| Summary: | 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 give a simple way to construct efficient circuits for diagonal unitaries without ancillas, using a correspondence between Walsh functions and a basis for diagonal operators. This correspondence reduces the problem of constructing the minimal-depth circuit within a given error tolerance, for an arbitrary diagonal unitary ${{e}^{if\left( \hat{x}\, \right)}}$ in the $\left| x \right\rangle$ 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.
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