Quantum Computation for Physical Modeling

FORWARD posed by Feynman himself to show that the one dimensional Dirac equation could be modeled by a singleOne of the most famous American physicists of the speed particle traveling in a two-dimensional spacetwentieth century, Richard Feynman, in 1982 was the time as a sum over zigzag paths of straight line elefirst to propose using a quantum mechanical computments [6], with the amplitude of a particular path coning device to efficiently simulate quantum mechanical tributing to the kernel by the number of "collisions" or many-body dynamics [1, 2, 3], a task that is exponencorners along that zigzag path. This quantum lattice tially complex in the number of particles treated and gas representation of quantum mechanics is equivalent is completely intractable by any classical computing to the well known path integral representation. 2 A means for large systems of many particles. In the two quantum lattice gas accounts for all contributing paths decades following his work, remarkable progress has by simultaneously evolving many particles in a unitary been made both theoretically and experimentally in fashion. Therefore, instead of summing (or integratthe new field of quantum computation [4, 5]. Ironing over) paths as individual entities, all contributing ically, however, most of the theoretical progress in paths are effectively simulated in one fell swoop as a quantum computing has developed within the purview combined field quantity. In the end, the collisional of the computer scientist with the principle applicainteraction between particles in the quantum lattice tions of efficient quantum information processing rerepresentation can be described by an effective field lated to cryptography and secure quantum commutheory (the Dirac equation in this particular case) at nication'. In an effort return to Feynman's original the large-scale called the continuum limit. Sdirection, the Air Force Research Laboratory and the Beginning in the mid 1990's, a contemporaneous Air Force Office of Scientific Research has established a series of quantum lattice-gas algorithms to model the multidisciplinary basic research theme called Quantum relativistic Dirac equation, equivalent to Feynman's Computation for Physical Modeling to explore quanoriginal algorithm, were published by Succi [8, 9], tum algorithms to model dynamical physical systems. Bialynicki-Birula [10], and Meyer [11, 12, 13, 14, 15]. Our goal is to establish a practical and generic means Furthermore, a series of papers on modeling the nonby which the power of quantum mechanics (that is, relativisitic Schroedinger equation were published by quantum parallelism due to the superposition and enBoghosian and Taylor [16, 17, 18] and by Yepez and tanglement of states) can be used to speedup numeriBoghosian [19], the latter article appearing in this iscal simulations of interest to computational physicists. sue. Our present goal in the Quantum Computation Notwithstanding the veritable stampede towards for Physical Modeling project is accelerate this algocomputer science related applications by most rerithmic developmental effort that has occurred over searchers in the field of quantum computing, a few the past decade. maverick physicists have developed some quantum alIn. fact, we hope to go further in the application gorithms to model quantum mechanical systems. A of this quantum algorithmic method. We have destarting point for this development was a problem veloped new efficient quantum lattice-gas algorithms to model classical dynamical systems [20, 21, 22, 23]. References to specific publications in these subjects are so Meyer also addresses this subject in his article on physubiquitous in the quantum computing literature that we do not include any here. Comprehensive collections of quantum comA solution to Feynman's "quantum lattice gas" problem was puting papers have been recently published [4, 5]. published in 1984 by Jacobson and Schulman [7].