Tapered Quantum Phase Estimation*

The quantum phase estimation (QPE) algorithm has been used in algorithmic settings to determine the phase of an input state, solve systems of linear equations, estimate quantum amplitudes, and for quantum principal component analysis. In standard QPE, ancilla qubits are prepared in an equal superposition state. In tapered quantum phase estimation (tQPE) we allow for an arbitrary initial ancilla state and optimize this state using modern signal analysis methods. In particular, given a number of ancilla qubits, we minimize the probability of outputting an estimate that deviates from the true phase by a certain amount in the worst-case and average error settings. Using frequency- (equivalently, phase-) concentrated tapered estimates, we show that the number of extra qubits in tQPE required to guarantee that the output of the algorithm is $delta$-close to the true phase with probability at least $1 - epsilon$ can be reduced asymptotically to $m =O(loglog(1/epsilon))$, which is optimal. We then show that the worst-case success probability for QPE (with no extra qubits, $m = 0$) can be improved by approximately $20\%$ using an optimal taper. Finally, we demonstrate that the mean success probability can be improved, relative to standard QPE, in all cases.