A Note on Fine-Grained Quantum Reductions for Linear Algebraic Problems

We observe that any $T(n)$ time algorithm (quantum or classical) for several central linear algebraic problems, such as computing $\det(A)$, $tr(A^3)$, or $tr(A^{-1})$ for an $n \times n$ integer matrix $A$, yields a $O(T(n)) + \tilde O(n^2)$ time \textit{quantum algorithm} for $n \times n$ matrix-matrix multiplication. That is, on quantum computers, the complexity of these problems is essentially equivalent to that of matrix multiplication. Our results follow by first observing that the Bernstein-Vazirani algorithm gives a direct quantum reduction from matrix multiplication to computing $tr(ABC)$ for $n \times n$ inputs $A,B,C$. We can then reduce $tr(ABC)$ to each of our problems of interest. For the above problems, and many others in linear algebra, their fastest known algorithms require $\Theta(n^\omega)$ time, where $\omega \approx 2.37$ is the current exponent of fast matrix multiplication. Our finding shows that any improvements beyond this barrier would lead to faster quantum algorithms for matrix multiplication. Our results complement existing reductions from matrix multiplication in algebraic circuits [BCS13], and reductions that work for standard classical algorithms, but are not tight -- i.e., which roughly show that an $O(n^{3-\delta})$ time algorithm for the problem yields an $O(n^{3-\delta/3})$ matrix multiplication algorithm [WW10].