The Computational Complexity of Quantum Determinants

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

Multispace & Multistructure. Neutrosophic Transdisciplinarity (100 Collected Papers of Sciences), Vol. IV

The fourth volume, in my book series of “Collected Papers”, includes 100 published and unpublished articles, notes, (preliminary) drafts containing just ideas to be further investigated, scientific souvenirs, scientific blogs, project proposals, small experiments, solved and unsolved problems and conjectures, updated or alternative versions of previous papers, short or long humanistic essays, letters to the editors - all collected in the previous three decades (1980-2010) – but most of them are from the last decade (2000-2010), some of them being lost and found, yet others are extended, diversified, improved versions. This is an eclectic tome of 800 pages with papers in various fields of sciences, alphabetically listed, such as: astronomy, biology, calculus, chemistry, computer programming codification, economics and business and politics, education and administration, game theory, geometry, graph theory, information fusion, neutrosophic logic and set, non-Euclidean geometry, number theory, paradoxes, philosophy of science, psychology, quantum physics, scientific research methods, and statistics. It was my preoccupation and collaboration as author, co-author, translator, or cotranslator, and editor with many scientists from around the world for long time. Many topics from this book are incipient and need to be expanded in future explorations