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 $n\times n$ matrix $X$, the $q$-permanent of $X$ is defined as $$\mathrm{Per}_q(X) = \sum_{\sigma\in S_n} q^{\ell(\sigma)}X_{1,\sigma(1)}\ldots X_{n,\sigma(n)},$$ where $\ell(\sigma)$ is the inversion number of permutation $\sigma$ 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 $\mathsf{Mod}_p\mathsf{P}$-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.

翻译：本文研究了量子行列式的计算复杂性，即矩阵积和式的一种$q$-形变：给定单位圆上的复数$q$和一个$n\times n$矩阵$X$，$X$的$q$-积和式定义为$$\mathrm{Per}_q(X) = \sum_{\sigma\in S_n} q^{\ell(\sigma)}X_{1,\sigma(1)}\ldots X_{n,\sigma(n)},$$其中$\ell(\sigma)$是$n$元对称群$S_n$中置换$\sigma$的逆序数。该函数族推广了行列式和积和式，分别对应$q=-1$和$q=1$的情形。在最坏情况复杂性方面，利用Liouville逼近定理和代数数论中的结论，我们证明对于奇素数幂$m=p^k$的本原$m$次单位根$q$，精确计算$q$-积和式是$\mathsf{Mod}_p\mathsf{P}$-难的。这意味着存在高效算法计算$q$-积和式将导致多项式层级坍缩。进一步，我们证明使用多项式乘法误差逼近预言机和有限代数整数集合的成员关系预言机可以计算$q$-积和式。由此，存在高效近似算法同样意味着多项式层级坍缩。通过随机自归约性，计算$q$-积和式对于满足强自相关性质的大量分布仍然是困难的。具体地，这一结论是通过从$1$-积和式到单位圆上$O(1/n^2)$个点$z$处的$q$-积和式的归约证明的。由于积和式函数族共享共同的代数结构，为积和式困难性发展的多种技术可以推广到$q$-积和式。