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Implementation of a quantum walk in a cycle of four nodes using an alternative method based on the Swap gate

Dagoberto S. Freitas, Aisis Rodrigues Barbosa

Published Oct 5, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.13102/sscf.v19i.8872

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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.
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.
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 have been extensively studied recently, mainly due to their vast difference in behavior to classical random walks. This paper is concerned with discrete time and space quantum walks of particles that propagate through a one-dimensional line. This line can be either a lattice or a graph or any other form of mathematical structure that can be viewed as a one-dimensional line. First is defined a concrete way to describe the unitary evolution of a quantum walk through a balanced coin operator and a shift operator. Then follows the implementation of the quantum walk on an 888-cycle, i.e a cycle graph with 888 nodes, which is then run locally as a simulation and on IBM's quantum computer. The paper explores two implementations of the quantum walk as a quantum circuit: the first one consists of generalised controlled inversions, as introduced in \cite{EffWalk}, whereas the second one tries to replace them with rotation operators around the basis states. The main aim is to find a way around the caveat resulting from the large amount of ancilla qubits required to carry out the computation in the case where only generalised inverters are used. Next, another three experiments are computed, involving cycles with a larger state space, more specifically 161616, 323232 and 646464 possible positions. In order to measure the magnitude of the error of the circuit we use the cross entropy benchmarking method, calculated through the Hellinger distance. Finally, a derivation of the variance of the quantum walk is provided along with a calculation of the variance for our experiment.
Quantum coin is an essential element of the discrete-time quantum walk, the quantum analog of random walk. Most existing papers relating to quantum coin discussed the implementation and performance of the coin on power-of-two states and degrees of graphs. In this paper, I proposed and analyzed three alternative coins–binary state coin, W state coin, and Walsh-Hadamard coin–that possessed the property to perform on all states and degrees of graphs. I discussed the implementation and the performance of the three coins in the application of Quantum Walk and Quantum Walk Search. I compared and discussed the advantages, disadvantages, and errors of the three coins on a quantum simulator and a NISQ computer (i.e. IBM Q). In the end, I presented further applications for the three coins in quantum-walk-based algorithms and other potential topics in quantum computing.
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.
Signal processing is important for modern technologies such as digital communication systems and sensor networks, and is used to enhance and optimize the quality of transmitted signals. However, traditional methods and design frameworks are insufficient for the demands of future communication networks, which are expected to be highly complex and constantly changing. Our research focuses on a mathematical model known as a Markov model, which represents a system as a series of states linked by transitions. This model can be applied to various signal processing challenges in telecommunications, such as the modeling of communication channels, sources of information, and the correction of distortion and noise in transmitted signals. We examine the random walk algorithm, a simple and efficient way to simulate the behavior of a Markov chain, as it can provide valuable insights into the performance of communication systems and help to optimize their design. However, it may not be able to accurately model systems with complex transitions or systems that exhibit long-term correlations. In order to address the challenges and limitations of classical random walk, we propose the use of quantum random walk, which is a quantum mechanical version of classical random walk. It describes the motion of particles in a probabilistic way, similar to classical random walk, but with the added complexity of quantum mechanics. We also provide quantum walk algorithm with Hadamard gate implementation for coin operator.