Experimental demonstration and exploration of quantum lattice gas algorithms

Abstract

Recently, it has been suggested that an array of small quantum information processors sharing classical information can be used to solve selected computational problems, referred to as a type-II quantum computer. The first concrete implementation demonstrated here solves the diffusion equation, and it provides a test example from which to probe the strengths and limitations of this new computation paradigm. The NMR experiment consists of encoding a mass density onto an array of 16 two-qubit quantum information processors and then following the computation through 7 time steps of the algorithm. The results show a good agreement with the analytic solution for diffusive dynamics. From the numerical simulations of the NMR implementations, we explore two major error sources (1) the systematic error in the collision operator and (2) the linear approximation in the initialization. Since the mass density evolving under the Burgers equation develops sharp features over time, this is a stronger test of liquid state NMR implementations of type-II quantum computers than the previous example using the diffusion equation. Small systematic errors in the collision operator accumulate and swamp all other errors. We propose, and demonstrate, that the accumulation of this error can be avoided to a large extent by replacing the single collision operator with a set of operators, that have random errors and similar fidelities.

(cont.) Experiments have been implemented on 16 two-qubit sites for eight successive time steps for the Burgers equation. The improvement in the experimental results suggests that more complicated modulation of error terms may offer further improvement. An alternative approach has been suggested to encode in the Fourier space (k-space) to remove the usage of this linear approximation. This new method also provides us a simple means to implement the streaming operation quantum mechanically by controlling magnetic field gradients sandwiched with RF pulses. Therefore, this method might serve as a new tool to probe the implementations of quantum lattice gas (QLG) algorithms. Experimental demonstration of the diffusion equation has been performed on 16 two-qubit sites for four successive time steps. Recently, much attention has been focused on constructing many identical simple processing elements arranged in a cellular automata architecture recently. It is likely that the early quantum hardware will be built in a similar manner. Quantum lattice gas algorithms therefore provide a bridge between such hardware and potential early algorithms. We propose a quantum lattice gas model similar to the one proposed by Margolus for the classical setting.

(cont.) This quantum algorithm simulates the one-particle quantum random walk. The preliminary experimental design associated with the lattice gas model on a ring molecule is presented. The searches for the suitable pulses to construct the unitary operators, used in the implementations of the lattice gas model, are done and the results are encouraging.