From Bit-Parallelism to Quantum: Breaking the Quadratic Barrier

Many problems that can be solved in quadratic time have bit-parallel speed-ups with factor $w$, where $w$ is the computer word size. For example, edit distance of two strings of length $n$ can be solved in $O(n^2/w)$ time. In a reasonable classical model of computation, one can assume $w=\Theta(\log n)$. There are conditional lower bounds for such problems stating that speed-ups with factor $n^\epsilon$ for any $\epsilon>0$ would lead to breakthroughs in complexity theory. However, these conditional lower bounds do not cover quantum models of computing. Moreover, it is open if problems like edit distance can be solved in truly sub-quadratic time using quantum computing. To partially address this question, we study another bit-parallel algorithm for a problem that admits a quadratic conditional lower bound, and show how to convert its bit-parallelism into a realistic quantum algorithm that attains speed-up with factor $n$. The technique we use is simple and general enough to apply to many similar bit-parallel algorithms, where dependencies are local. However, it does not immediately yield a faster algorithm for more complex problems like edit distance, whose bit-parallel dynamic programming solutions require breaking more global dependencies. We hope that this initial study sheds some light on how, in general, bit-parallelism could be converted to quantum parallelism.