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FROM GAUGE TRANSFORMATIONS TO TOPOLOGY COMPUTATION IN QUANTUM LATTICE GAS AUTOMATA

Published Dec 4, 2008• 0 citations
OpenAlex

Abstract

ABSTRACT The evolution of a quantum lattice gas automaton (LGA) for a single charged particle isinvariant under multiplication of the wave function by a global phase. Requiring invariance under the corresponding local gauge transformations determines the rule for minimalcoupling to an arbitrary external electromagnetic field. We develop the Aharonov-Bohm effect in the resulting model into a constant time algorithm to distinguish a one dimen-sional periodic lattice from one with boundaries; any classical deterministic LGA algorithm distinguishing these two spatial topologies would have expected running time on the orderof the cardinality of the lattice. PACS numbers: 03.65.Lx, 03.65.Vf, 03.65.Pm.KEY WORDS: quantum lattice gas; quantum computation; Aharonov-Bohm effect; spatial topology.

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ABSTRACT The evolution of a quantum lattice gas automaton (LGA) for a single charged particle isinvariant under multiplication of the wave function by a global phase. Requiring invariance under the corresponding local gauge transformations determines the rule for minimalcoupling to an arbitrary external electromagnetic field. We develop the Aharonov-Bohm effect in the resulting model into a constant time algorithm to distinguish a one dimen-sional periodic lattice from one with boundaries; any classical deterministic LGA algorithm distinguishing these two spatial topologies would have expected running time on the orderof the cardinality of the lattice. PACS numbers: 03.65.Lx, 03.65.Vf, 03.65.Pm.KEY WORDS: quantum lattice gas; quantum computation; Aharonov-Bohm effect; spatial topology.
The evolution of a quantum lattice gas automaton (QLGA) for a single charged particle is invariant under multiplication of the wave function by a global phase. Requiring invariance under the corresponding local gauge transformations determines the rule for minimal coupling to an arbitrary external electromagnetic field. We develop the Aharonov–Bohm effect in the resulting model into a constant time algorithm to distinguish a one-dimensional periodic lattice from one with boundaries; any classical deterministic lattice gas automaton (LGA) algorithm distinguishing these two spatial topologies would have expected running time on the order of the cardinality of the lattice.
A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, in this paper we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one parameter family of evolution rules which are best interpreted as those for a one particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second of which, to multiple interacting particles, is the correct definition of a quantum lattice gas. KEY WORDS: quantum cellular automaton; quantum lattice gas; quantum computation. to appear in J. Stat. Phys.
We continue our analysis of the physics of quantum lattice gas automata (QLGA). Previous work has been restricted to periodic or infinite lattices; simulation of more realistic physical situations requires finite sizes and nonperiodic boundary conditions. Furthermore, envisioning a QLGA as a nanoscale computer architecture motivates consideration of inhomogeneities in the `substrate'; this translates into inhomogeneities in the local evolution rules. Concentrating on the one-particle sector of the model, we determine the various boundary conditions and rule inhomogeneities which are consistent with unitary global evolution. We analyse the reflection of plane waves from boundaries, simulate wavepacket refraction across inhomogeneities, and conclude by discussing the extension of these results to multiple particles.
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.
We define quantum cellular automata as infinite quantum lattice systems with discrete time dynamics, such that the time step commutes with lattice translations and has strictly finite propagation speed. In contrast to earlier definitions this allows us to give an explicit characterization of all local rules generating such automata. The same local rules also generate the global time step for automata with periodic boundary conditions. Our main structure theorem asserts that any quantum cellular automaton is structurally reversible, i.e., that it can be obtained by applying two blockwise unitary operations in a generalized Margolus partitioning scheme. This implies that, in contrast to the classical case, the inverse of a nearest neighbor quantum cellular automaton is again a nearest neighbor automaton. We present several construction methods for quantum cellular automata, based on unitaries commuting with their translates, on the quantization of (arbitrary) reversible classical cellular automata, on quantum circuits, and on Clifford transformations with respect to a description of the single cells by finite Weyl systems. Moreover, we indicate how quantum random walks can be considered as special cases of cellular automata, namely by restricting a quantum lattice gas automaton with local particle number conservation to the single particle sector.
On the basis of the principle that topological quantum phases arise from the scattering around space-time defects in higher dimensional unification, a geometric model is presented that associates with each quantum phase an element of a transformation group. 1 Introduction Within a physical theory there are often effects or phenomena studied in idealized form in order to gain insight into the peculiarities of the theory. In Quantum Mechanics one such effect is the appearance of topological quantum phases in the wave functions of particles moving freely in multiply connected space-times the prototype of this effect being the Aharonov-Bohm (AB) effect [1], the appearance of a phase factor in the wave function of an electron which moves around a magnetic flux line. A similar effect is the Aharonov-Casher (AC) effect [2] which is obtained from the AB effect by replacing the flux line and the electron by a charged line and a neutral particle with magnetic moment, respectively. There have al...