Practical computational advantage from the quantum switch on a generalized family of promise problems

TL;DR

This paper demonstrates that the quantum switch offers computational advantages for a broad class of promise problems, including generalized cases using complex Hadamard matrices, and highlights the necessity of continuous variables for the most general scenarios.

Contribution

It generalizes promise problems using complex Hadamard matrices and proves quantum switch advantages in both finite and infinite-dimensional systems.

Findings

Quantum switch reduces query complexity for generalized promise problems.

Continuous variable systems are necessary for the most general promise problems.

Quantum advantage holds for both finite and infinite-dimensional cases.

Abstract

The quantum switch is a quantum computational primitive that provides computational advantage by applying operations in a superposition of orders. In particular, it can reduce the number of gate queries required for solving promise problems where the goal is to discriminate between a set of properties of a given set of unitary gates. In this work, we use Complex Hadamard matrices to introduce more general promise problems, which reduce to the known Fourier and Hadamard promise problems as limiting cases. Our generalization loosens the restrictions on the size of the matrices, number of gates and dimension of the quantum systems, providing more parameters to explore. In addition, it leads to the conclusion that a continuous variable system is necessary to implement the most general promise problem. In the finite dimensional case, the family of matrices is restricted to the so-called Butson-Hadamard type, and the complexity of the matrix enters as a constraint. We introduce the ``query per gate'' parameter and use it to prove that the quantum switch provides computational advantage for both the continuous and discrete cases. Our results should inspire implementations of promise problems using the quantum switch where parameters and therefore experimental setups can be chosen much more freely.