  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.
