The Computational Complexity of Quantum Determinants
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In this work, we study the computational complexity of quantum determinants, a q-deformation of matrix permanents: Given a complex number q on the unit circle in the complex plane and an n× n matrix X, the q-permanent of X is defined as Per_q(X) = ∑_σ∈ S_n q^ℓ(σ)X_1,σ(1)… X_n,σ(n), where ℓ(σ) is the inversion number of permutation σ in the symmetric group S_n on n elements. The function family generalizes determinant and permanent, which correspond to the cases q=-1 and q=1 respectively. For worst-case hardness, by Liouville's approximation theorem and facts from algebraic number theory, we show that for primitive m-th root of unity q for odd prime power m=p^k, exactly computing q-permanent is 𝖬𝗈𝖽_p𝖯-hard. This implies that an efficient algorithm for computing q-permanent results in a collapse of the polynomial hierarchy. Next, we show that computing q-permanent can be achieved using an oracle that approximates to within a polynomial multiplicative error and a membership oracle for a finite set of algebraic integers. From this, an efficient approximation algorithm would also imply a collapse of the polynomial hierarchy. By random self-reducibility, computing q-permanent remains to be hard for a wide range of distributions satisfying a property called the strong autocorrelation property. Specifically, this is proved via a reduction from 1-permanent to q-permanent for O(1/n^2) points z on the unit circle. Since the family of permanent functions shares common algebraic structure, various techniques developed for the hardness of permanent can be generalized to q-permanents.
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