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

Noah Linden1 and Ronald de Wolf2

1School of Mathematics, University of Bristol. n.linden@bristol.ac.uk
2QuSoft, CWI and University of Amsterdam, the Netherlands. rdewolf@cwi.nl

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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.

The quantum Fourier transform (QFT) is a key primitive that is typically used as a subroutine within a larger quantum computation. As such, we may have little control over the state that is input to the QFT. We show that good performance of the QFT on an $average$ input state (1) is efficiently testable, and (2) suffices to achieve good $worst$-$case$ performance for QFT-based tasks such as phase estimation, period finding and amplitude estimation.

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Cited by

[1] Joran van Apeldoorn, Arjan Cornelissen, András Gilyén, and Giacomo Nannicini, "Quantum tomography using state-preparation unitaries", arXiv:2207.08800, (2022).

[2] Vahid R. Asadi, Alexander Golovnev, Tom Gur, Igor Shinkar, and Sathyawageeswar Subramanian, "Quantum Worst-Case to Average-Case Reductions for All Linear Problems", arXiv:2212.03348, (2022).

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