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IEEE Computer Society Annual Symposium on VLSI

Design of Quantum Circuits for Random Walk Algorithms

Amlan Chakrabarti, Chia-Chun Lin, Niraj Kumar Jha

Published Aug 1, 2012• 9 citations
Publisher

Abstract

A quantum algorithm is defined by a sequence of operations that runs on a realistic model of quantum computation. Since the first quantum algorithm proposed by David Deutsch(1985), a large number of impressive quantum algorithms have been developed. Quantum random walks on a graph, which are analogous to classical stochastic walk, form the basis for some of the recent quantum algorithms that promise to significantly outperform existing classical random walk algorithms. Though research has been done on the application of quantum random walk to important computational problems, very little work has been done on its quantum circuit design. There are two types of quantum random walk algorithms: discrete-time and continuous-time. In this paper, we propose quantum circuit designs for both types of random walk algorithms that operate on various graphs. We consider in detail two important problems to which random walk algorithms are applicable: the triangle finding problem and binary welded tree problem. Though there exist a few research works related to quantum circuit design for random walk on graphs, to the best of our knowledge, the circuit designs we present here are first of their kind. We also provide an estimate of the quantum cost of these circuits for several physical machine descriptions (PMDs)of quantum systems, based on the number of quantum operations and execution cycles.

DOI
10.1109/isvlsi.2012.45

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A quantum algorithm is defined by a sequence of operations that runs on a realistic model of quantum computation. Since the first quantum algorithm proposed by David Deutsch(1985), a large number of impressive quantum algorithms have been developed. Quantum random walks on a graph, which are analogous to classical stochastic walk, form the basis for some of the recent quantum algorithms that promise to significantly outperform existing classical random walk algorithms. Though research has been done on the application of quantum random walk to important computational problems, very little work has been done on its quantum circuit design. There are two types of quantum random walk algorithms: discrete-time and continuous-time. In this paper, we propose quantum circuit designs for both types of random walk algorithms that operate on various graphs. We consider in detail two important problems to which random walk algorithms are applicable: the triangle finding problem and binary welded tree problem. Though there exist a few research works related to quantum circuit design for random walk on graphs, to the best of our knowledge, the circuit designs we present here are first of their kind. We also provide an estimate of the quantum cost of these circuits for several physical machine descriptions (PMDs)of quantum systems, based on the number of quantum operations and execution cycles.
Some interesting findings can be achieved by modeling and visualizing networks which can include complex networks and large scale networks to name a few. These networks have been analyzed with the help of classical stochastic walks before. So, here in our proposed work, we provide the quantum version of the walk which basically involves making use of properties found in quantum computing. These properties can include such as quantum interference, no-cloning theorem, and restriction on intermediate measurement. In this proposed work, we present quantum circuit designs for discrete time random walk algorithms that operate on various networks.To demonstrate our ideas we have considered two different types of networks, the star network and an extended version of the star network.Though there exist a few research works related to quantum approach for random walk on networks, to the best of our knowledge, the ideas and approach we present here are unique. These ideas which we have discussed can be extended to more complex networks and systems as well. As an application of quantum walk we have shown, how results got after performing quantum walk can be used to identify critical nodes in such networks. Therefore, we have also provided some analysis of complex networks. Identification of critical nodes is important for robustness and survivability of the network.
Abstract A random walk is one of the widely adopted random processes for simulation and approximation in multiple areas of science and engineering. Quantum random walk is an analog version of the classical random walk. It was first introduced in 1993 by Y. Aharonov et al. They presented that, with the nature of quantum characteristic, the average length of the walking path on a line is possible to larger than a classical random walk can produce. Therefore, the quantum random walk can be used as a tool to construct many other quantum algorithms. Moreover, it can solve graph problems that many real-world problems can be formulated. However, quantum algorithms are only useful in practice if we can implement them efficiently on a quantum computer. In this study, we are then interested in designing and implementing a quantum circuit that can be run on a real quantum computing device. We firstly focus on the most straightforward, which is a one-dimension, quantum random walk algorithm. Then, the quantum circuits are developed and implemented on a real quantum computer and a quantum computing simulator using software development kits provided by IBM. Lastly, the performance and results of the quantum circuits tested on both computing platforms are presented.
Quantum walks, being the quantum analog of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the ``glued trees'' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed and discuss potential applications to quantum search for marked vertices along these graphs.
Discrete quantum walks are quantum analogues of classical random walks. They are an important tool in quantum computing and a number of algorithms can be viewed as discrete quantum walks, in particular Grover's search algorithm. These walks are constructed on an underlying graph, and so there is a relation between properties of walks and properties of the graph. This book studies the mathematical problems that arise from this connection, and the different classes of walks that arise. Written at a level suitable for graduate students in mathematics, the only prerequisites are linear algebra and basic graph theory; no prior knowledge of physics is required. The text serves as an introduction to this important and rapidly developing area for mathematicians and as a detailed reference for computer scientists and physicists working on quantum information theory.
Using numerical simulation, we measured the performance of several potential quantum algorithms, based on quantum random walks, to solve Boolean satisfiability (SAT) problems. We develop the fundamentals of quantum computing and the theory of classical algorithms to indicate how these algorithms could be implemented. We also discuss the development of quantum random walks and the principles that go into creating them, presenting existing work on search algorithms using coined discrete-time quantum random walks. Although our quantum algorithms do not solve SAT problems faster than may be done classically, as a step toward that objective, we present the first example of a quantum walk on a directed graph that has a significant increase in speed over the analogous classical walk.
The random walk formalism is used across a wide range of applications, from modelling share prices to predicting population genetics. Likewise, quantum walks have shown much potential as a framework for developing new quantum algorithms. Here we present explicit efficient quantum circuits for implementing continuous-time quantum walks on the circulant class of graphs. These circuits allow us to sample from the output probability distributions of quantum walks on circulant graphs efficiently. We also show that solving the same sampling problem for arbitrary circulant quantum circuits is intractable for a classical computer, assuming conjectures from computational complexity theory. This is a new link between continuous-time quantum walks and computational complexity theory and it indicates a family of tasks that could ultimately demonstrate quantum supremacy over classical computers. As a proof of principle, we experimentally implement the proposed quantum circuit on an example circulant graph using a two-qubit photonics quantum processor.