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

Jorge Escandón-Monardes, Aldo Delgado, and Stephen P. Walborn

Millennium Institute for Research in Optics and Physics Department, Universidad de Concepción, 160-C Concepción, Chile

### 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.

A set of quantum operations can be applied on a target system in different orders. In the simplest case, an operation $A$ can be followed by another operation $B$ or, conversely, $B$ can be followed by $A$. Interestingly, in quantum mechanics those orders can be coherently controlled by an additional quantum system, leading to a "superposition" of different gate orders. This can be achieved using a device known as quantum switch, which has seen a broad range of applications in recent years.

In particular, the quantum switch provides computational advantage in solving some promise problems, such as the Fourier Promise Problem. However, experimental implementations of this task are technically difficult, since they require the dimension of the quantum systems to scale factorially with the number of gates.

Here, we generalise previous approaches by introducing the Complex Hadamard Promise Problem and prove that this family exists for every finite dimension, removing the unfavourable scaling of the Fourier Promise Problem. Moreover, we take its study to the continuous variable regime and loosen restrictions on a number of parameters. This should inspire new practical implementations of promise problems using the quantum switch.

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