Threshold Fourier-coefficient anti-concentration

Establish the threshold Fourier-coefficient anti-concentration conjecture for the randomized partial-rotation construction in threshold BosonSampling, namely that the top Fourier coefficient is at least the Haar-averaged threshold output probability divided by a polynomial, except with inverse-polynomial probability over threshold outcomes, Haar-random unitaries, and rotation data.

Background

The alternative leading-coefficient extraction approach reduces the precision loss from exp(O(n log n)) to exp(O(n^λ)), but it requires anti-concentration of the relevant threshold Fourier coefficient. Specifically, Conjecture ass:tbs-tfca requires the top coefficient associated with a typical threshold outcome and randomized partial rotations to avoid being exponentially smaller than the corresponding Haar-averaged threshold probability.

The conjecture is essential because the reduction estimates a ratio whose denominator is this Fourier coefficient. A proof would make the conditional improved-imprecision result unconditional for threshold BosonSampling.