Quantum lattice-gas model for the many-particle Schrödinger equation inddimensions

Abstract

We consider a general class of discrete unitary dynamical models on the lattice. We show that generically such models give rise to a wavefunction satisfying a Schrödinger equation in the continuum limit, in any number of dimensions. There is a simple mathematical relationship between the mass of the Schrödinger particle and the eigenvalues of a unitary matrix describing the local evolution of the model. Second quantized versions of these unitary models can be defined, describing in the continuum limit the evolution of a nonrelativistic quantum many-body theory. An arbitrary potential is easily incorporated into these systems. The models we describe fall in the class of quantum lattice gas automata, and can be implemented on a quantum computer with a speedup exponential in the number of particles in the system. This gives an efficient algorithm for simulating general nonrelativistic interacting quantum many-body systems on a quantum computer.

Random walk in generalized quantum theory

“…To our knowledge, this question remains open (cf. [8,9]), but one might hope to settle it using the techniques we develop herein. Indeed, we hope that one of the processes we construct below will have the desired limit.…”

“…By allowing a one-step memory, or equivalently, endowing the walker with a two-valued internal state, one can salvage unitarity, and several models of this sort have been discussed in the literature [8,9]. 3 Quantal will be used to refer to the generalised quantum measure theories studied in this paper, of which unitary quantum mechanics is only one particular case.…”

Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms

“…The discrete time quantum random walk is just one example of the broader class of quantum cellular automata and quantum lattice gases [25,36,[39][40][41]. These systems replace the Boolean variables of classical cellular automata with complex amplitudes, allowing the state of the quantum automaton to be a superposition of the states of the classical automaton.…”

Self-protected quantum algorithms based on quantum state tomography

“…Yet another class consists of algorithms for simulating quantum systems. Simulation algorithms can provide an exponential speed-up over any known classical algorithm for a variety of quantum systems [5,6,7,8,9,10,11,12,13,14,15] and are very promising for many applications in the physical sciences. These include atomic, molecular, solid state, and nuclear simulations and do not necessarily require a fully scalable quantum computing device [16].…”