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QWASI: The Quantum Walk Simulator

Warren J. Wilson

Published Jan 1, 2020• 0 citations
OpenAlex

Abstract

As quantum computing continues to evolve, the ability to design and analyze novel quantum algorithms becomes a necessary focus for research. In many instances, the virtues of quantum algorithms only become evident when compared to their classical counterparts, so a study of the former often begins with a consideration of the latter. This is very much the case with quantum walk algorithms, as the success of random walks and their many, varied applications have inspired much interest in quantum correlates. Unfortunately, finding purely algebraic solutions for quantum walks is an elusive endeavor. At best, and when solvable, they require simple graphical structure, exploitable symmetries, and case by case analyses. In more complex instances, as with most arbitrary graphs, the pursuit of a general solution is untenable. Enter Qwasi (Quantum Walk Simulator), a software tool that allows for the composition of arbitrary graphs by a user, includes a mathematical backend that can construct the Hilbert Space from these graphs, and simulates the evolution of the quantum walk as a set of numerical data. This simulation can be observed in real time, compiled into graphical plots, or exported as a CSV file. At present, the software focuses upon discrete-time scattering quantum walks, but expansion to other graph-based algorithms – anything that operates on a collection of vertices and connected edges – would not be a demanding task.

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As quantum computing continues to evolve, the ability to design and analyze novel quantum algorithms becomes a necessary focus for research. In many instances, the virtues of quantum algorithms only become evident when compared to their classical counterparts, so a study of the former often begins with a consideration of the latter. This is very much the case with quantum walk algorithms, as the success of random walks and their many, varied applications have inspired much interest in quantum correlates. Unfortunately, finding purely algebraic solutions for quantum walks is an elusive endeavor. At best, and when solvable, they require simple graphical structure, exploitable symmetries, and case by case analyses. In more complex instances, as with most arbitrary graphs, the pursuit of a general solution is untenable. Enter Qwasi (Quantum Walk Simulator), a software tool that allows for the composition of arbitrary graphs by a user, includes a mathematical backend that can construct the Hilbert Space from these graphs, and simulates the evolution of the quantum walk as a set of numerical data. This simulation can be observed in real time, compiled into graphical plots, or exported as a CSV file. At present, the software focuses upon discrete-time scattering quantum walks, but expansion to other graph-based algorithms – anything that operates on a collection of vertices and connected edges – would not be a demanding task.
Several research groups are giving special attention to quantum walks recently, because this research area have been used with success in the development of new efficient quantum algorithms. A general simulator of quantum walks is very important for the development of this area, since it allows the researchers to focus on the mathematical and physical aspects of the research instead of deviating the efforts to the implementation of specific numerical simulations. In this paper we present QWalk, a quantum walk simulator for one- and two-dimensional lattices. Finite two-dimensional lattices with generic topologies can be used. Decoherence can be simulated by performing measurements or by breaking links of the lattice. We use examples to explain the usage of the software and to show some recent results of the literature that are easily reproduced by the simulator.Program title: QWalkCatalogue identifier: AEAX_v1_0Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAX_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: GNU General Public LicenceNo. of lines in distributed program, including test data, etc.: 10 010No. of bytes in distributed program, including test data, etc.: 172 064Distribution format: tar.gzProgramming language: CComputer: Any computer with a C compiler that accepts ISO C99 complex arithmetic (recent versions of GCC, for instance). Pre-compiled Windows versions are also providedOperating system: The software should run in any operating system with a recent C compiler. Successful tests were performed in Linux and WindowsRAM: Less than 10 MB were required for a two-dimensional lattice of size 201×201. About 400 MB, for a two-dimensional lattice of size 1601×1601Classification: 16.5Nature of problem: Classical simulation of discrete quantum walks in one- and two-dimensional lattices.Solution method: Iterative approach without explicit representation of evolution operator.Restrictions: The available amount of RAM memory imposes a limit on the size of the simulations.Unusual features: The software provides an easy way of simulating decoherence through detectors or random broken links. In the two-dimensional simulations it also allows the definition of permanent broken links, besides calculation of total variation distance (from the uniform and from an approximate stationary distribution) and the choice between two different physical lattices. It also provides an easy way of performing measurements on specific sites of the 2D lattice and the analysis of observation screens. In one-dimensional simulations it allows the choice between three different lattices. Both one- and two-dimensional simulations facilitates the generation of graphics by automatically generating gnuplot scrips.Additional comments:•An earlier version of QWalk was first presented in [1].•The simulator generates gnuplot scripts that can be used to make graphics of the output data.•Several examples of input files are provided.Running time: The simulation of 100 steps for a two-dimensional lattice of size 201×201 took less than 2 seconds on a Pentium IV 2.6 GHz with 512 MB of RAM memory, 512 KB of cache memory and under Linux. It also took about 15 minutes for a lattice of size 1601×1601 on the same computer. Optimization option -O2 was used during compilation for these tests.References: [1] F.L. Marquezino, R. Portugal, QWalk: Simulador de caminhadas quânticas, in: Proceedings of 2nd WECIQ, Campina Grande, Brazil, IQuanta, 2007, pp. 123–132.
Quantum walks provide a versatile framework for quantum algorithms across a wide range of applications. We develop efficient classical simulation methods for Szegedy quantum walks that avoid explicit construction of the full unitary evolution operator. Unlike previous approaches restricted to a particular walk formulation, our framework is built from fundamental update and reflection operators, enabling the simulation of a broader class of Szegedy walk formulations. We further extend these methods to phase-estimation-based algorithms coupled to the walk, including implementations suitable for large sparse graphs. The resulting methods achieve optimal O(N2)O(N^2)O(N2) complexity for dense graphs with NNN nodes. For sparse graphs, the computational cost scales linearly with the number of edges, which is O(N)O(N)O(N) in many cases. We implement the framework in the Python package SQWLib and illustrate its capabilities through simulations of representative algorithms, including quantum simulated annealing and quantum search on graphs. These results provide a practical tool for studying Szegedy-walk-based algorithms numerically beyond purely analytical treatments.
Abstract Quantum walks have been shown to have a wide range of applications, from artificial intelligence, to photosynthesis, and quantum transport. Quantum stochastic walks (QSWs) generalize this concept to additional non-unitary evolution. In this paper, we propose a trajectory-based quantum simulation protocol to effectively implement a family of discrete-time QSWs in a quantum device. After deriving the protocol for a 2-vertex graph with a single edge, we show how our protocol generalizes to a graph with arbitrary topology and connectivity. The straightforward generalization leads to simple scaling of the protocol to complex graphs. Finally, we show how to simulate a restricted class of continuous-time QSWs by a discrete-time QSW, and how this is amenable to our simulation protocol for discrete-time QSWs.
Random walks are a powerful tool for the efficient implementation of algorithms in classical computation. Their quantum-mechanical analogues, called quantum walks, hold similar promise. Quantum walks provide a model of quantum computation that has recently been shown to be equivalent in power to the standard circuit model. As in the classical case, quantum walks take place on graphs and can undergo discrete or continuous evolution, though quantum evolution is unitary and therefore deterministic until a measurement is made. This thesis considers the usefulness of continuous-time quantum walks to quantum computation from the perspectives of both their fundamental power under various formulations, and their applicability in practical experiments. In one extant scheme, logical gates are effected by scattering processes. The results of an exhaustive search for single-qubit operations in this model are presented. It is shown that the number of distinct operations increases exponentially with the number of vertices in the scattering graph. A catalogue of all graphs on up to nine vertices that implement single-qubit unitaries at a specific set of momenta is included in an appendix. I develop a novel scheme for universal quantum computation called the discontinuous quantum walk, in which a continuous-time quantum walker takes discrete steps of evolution via perfect quantum state transfer through small `widget' graphs. The discontinuous quantum-walk scheme requires an exponentially sized graph, as do prior discrete and continuous schemes. To eliminate the inefficient vertex resource requirement, a computation scheme based on multiple discontinuous walkers is presented. In this model, n interacting walkers inhabiting a graph with 2n vertices can implement an arbitrary quantum computation on an input of length n, an exponential savings over previous universal quantum walk schemes. This is the first quantum walk scheme that allows for the application of quantum error correction. The many-particle quantum walk can be viewed as a single quantum walk undergoing perfect state transfer on a larger weighted graph, obtained via equitable partitioning. I extend this formalism to non-simple graphs. Examples of the application of equitable partitioning to the analysis of quantum walks and many-particle quantum systems are discussed.
Graph partitioning is a crucial problem in computer science with applications in diverse areas like VLSI design, image segmentation, and network analysis. Traditional classical algorithms often face significant challenges when dealing with large and complex graphs, leading to increased computational time and memory requirements. This paper explores the potential of quantum walk algorithms to address this issue. We propose a framework for leveraging quantum walk algorithms to efficiently explore the connectivity structure of a graph and identify optimal partitions. The core idea involves mapping the graph's nodes and edges onto a quantum system and utilizing the inherent parallelism of quantum walks to traverse the graph. We present a theoretical analysis of the algorithm's potential speedup compared to classical approaches, focusing on scenarios where quantum computation offers a significant advantage. Specifically, we formulate the graph partitioning problem as a search problem on a graph represented as a quantum walk, considering the Hamiltonian operator and its eigenstates for efficient traversal. The effectiveness of the algorithm is dependent on the graph's structure and the ability to accurately simulate the quantum walk. The presented methodology provides a novel approach to graph partitioning, potentially leading to substantial performance improvements for large, complex graphs.
This paper investigates the application of quantum walk algorithms to large-scale graph search problems. Classical algorithms often struggle with the computational complexity of exploring vast graphs, particularly when searching for specific patterns or nodes. Quantum walks, inspired by the probabilistic nature of random walks, provide a promising alternative by utilizing quantum superposition and interference to accelerate the search process. We explore the theoretical framework and potential advantages of employing quantum walks for graph traversal, focusing on scenarios where exponential speedups are theoretically possible. This work aims to demonstrate the potential of quantum computing to tackle computationally demanding graph search tasks that are currently intractable for classical methods. The core concept leverages the ability to simultaneously explore multiple paths within a graph, drastically reducing the search time compared to sequential exploration. We present a foundational understanding of quantum walk implementation and analyze its suitability for various graph search algorithms.