Efficient Quantum Circuits for Diagonal Unitaries Without Ancillas

TL;DR

This paper introduces a method to construct efficient, ancilla-free quantum circuits for diagonal unitaries by leveraging Walsh function approximations, enabling resource-efficient quantum simulations.

Contribution

It establishes a novel correspondence between Walsh functions and diagonal operators, simplifying the design of minimal-depth circuits without ancillas for arbitrary diagonal unitaries.

Findings

Achieves high-fidelity quantum simulations with few qubits

Reduces circuit depth and resource requirements

Demonstrates practical application to quantum chemistry problem

Abstract

The accurate evaluation of diagonal unitary operators is often the most resource-intensive element of quantum algorithms such as real-space quantum simulation and Grover search. Efficient circuits have been demonstrated in some cases but generally require ancilla registers, which can dominate the qubit resources. In this paper, we point out a correspondence between Walsh functions and a basis for diagonal operators that gives a simple way to construct efficient circuits for diagonal unitaries without ancillas. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitary $e^{if(\hat{x})}$ in the $|x>$ basis, to that of finding the minimal-length Walsh-series approximation to the function $f(x)$. We apply this approach to the quantum simulation of the classical Eckart barrier problem of quantum chemistry, demonstrating that high-fidelity quantum simulations can be achieved with few qubits and low depth.