Testing Permanent Oracles -- Revisited

TL;DR

This paper presents a polynomial-time algorithm to test the accuracy of oracles claiming to approximate the permanent of Gaussian matrices, addressing a key challenge in complexity theory and quantum computing.

Contribution

It introduces a novel polynomial-time testing method for permanent approximation oracles over complex numbers, extending previous finite field techniques.

Findings

The testing algorithm works efficiently for Gaussian matrices.

It bridges a gap in verifying permanent approximation oracles.

The approach may influence future research in quantum complexity theory.

Abstract

Suppose we are given an oracle that claims to approximate the permanent for most matrices X, where X is chosen from the Gaussian ensemble (the matrix entries are i.i.d. univariate complex Gaussians). Can we test that the oracle satisfies this claim? This paper gives a polynomial-time algorithm for the task. The oracle-testing problem is of interest because a recent paper of Aaronson and Arkhipov showed that if there is a polynomial-time algorithm for simulating boson-boson interactions in quantum mechanics, then an approximation oracle for the permanent (of the type described above) exists in BPP^NP. Since computing the permanent of even 0/1 matrices is #P-complete, this seems to demonstrate more computational power in quantum mechanics than Shor's factoring algorithm does. However, unlike factoring, which is in NP, it was unclear previously how to test the correctness of an approximation oracle for the permanent, and this is the contribution of the paper. The technical difficulty overcome here is that univariate polynomial self-correction, which underlies similar oracle-testing algorithms for permanent over finite fields --- and whose discovery led to a revolution in complexity theory --- does not seem to generalize to complex (or even, real) numbers. We believe that this tester will motivate further progress on understanding the permanent of Gaussian matrices.