Windowed Quantum Phase Estimation: Signal Processing Approach to a Quantum Algorithm

Quantum phase estimation (QPE) is a key ingredient to some of the most important results of quantum computing. In this paper we examine the QPE algorithm through the looking glass of signal processing. This perspective helps to reveal a relation between the distribution of quantum phase estimates and the shape of the discrete Fourier transform (DFT) of a pure complex sinusoid. Guided by spectral shaping theory, we design a class of quantum circuits that are the quantum equivalents to classical windowing operators, for a family of window functions that have compact support in the DFT. These circuits have complexity linear in the number of target qubits, strictly lower than QPE itself. Using these circuits we are able to reshape QPE's performance characteristics. In particular, the asymptotic marginal cost for 2x reduction in fail rate (2x improvement in reliability) is reduced from one qubit to 1/3 qubit or lower, which can help reduce the qubit budget for a preset accuracy-reliability target, especially in high-reliability scenarios. This makes a demonstration of the value of signal processing principles in designing quantum algorithms.