\rangle= & {} C_{1,\ldots ,k-1}(X_{0})|q_{0}\rangle \cdots C_{k-2,k-1}(X_{k-3})\\&|q_{k-3}\rangle \,\,C_{k-1}(X_{k-2})|q_{k-2}\rangle \,\,X_{k-1}|q_{k-1}\rangle . \end{aligned}$$

Simplifying the right-hand side by using Eq. (15) gives

$$\begin{aligned} |q_{0}\oplus (q_1\cdots q_{k-1})\rangle \ldots |q_{k-3}\oplus (q_{k-2}\cdot q_{k-1})\rangle |q_{k-2}\oplus q_{k-1}\rangle |q_{k-1}\oplus 1\rangle . \end{aligned}$$

On the other hand, the addition $q+1$ in the binary representation, namely $(q_0 \cdots q_{k-1})_2\oplus 1$ yields

where the gray-colored bits in the first line of the table show the carries. The result (given in the fourth line of the table) is obtained by performing the addition of the rightmost bits of the table, that is, adding bits $q_{k-1}$ and 1. The result is $q_{k-1}\oplus 1$ and the carry is $q_{k-1}$, which is placed over $q_{k-2}$ as a gray-colored bit. Then, bits $q_{k-1}$ and $q_{k-2}$ are added that gives $q_{k-2}\oplus q_{k-1}$ and the carry is $q_{k-2}\cdot q_{k-1}$, which is placed over $q_{k-3}$. The addition goes on until the leftmost bit is reached. The final result coincides with the action of P on $|q_0\ldots q_{k-1}\rangle$, which proves the proposition. $\square$

B Appendix

This appendix describes function $\texttt {new}\_\texttt {mcrz}$, which decomposes the multi-controlled $R_z$ gate, and function $\texttt {new}\_\texttt {mcz}$, which decomposes the multi-controlled Z gate. Those functions use the same syntax of functions $\texttt {mcrz}$ and $\texttt {mcz}$ implemented in Qiskit. Note that our implementation uses fewer CNOTs.

C Appendix

Figure 13 depicts our results for a staggered quantum walk on the 8-cycle with $\theta =\pi /4$ and initial state $(|3\rangle +|4\rangle )/\sqrt{2}$ using three qubits of the ourense quantum computer. The two high peaks moving in opposite direction display the well-known signature of quantum walks on the one-dimensional lattice.

Fig. 13

Probability distribution of eight steps using exact calculations (blue) and the ourense quantum computer (red) using 3 qubits. The first plot refers to the preparation of the initial state (i.s.) $(|3\rangle +|4\rangle )/\sqrt{2}$. The following plots are successive steps

Table 6 shows the corresponding fidelities, where d and h are the total variation and Hellinger distances, respectively. After the sixth step, the fidelity is high, but the output of the quantum computer is worthless. This shows that the fidelity is not a good measure when the exact probability distribution is almost flat.

Table 6 Fidelities between the probability distributions generated by the ourense quantum computer and the exact simulation for one walker on a 8-vertex cycle up to step 8