The Deutsch-Jozsa Problem: De-quantisation and Entanglement

TL;DR

This paper investigates the Deutsch-Jozsa problem, demonstrating how quantum solutions can sometimes be replaced with classical algorithms, and explores the conditions under which such de-quantisation is possible, extending understanding of quantum-classical boundaries.

Contribution

The paper analyzes when the quantum Deutsch-Jozsa algorithm can be de-quantised into classical solutions, providing insights into the boundaries between quantum and classical computation.

Findings

Quantum solutions can sometimes be replaced by classical algorithms.

De-quantisation depends on specific properties of the Boolean function.

Extended analysis of de-quantisation applicability to broader cases.

Abstract

The Deustch-Jozsa problem is one of the most basic ways to demonstrate the power of quantum computation. Consider a Boolean function f : {0,1}^n to {0,1} and suppose we have a black-box to compute f. The Deutsch-Jozsa problem is to determine if f is constant (i.e. f(x) = const forall x in {0,1}^n) or if f is balanced (i.e. f(x) = 0 for exactly half the possible input strings x in {0,1}^n) using as few calls to the black-box computing f as is possible, assuming f is guaranteed to be constant or balanced. Classically it appears that this requires at least 2^{n-1}+1 black-box calls in the worst case, but the well known quantum solution solves the problem with probability one in exactly one black-box call. It has been found that in some cases the algorithm can be de-quantised into an equivalent classical, deterministic solution. We explore the ability to extend this de-quantisation to further cases, and examine with more detail when de-quantisation is possible, both with respect to the Deutsch-Jozsa problem, as well as in more general cases.