The polynomial degree of recursive Fourier sampling

Authors:
Benjamin Johnson
Affiliations:
School of Information, University of California, Berkeley
Venue:
TQC'10 Proceedings of the 5th conference on Theory of quantum computation, communication, and cryptography
Year:
2010

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Abstract

We present matching upper and lower bounds for the "weak" polynomial degree of the recursive Fourier sampling problem from quantum complexity theory. The degree bound is h + 1, where h is the order of recursion in the problem's definition, and this bound is exponentially lower than the bound implied by the existence of a BQP algorithm for the problem. For the upper bound we exhibit a degree-h + 1 real polynomial that represents the problem on its entire domain. For the lower bound, we show that any non-zero polynomial agreeing with the problem, even on just its zero-inputs, must have degree at least h + 1. The lower bound applies to representing polynomials over any Field.