Quantum Query Complexity of the Hyperoctahedral Group

Abstract

We determine the quantum query complexity of oracle identification on the hyperoctahedral group $B_N=\{\pm 1{\}}^N⋊S_N$ 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 $\epsilon$-parity obstruction that restricts the bottleneck representation $\mathrm{sgn}(\ddot{I}ƒ)$ to even tensor powers. The proof combines a reduction to $S_N$ Kronecker products via Rademacher moment polynomials with the bipartition distance formula $d_T(((N),\varnothing ),(\hat{I}\pm ,{\hat{I}}^2))=2(N-\hat{I}{\pm }_1)-|{\hat{I}}^2|$ in the tensor product graph. A closed-form generating function yields the first-appearance multiplicity $(2N-3)!!$. We also show $Q_{\mathrm{decomp}}(\ddot{I}†)\le 2Q_{\mathrm{signed}}(\ddot{I}†)$, with equality on $B_2$, and conjecture a link between the adversary bound and the graph eccentricity.

33 pages