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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2604.13554 |
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| _version_ | 1866913033567076352 |
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| author | Bae, Ji Ho |
| author_facet | Bae, Ji Ho |
| contents | 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 $\operatorname{sgn}(σ)$ to even tensor powers. The proof combines a reduction to $S_N$ Kronecker products via Rademacher moment polynomials with the bipartition distance formula $d_T(((N),\varnothing),(α,β)) = 2(N-α_1)-|β|$ in the tensor product graph. A closed-form generating function yields the first-appearance multiplicity $(2N-3)!!$. We also show $Q_{\mathrm{decomp}}(φ) \le 2\,Q_{\mathrm{signed}}(φ)$, with equality on $B_2$, and conjecture a link between the adversary bound and the graph eccentricity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13554 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum Query Complexity of the Hyperoctahedral Group Bae, Ji Ho Combinatorics 81P68, 20C30, 68Q12 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 $\operatorname{sgn}(σ)$ to even tensor powers. The proof combines a reduction to $S_N$ Kronecker products via Rademacher moment polynomials with the bipartition distance formula $d_T(((N),\varnothing),(α,β)) = 2(N-α_1)-|β|$ in the tensor product graph. A closed-form generating function yields the first-appearance multiplicity $(2N-3)!!$. We also show $Q_{\mathrm{decomp}}(φ) \le 2\,Q_{\mathrm{signed}}(φ)$, with equality on $B_2$, and conjecture a link between the adversary bound and the graph eccentricity. |
| title | Quantum Query Complexity of the Hyperoctahedral Group |
| topic | Combinatorics 81P68, 20C30, 68Q12 |
| url | https://arxiv.org/abs/2604.13554 |