Average-Case Verification of the Quantum Fourier Transform Enables Worst-Case Phase Estimation

TL;DR

This paper demonstrates that ensuring average-case performance of the quantum Fourier transform suffices for reliable worst-case phase estimation, and provides an efficient method to verify this average-case behavior.

Contribution

It shows that average-case verification of the QFT guarantees worst-case accuracy in key quantum algorithms and introduces an efficient verification procedure.

Findings

Average-case performance implies worst-case accuracy for phase estimation.

Efficient verification method for average-case QFT performance.

Reduces the complexity of validating quantum Fourier transform implementations.

Abstract

The quantum Fourier transform (QFT) is a key primitive for quantum computing that is typically used as a subroutine within a larger computation, for instance for phase estimation. As such, we may have little control over the state that is input to the QFT. Thus, in implementing a good QFT, we may imagine that it needs to perform well on arbitrary input states. Verifying this worst-case correct behaviour of a QFT-implementation would be exponentially hard (in the number of qubits) in general, raising the concern that this verification would be impossible in practice on any useful-sized system. In this paper we show that, in fact, we only need to have good average-case performance of the QFT to achieve good worst-case performance for key tasks -- phase estimation, period finding and amplitude estimation. Further we give a very efficient procedure to verify this required average-case behaviour of the QFT.