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IEEE/ACM International Workshop on Quantum Software Engineering (Q-SE)

Experimental Implementation of Discrete Time Quantum Walk with the IBM Qiskit Library

Pierriccardo Olivieri, Mehrnoosh Askarpour, Elisabetta Di Nitto

Published Jun 1, 2021• 4 citations
Publisher

Abstract

This paper presents an implementation of quantum random walk using the Qiskit library, given the quantum circuits extracted from the literature. We started by implementing an instance of Coined Discrete Quantum Walk on an 8-node cyclic graph. Observing the limitations of this implementation, we experimented a different approach by proposing an implementation for Szegedy Quantum Walk in the IBM Qiskit library [1]. The proposed solution could be used as a building block to construct quantum implementation of other classic algorithms.

DOI
10.1109/q-se52541.2021.00014

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This paper presents an implementation of quantum random walk using the Qiskit library, given the quantum circuits extracted from the literature. We started by implementing an instance of Coined Discrete Quantum Walk on an 8-node cyclic graph. Observing the limitations of this implementation, we experimented a different approach by proposing an implementation for Szegedy Quantum Walk in the IBM Qiskit library [1]. The proposed solution could be used as a building block to construct quantum implementation of other classic algorithms.
This work has the objective of implementing quantum random walks in a graph with 4 nodes represented by a circle. It was observed that during the implementation of a discrete-time quantum walk, based on Hadamard's coin, the simulated results diverge from the theoretically expected results. In order to correct the discrepancies between the processed and theoretically expected results we use an alternative method based on the Swap port. To perform the simulation of the algorithms, the Qiskit framework and a real processor provided by IBM through remote access was used. The simulations performed in the real processor showed small fluctuations, however the quantum states were obtained with satisfactory probabilities.
Abstract The rapid development of quantum computing has led to increasing interest in quantum algorithms for a variety of different applications. Quantum walks have also experienced a surge in interest due to their potential use in quantum algorithms. Using the qiskit software package, we test how accurately the current generation of quantum computers provided by IBM can simulate a cycle discrete-time quantum walk. Implementing an 8-node, 8-step walk and a simpler 4-node, 4-step discrete-time quantum walk on an IBM quantum device known as , the results for each step of the respective walks are presented. A custom noise model is developed in order to estimate that noise levels in the quantum device would need to be reduced by at least 94% in order to execute a 16-node, 16-step cycle discrete-time quantum walk to a reasonable level of fidelity. Graphic abstract
Quantum walks are promising tools based on classical random walks, with plenty of applications such as many variants of optimization. Here we introduce the semiclassical walks in discrete time, which are algorithms that combines classical and quantum dynamics. Specifically, a semiclassical walk can be understood as a classical walk where the transition matrix encodes the quantum evolution. We have applied this algorithm to Szegedy’s quantum walk, which can be applied to any arbitrary weighted graph. We first have solved the problem analytically on regular 1D cycles to show the performance of the semiclassical walks. Next, we have simulated our algorithm in a general inhomogeneous symmetric graph, finding that the inhomogeneity drives a symmetry breaking on the graph. Moreover, we show that this phenomenon is useful for the problem of ranking nodes in symmetric graphs, where the classical PageRank fails. We have demonstrated experimentally that the semiclassical walks can be applied on real quantum computers using the platform IBM Quantum.
We demonstrate an implementation of the hitting time of a discrete time quantum random walk on cubelike graphs using IBM's Qiskit platform. Our implementation is based on efficient circuits for the Grover and Shift operators. We verify the results about the one-shot hitting time of quantum walks on a hypercube as proved in [https://link.springer.com/article/10.1007/s00440-004-0423-2]. We extend the study to another family of cubelike graphs called the augmented cubes [https://onlinelibrary.wiley.com/doi/abs/10.1002/net.10033]. Based on our numerical study, we conjecture that for all families of cubelike graphs there is a linear relationship between the degree of a cubelike graph and its hitting time which holds asymptotically. That is, for any cubelike graph of degree ΔΔΔ, the probability of finding the quantum random walk at the target node at time πΔ2\frac{πΔ}{2}2πΔ​ approaches 1 as the degree ΔΔΔ of the cubelike graph approaches infinity.
In this paper, we give a procedure to construct quantum circuits for implementing discrete-time quantum walks on a family of Cayley graphs called cubelike graphs. We construct these circuits on IBM’s Qiskit platform and demonstrate an implementation of the quantum hitting times on cubelike graphs. Based on our numerical study, we conjecture that for all families of cubelike graphs there is a linear relationship between the degree of a cubelike graph and its hitting time which holds asymptotically. This conjecture, if proved, will generalize the result of hitting times of discrete-time quantum walks on hypercubes to general family of cubelike graphs.
We investigate the counterparts of random walks in universal quantum computing and their implementation using standard quantum circuits. Quantum walks have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1D graph, namely a circle, and show how to implement a discrete-time quantum walk in quantum circuits built with universal CNOT and single qubit gates. We review elementary quantum gates and circuit decomposition techniques and propose a generalized version of all CNOT-based circuits of the quantum walk. We simulated these circuits on five different qubits IBM-Q quantum devices. This quantum computer has nonperfect gates based on superconducting qubits, and, therefore, we analyzed the impact of the CNOT errors and CNOT-depth on the fidelity of the circuit.