Quantum Query Complexity of the Hyperoctahedral Group

TL;DR

This paper determines the quantum query complexity of oracle identification on the hyperoctahedral group, revealing it is twice that of the symmetric group due to an e9-parity obstruction, and introduces new combinatorial and algebraic tools.

Contribution

It provides an exact quantum query complexity for the hyperoctahedral group and develops a novel reduction technique and multiplicity formulas for analyzing such complexities.

Findings

Quantum query complexity of B_N is 2(N-1).

Doubling of complexity compared to S_N is due to an e9-parity obstruction.

A closed-form generating function for multiplicities is derived.

Abstract

We determine the quantum query complexity of oracle identification on the hyperoctahedral group $B_N = \{\pm 1\}^N \rtimes S_N$ with respect to the natural representation: $Q_{LV}(B_N) = 2(N-1)$ for all $N \ge 2$. This is twice the symmetric-group value $Q_{LV}(S_N) = N-1$; the doubling arises from an $\varepsilon$-parity obstruction that restricts the bottleneck representation $\operatorname{sgn}(\sigma)$ to even tensor powers. The proof combines a reduction to $S_N$ Kronecker products via Rademacher moment polynomials with the bipartition distance formula $d_T(((N),\varnothing),(\alpha,\beta)) = 2(N-\alpha_1)-|\beta|$ in the tensor product graph. A closed-form generating function yields the first-appearance multiplicity $(2N-3)!!$. We also show $Q_{\mathrm{decomp}}(\varphi) \le 2\,Q_{\mathrm{signed}}(\varphi)$, with equality on $B_2$, and conjecture a link between the adversary bound and the graph eccentricity.