On the hardness of the noncommutative determinant

TL;DR

This paper investigates the computational difficulty of calculating the noncommutative determinant, establishing its deep connections to the permanent problem and demonstrating its inherent complexity through various reductions.

Contribution

It proves that small circuits for the noncommutative determinant imply small circuits for the permanent, highlighting the problem's computational hardness.

Findings

Small noncommutative determinant circuits imply small permanent circuits.

Computing the permanent reduces to computing a large noncommutative determinant.

Permanent over rationals reduces to noncommutative determinant over Clifford algebras.

Abstract

In this paper we study the computational complexity of computing the noncommutative determinant. We first consider the arithmetic circuit complexity of computing the noncommutative determinant polynomial. Then, more generally, we also examine the complexity of computing the determinant (as a function) over noncommutative domains. Our hardness results are summarized below: 1. We show that if the noncommutative determinant polynomial has small noncommutative arithmetic circuits then so does the noncommutative permanent. Consequently, the commutative permanent polynomial has small commutative arithmetic circuits. 2. For any field F we show that computing the n X n permanent over F is polynomial-time reducible to computing the 2n X 2n (noncommutative) determinant whose entries are O(n^2) X O(n^2) matrices over the field F. 3. We also derive as a consequence that computing the n X n permanent over nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of n^{O(1)} dimension. Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials.