On continuous variable quantum algorithms for oracle identification problems

TL;DR

This paper develops a framework for continuous variable quantum algorithms applied to oracle identification problems, demonstrating that the Deutsch-Jozsa algorithm in this setting is probabilistic and cannot achieve exponential speed-up due to an uncertainty principle.

Contribution

It introduces a formalism for continuous variable quantum algorithms in oracle problems and clarifies the limitations of the Deutsch-Jozsa algorithm in this context.

Findings

The continuous variable Deutsch-Jozsa algorithm is probabilistic.

An uncertainty relation prevents exponential speed-up.

Contradicts previous claims of exponential advantage.

Abstract

We establish a framework for oracle identification problems in the continuous variable setting, where the stated problem necessarily is the same as in the discrete variable case, and continuous variables are manifested through a continuous representation in an infinite-dimensional Hilbert space. We apply this formalism to the Deutsch-Jozsa problem and show that, due to an uncertainty relation between the continuous representation and its Fourier-transform dual representation, the corresponding Deutsch-Jozsa algorithm is probabilistic hence forbids an exponential speed-up, contrary to a previous claim in the literature.