Effects of noise, correlations, and errors in the preparation of initial states in quantum simulations

Abstract

In principle, a quantum system could be used to simulate another quantum system. The purpose of such a simulation would be to obtain information about problems which are difficult to simulate on a classical computer due to the exponential increase of the Hilbert space with the size of the system and which cannot be readily measured or controlled in an experiment. The system will interact with the surrounding environment and with the other particles in the system, and be implemented using imperfect controls, making it subject to noise. It has been suggested that noise does not need to be controlled to the same extent as it must be for general quantum computing. However, the effects of noise in quantum simulations are not well understood and how best to treat them in most cases is not known. In this paper we study an existing quantum algorithm for simulating the one-dimensional Fano-Anderson model using a liquid-state NMR device. We examine models of noise in the evolution using different initial states in the original model. We also add interacting spins to simulate a realistic situation where an environment of spins is present. We find that states which are entangled with their environment, and sometimes correlated but not necessarily entangled, have an evolution which is described by maps which are not completely positive. We discuss the conditions for this to occur and also the implications.