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Physical Review A

Scalable networks for discrete quantum random walks

Shingo Fujiwara, H. Osaki, I. M. Buluta, Shuichi Hasegawa

Published Sep 23, 2005• 14 citations
Publisher

Abstract

Recently, quantum random walks (QRWs) have been thoroughly studied in order to develop new quantum algorithms. In this paper we propose scalable quantum networks for discrete QRWs on circles, lines, and also in higher dimensions. In our method the information about the position of the walker is stored in a quantum register and the network consists of only one-qubit rotation and ${(\text{controlled})}^{n}\text{\ensuremath{-}}\mathrm{NOT}$ gates, therefore it is purely computational and independent of the physical implementation. As an example, we describe the experimental realization in an ion-trap system.

DOI
10.1103/physreva.72.032329

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Recently, quantum random walks (QRWs) have been thoroughly studied in order to develop new quantum algorithms. In this paper we propose scalable quantum networks for discrete QRWs on circles, lines, and also in higher dimensions. In our method the information about the position of the walker is stored in a quantum register and the network consists of only one-qubit rotation and (controlled)n\ensuremath-NOT{(\text{controlled})}^{n}\text{\ensuremath{-}}\mathrm{NOT}(controlled)n\ensuremath-NOT gates, therefore it is purely computational and independent of the physical implementation. As an example, we describe the experimental realization in an ion-trap system.
Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. Quantum walks represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves onto the multi-qubit states in a quantum processor, we present different configurations of quantum circuits for the implementation of discrete-time quantum walks in one-dimensional position space. With example circuits for a five qubit machine we address scalability to higher dimensions and larger quantum processors.
Quantum walks are a promising framework for developing quantum algorithms and quantum simulations. They represent an important test case for the application of quantum computers. Here we present different forms of discrete-time quantum walks (DTQWs) and show their equivalence for physical realizations. Using an appropriate digital mapping of the position space on which a walker evolves to the multiqubit states of a quantum processor, we present different configurations of quantum circuits for the implementation of DTQWs in one-dimensional position space. We provide example circuits for a five-qubit processor and address scalability to higher dimensions as well as larger quantum processors.
Recently, several groups have investigated quantum analogues of random walk algorithms, both on a line and on a circle. It has been found that the quantum versions have markedly different features to the classical versions. Namely, the variance on the line, and the mixing time on the circle increase quadratically faster in the quantum versions as compared to the classical versions. Here, we propose a scheme to implement the quantum random walk on a line and on a circle in an ion trap quantum computer. With current ion trap technology, the number of steps that could be experimentally implemented will be relatively small. However, we show how the enhanced features of these walks could be observed experimentally. In the limit of strong decoherence, the quantum random walk tends to the classical random walk. By measuring the degree to which the walk remains ``quantum,'' this algorithm could serve as an important benchmarking protocol for ion trap quantum computers.
We investigate the counterparts of random walk in universal quantum computing and their implementation using standard quantum circuits. Quantum walk have been recently well investigated for traversing graphs with certain oracles. We focus our study on traversing a 1-D graph, namely a circle, and show how to implement discrete circular quantum walk in quantum circuits built with universal CNOT and single quit gates. We review elementary quantum gates and circuit decomposition and propose a a generalized version of the all CNOT based quantum discrete circular walk. We simulated these circuits on an IBM quantum supercomputer London IBM-Q with 5 qubits. This quantum computer has non perfect gates based on superconducting qubits, therefore we analyze the impact of errors on the fidelity of the Walker circuit.
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 networks are complex systems formed by the interaction among quantum processors through quantum channels. Analogous to classical computer networks, quantum networks allow for the distribution of quantum computation among quantum computers. In this work, we describe a quantum walk protocol to perform distributed quantum computing in a quantum network. The protocol uses a quantum walk as a quantum control signal to perform distributed quantum operations. We consider a generalization of the discrete-time coined quantum walk model that accounts for the interaction between a quantum walker system in the network graph with quantum registers inside the network nodes. The protocol logically captures distributed quantum computing, abstracting hardware implementation and the transmission of quantum information through channels. Control signal transmission is mapped to the propagation of the walker system across the network, while interactions between the control layer and the quantum registers are embedded into the application of coin operators. We demonstrate how to use the quantum walker system to perform a distributed CNOT operation, which shows the universality of the protocol for distributed quantum computing. Furthermore, we apply the protocol to the task of entanglement distribution in a quantum network.