Hidden variable simulation of Deutsch algorithm

Abstract

Quantum algorithm is to compute a given task using quantum resources, such as superposition and entanglement, much faster than its counter-part, classical algorithm. Deutsch’s algorithm, one of well-known quantum algorithm, can answer the type of a given function, that is balanced or constant, with less queries than any other classical methods. Even though Deutsch’s algorithm is a quantum algorithm, it is interesting to ask whether it can be simulated by classical theory, if so, how it works. At the point-view of hidden variable theory, quantum operators are modeled as stochastic operations that change the distribution of hidden variables, those for simulating quantum computation. It is likely that if we allow an arbitrary number of such variables, the classical model can simulate every quantum computation. We ask if this is the case. If so, we further ask how many hidden variables and/or how much classical communication suffices for the classical model to simulate a given quantum algorithm. In this work, we show that a hidden variable theory can in particular simulate Deutsch’s algorithm as far as further resources are provided. The classical simulation requires four classical bits and two-bit classical communication whereas two quantum bits and a single quantum communication suffices to the original Deutsch’s algorithm. This work will open a new perspective of quantum information science with respect to computation complexity and communication complexity.