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arXiv:quant-ph/0111069 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Quantum computing classical physics

DAVID A. MEYER

Published Mar 15, 2002• 33 citations
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Abstract

In the past decade, quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests that they may also speed up the simulation of some classical systems. I describe one class of discrete quantum algorithms which do so--quantum lattice-gas automata--and show how to implement them efficiently on standard quantum computers.

DOI
10.1098/rsta.2001.0936

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In the past decade, quantum algorithms have been found which outperform the best classical solutions known for certain classical problems as well as the best classical methods known for simulation of certain quantum systems. This suggests that they may also speed up the simulation of some classical systems. I describe one class of discrete quantum algorithms which do so--quantum lattice-gas automata--and show how to implement them efficiently on standard quantum computers.
The quantum lattice-gas automata (QLGA) have been studied to simulate the time-dependent Schrodinger equation. It is particularly well suited to implementation on a quantum computer; the many-body quantum system can be simulated very efficiently. On the other hand, its emulation on a classical computer would also have an advantage, because its unitarity might lead better behavior than standard finite-difference methods. In this study, we apply the QLGA method to the analysis of the one-particle quantum wave propagation in nanoscale devices. To this end, we develop a new algorithm to implement the open boundary condition into the QLGA method. In the QLGA simulation, the space is discretized and qubits are assigned to each node of the lattice. The each qubit contains the probability amplitude of whether the lattice node is occupied or not by the simulated particles. We have assigned the qubit array outside the simulation region to store the information about the particles which have flown out across the boundary. Then it becomes possible to simulate the open systems with the QLGA simulation.
Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.
Abstract Quantum computers are designed to outperform standard computers by running quantum algorithms. Areas in which quantum algorithms can be applied include cryptography, search and optimisation, simulation of quantum systems and solving large systems of linear equations. Here we briefly survey some known quantum algorithms, with an emphasis on a broad overview of their applications rather than their technical details. We include a discussion of recent developments and near-term applications of quantum algorithms.
Lattice Gas Automata (LGA) is a classical method for simulating physical phenomena, including Computational Fluid Dynamics (CFD). Quantum LGA (QLGA) is the family of methods that implement LGA schemes on quantum computers. In recent years, QLGA has garnered attention from researchers thanks to its potential of efficiently modeling CFD processes by either reducing memory requirements or providing simultaneous representations of exponentially many LGA states. In this work, we introduce novel building blocks for QLGA algorithms that rely on computational basis state encodings. We address every step of the algorithm, from initial conditions to measurement, and provide detailed complexity analyses that account for all discretization choices of the system under simulation. We introduce multiple ways of instantiating initial conditions, efficient boundary condition implementations for novel geometrical patterns, a novel collision operator that models less restricted interactions than previous implementations, and quantum circuits that extract quantities of interest out of the quantum state. For each building block, we provide intuitive examples and open-source implementations of the underlying quantum circuits.
This paper is a gentle but rigorous introduction to quantum computing intended for discrete mathematicians. Starting from a small set of assumptions on the behavior of quantum computing devices, we analyze their main characteristics, stressing the differences with classical computers, and finally describe two well-known algorithms (Simon's algorithm and Grover's algorithm) using the formalism developed in previous sections. This paper does not touch on the physics of the devices, and therefore does not require any notion of quantum mechanics. Numerical examples based on an implementation of Grover's algorithm using open-source software are provided.