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arXiv · 2607.13186

Quantum determinants in polynomial time

Igor Pak·Daniel Soskin
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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.

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Igor Pak, Daniel Soskin. 2026-07-14. Quantum determinants in polynomial time. https://arxiv.org/abs/2607.13186

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