Exact quantum query complexity of computing Hamming weight modulo powers of two and three

Abstract

We study the problem of computing the Hamming weight of an $n$-bit string modulo $m$, for any positive integer $m \leq n$ whose only prime factors are 2 and 3. We show that the exact quantum query complexity of this problem is $\left\lceil n(1 - 1/m) \right\rceil$. The upper bound is via an iterative query algorithm whose core components are the well-known 1-query quantum algorithm (essentially due to Deutsch) to compute the Hamming weight a 2-bit string mod 2 (i.e., the parity of the input bits), and a new 2-query quantum algorithm to compute the Hamming weight of a 3-bit string mod 3. We show a matching lower bound (in fact for arbitrary moduli $m$) via a variant of the polynomial method [de Wolf, SIAM J. Comput., 32(3), 2003]. This bound is for the weaker task of deciding whether or not a given $n$-bit input has Hamming weight 0 modulo $m$, and it holds even in the stronger non-deterministic quantum query model where an algorithm must have positive acceptance probability iff its input evaluates to 1. For $m>2$ our lower bound exceeds $n/2$, beating the best lower bound provable using the general polynomial method [Theorem 4.3, Beals et al., J. ACM 48(4), 2001].