  Many quantum algorithms rely on the measurement of complex quantum
amplitudes. Standard approaches to obtain the phase information, such as the
Hadamard test, give rise to large overheads due to the need for global
controlled-unitary operations. We introduce a quantum algorithm based on
complex analysis that overcomes this problem for amplitudes that are a
continuous function of time. Our method only requires the implementation of
real-time evolution and a shallow circuit that approximates a short
imaginary-time evolution. We show that the method outperforms the Hadamard test
in terms of circuit depth and that it is suitable for current noisy quantum
computers when combined with a simple error-mitigation strategy.
