BQP_p = PP for integer p > 2

TL;DR

This paper proves that for all integers p greater than 2, the class BQP_p, defined by quantum computations with p-norm measurements, equals PP, extending previous results that were limited to even p > 2.

Contribution

It generalizes the equality BQP_p = PP to all integers p > 2, broadening the understanding of quantum computational classes under p-norm perturbations.

Findings

BQP_p = PP for all integers p > 2

Extends previous results from even p > 2 to all p > 2

Shows the robustness of the class equality under p-norm variations

Abstract

There's something really strange about quantum mechanics. It's not just that cats can be dead and alive at the same time, and that entanglement seems to violate the principle of locality; quantum mechanics seems to be what Aaronson calls "an island in theoryspace", because even slight perturbations to the theory of quantum mechanics seem to generate absurdities. In [Aar 04] and [Aar 05], he explores these perturbations and the corresponding absurdities in the context of computation. In particular, he shows that a quantum theory where the measurement probabilities are computed using p-norm instead of the standard 2-norm has the effect of blowing up the class BQP (the class of problems that can be efficiently solved on a quantum computer) to at least PP (the class of problems that can be solved in probabilistic polynomial time). He showed that PP \subseteq BQP_p \subseteq PSPACE for all constants p != 2, and that BQP_p = PP for even integers p > 2. Here, we show that this equality holds for all integers p > 2.