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arXiv:1003.3779 · Communications in computer and information science

GPGPU Based Simulations for One and Two Dimensional Quantum Walks

Marek Sawerwain, Roman Gielerak

Published Jan 1, 2010• 6 citations
arXivHTMLPDF
DOI
10.1007/978-3-642-13861-4_3

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Scientific Reports•Jul 4, 2017•12 citations
Quantum random walks are the natural counterpart of the classical random walk, arising when quantum effects are taken into consideration. Many Monte Carlo methods use the classical Brownian motion as a building block. Although it appears naturally in various problems of computational physics, it is also used extensively in the stochastic models used for pricing of financial assets. When quantum random walks are used instead, more complex behaviors can be observed. However, the simulation of a quantum random walk requires more computational resources. By using low-discrepancy sequences instead of pseudorandom numbers, one may hope to decrease the statistical error of the simulation. Another approach for improved efficiency is to use the power of computational accelerators, which are suitable for problems, where a well-ordered sequence of computations is to be performed repeatedly. In our work we concentrated on using GPGPU computing in order to speed-up the operations involved in the generation of the low-discrepancy sequence as well as the subsequent sampling of the quantum random walks. In this work we explain our sampling algorithm and demonstrate its efficiency on model problems from the domain of option pricing.
Quantum walks (QWs) provide a powerful tool as a quantum simulator to study topological phases. However, some interesting topological phenomena in the two dimensional (2D) QW that do not exist in the one dimensional case, e.g., the edge-state-enhanced transport, have not been demonstrated experimentally. Here we have observed 2D topological bound states with vanishing Chern numbers and confirmed the robustness of these bound states with respect to perturbations and disorder, which go beyond what has been known in static systems and are unique to periodically driven systems. Our studies open up an avenue to explore topological properties in multidimensional QWs.
Quantum walk (QW) provides a versatile platform for the realization of quantum algorithms. Due to the existence of the inevitable noises in the walk, the different quantum algorithms accommodating to different noises are demanded. Thus, the success of the algorithms based on the QW requires us to sense different noises in the walk. Until now, the way to distinguish different noises in the walk has been discussed rarely. Here, we propose an efficient way to sense the noises in the one and two dimensional QWs. The populations of the coin in the walk with or without decoherence are presented. By only detecting the populations of the coin in the QW, we can determine whether there exists the decoherence in the total QW system. Moreover, the non-Markovianity of the coin in the one and two dimensional QWs is revealed, in which the coin is taken as an open quantum system, and the other components of the QW system is taken as the large environment. With the measured value of the non-Markovianity for the coin, we can conjecture which kinds of noise emerges in the one and two dimensional QWs.