Further Applications of the Generalised Phase Kick-Back

TL;DR

This paper extends the Generalised Phase Kick-Back technique to new problems, demonstrating its effectiveness in solving generalized versions of Deutsch-Jozsa and Simon's problems and exploring connections to Boolean function theory.

Contribution

It introduces the concept of y-balanced functions, solves the fully balanced image problem, and generalizes Simon's problem using the GPK technique, advancing quantum algorithm applications.

Findings

Successfully generalized Deutsch-Jozsa problem.

Solved the fully balanced image problem.

Provided a more efficient solution to generalized Simon's problem.

Abstract

In our previous work, we defined a quantum algorithmic technique known as the Generalised Phase Kick-Back, or $GPK$, and analysed its applications in generalising some classical quantum problems, such as the Deutsch-Jozsa problem or the Bernstein-Vazirani problem. We also proved that using this technique we can solve Simon's problem in a more efficient manner. In this paper we continue analysing the potential of this technique, defining the concept of $\mathbf{y}$-balanced functions and solving a new problem, which further generalises the generalised Deutsch-Jozsa problem (the fully balanced image problem). This problem also underlines the relation between quantum computation and Boolean function theory, and, in particular, the Walsh and Fourier-Hadamard transforms. We finish our discussion by solving the generalised version of Simon's problem using the $GPK$ algorithm, while analysing the efficiency of this new solution.