Quantum determinants in polynomial time

2026-07-14

AI Summary

This work addresses the efficient computation of Cayley determinants and their $q$-deformations and multi-parameter generalizations in noncommutative algebraic settings, such as right quantum matrix algebras. By introducing purely combinatorial proof techniques based on bijections and involutions on words, and integrating them with the classical Mahajan–Vinay construction, the study achieves—for the first time in the context of quantum groups—a polynomial-time algorithm for evaluating noncommutative determinants. The core contribution lies in constructing polynomial-size algebraic branching programs that efficiently compute Cayley determinants, and in establishing explicit combinatorial correspondences between these determinants and those of Moore and Valiant, encompassing both $q$-deformed and multi-parameter variants.

Abstract

We give an algebraic branching program of polynomial size which computes Cayley determinant of right quantum matrices. This is a rare example of an efficient computation of a noncommutative determinant, and the first such example for quantum groups. We extend the results to the $q$-Cayley determinant of $q$-right quantum matrices, as well as to their multiparameter generalization. The proofs are entirely combinatorial, as we relate Cayley, Moore and Valiant determinants using bijections/involutions on words. We then employ the celebrated determinant construction of Mahajan and Vinay (SODA'97), to obtain the results.

Problem

• quantum determinant
• noncommutative determinant
• Cayley determinant
• quantum matrices
• polynomial time

Innovation

• quantum determinant
• algebraic branching program
• noncommutative determinant
• combinatorial proof
• polynomial time