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

We study the problem of computing the Hamming weight of an n-bit string modulo m, for any positive integer m ≤ n whose only prime factors are 2 and 3. We show that the exact quantum query complexity of this problem is ⌈ n(1 - 1/m) ⌉. 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].

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