Abstract

The quantum switch is a quantum com-putational primitive that provides compu-tational advantage by applying operations in a superposition of orders. In partic-ular, it can reduce the number of gate queries required for solving promise prob-lems where the goal is to discriminate be-tween 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 re-duce 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 con-straint. We introduce the "query per gate" parameter and use it to prove that the quantum switch provides computational advantage for both the continuous and dis-crete cases. Our results should inspire im-plementations of promise problems using the quantum switch where parameters and therefore experimental setups can be cho-sen much more freely.