On the hardness of the noncommutative determinant

  • Authors:

  • Vikraman Arvind;Srikanth Srinivasan

  • Affiliations:

  • Institute of Mathematical Sciences, Chennai, Chennai, India;Institute of Mathematical Sciences, Chennai, Chennai, India

  • Venue:
  • Proceedings of the forty-second ACM symposium on Theory of computing
  • Year:
  • 2010

Quantified Score

Hi-index 0.00

Visualization

Abstract

\begin{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 algorithms computing the determinant over noncommutative domains. Our hardness results are summarized below: 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. 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(n2) x O(n2) matrices over the field F. We also derive as a consequence that computing the n x n permanent over the nonnegative rationals is polynomial-time reducible to computing the noncommutative determinant over Clifford algebras of dimension nO(1). Our techniques are elementary and use primarily the notion of the Hadamard Product of noncommutative polynomials introduced in an earlier paper [AJS09].