Discrimination of unitary transformations in the Deutsch-Jozsa algorithm

TL;DR

This paper develops a framework for understanding quantum algorithms as tools for discriminating unitary transformations, applying it to the Deutsch-Jozsa problem, and analyzing the success probability of certain quantum algorithms with specific initial states.

Contribution

It introduces a general framework for quantum algorithms as discrimination tools and characterizes all algorithms solving the Deutsch-Jozsa problem with certainty under specific conditions.

Findings

All possible algorithms solving the problem with certainty are derived.

Quantum algorithms starting from thermal equilibrium states have limited advantage for large problem sizes.

Classical algorithms outperform certain quantum algorithms for problem sizes below approximately 10^5.

Abstract

We describe a general framework for regarding oracle-assisted quantum algorithms as tools for discriminating between unitary transformations. We apply this to the Deutsch-Jozsa problem and derive all possible quantum algorithms which solve the problem with certainty using oracle unitaries in a particular form. We also use this to show that any quantum algorithm that solves the Deutsch-Jozsa problem starting with a quantum system in a particular class of initial, thermal equilibrium-based states of the type encountered in solution state NMR can only succeed with greater probability than a classical algorithm when the problem size exceeds $n \sim 10^5.$