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This paper presents a novel quantum coding scheme that achieves exact multiuser unanimity testing with $O(\log n)$ complexity, significantly improving upon classical quadratic scaling limits. By leveraging specific angular pulse sequences within a serial quantum architecture, the method enables efficient verification of consensus states without requiring Bell entanglement, offering a robust primitive for distributed quantum networks.
Ordered-Angle Coding for Exact Multiuser Unanimity Testing
arXiv:2609.05645v1 Announce Type: new Abstract: We introduce ordered-angle coding for binary-unanimity testing among $n$ transformation-only users in a serial quantum architecture inherited from Loop-Back communication. User $B_i$ fixes one private sign across a logical word and applies $R(s_i\alpha_j)$ in trial $j$. If $w$ users choose the negative sign, serial composition gives $R[(n-2w)\alpha_j]$. We show that every fixed same-axis pulse that is deterministic for both unanimous inputs under the binary Bell readout has $\alpha_j=q_j\pi/(2n)$ with integer $q_j$. For a nonadaptive word $\mathbf q=(q_1,\ldots,q_m)$, a mixed Hamming weight imitates the unanimous signature with probability \[ M_{\mathbf q}(w)=\prod_{j=1}^{m}\cos^2\!\left(\frac{\pi q_jw}{n}\right). \] Within this complete deterministic-unanimity pulse family, a finite perfect word exists if and only if $n$ is a power of two. For $n=2^r$, the dyadic word $(1,2,4,\ldots,2^{r-1})$ is exact and pulse-minimal with $m_{\min}=r=\log_2n$ trials. It replaces the $O(n^2\log(1/\varepsilon))$ worst-case burden of repeated smallest-angle tests by exact $O(\log n)$ verification. Odd primes instead admit balanced statistical words with uniform mixed-weight imitation $2^{-(p-1)}$. For actual operations $R(s_i\alpha_j+\delta_{ij})$, an ideal rejecting position is lifted to $\sin^2\Delta_j$, where $\Delta_j=\sum_i\delta_{ij}$. This yields explicit robustness bounds with angular and binary-readout errors. Bell entanglement is not required for the additive algebra, but it keeps the traveler locally maximally mixed in every honest trial. We therefore present the result as a coded multiuser relation primitive with optional raw conference-key-candidate use, not as a composably secure conference-key protocol.
Abstract:We introduce ordered-angle coding for binary-unanimity testing among $n$ transformation-only users in a serial quantum architecture inherited from Loop-Back communication. User $B_i$ fixes one private sign across a logical word and applies $R(s_i\alpha_j)$ in trial $j$. If $w$ users choose the negative sign, serial composition gives $R[(n-2w)\alpha_j]$. We show that every fixed same-axis pulse that is deterministic for both unanimous inputs under the binary Bell readout has $\alpha_j=q_j\pi/(2n)$ with integer $q_j$. For a nonadaptive word $\mathbf q=(q_1,\ldots,q_m)$, a mixed Hamming weight imitates the unanimous signature with probability \[ M_{\mathbf q}(w)=\prod_{j=1}^{m}\cos^2\!\left(\frac{\pi q_jw}{n}\right). \] Within this complete deterministic-unanimity pulse family, a finite perfect word exists if and only if $n$ is a power of two. For $n=2^r$, the dyadic word $(1,2,4,\ldots,2^{r-1})$ is exact and pulse-minimal with $m_{\min}=r=\log_2n$ trials. It replaces the $O(n^2\log(1/\varepsilon))$ worst-case burden of repeated smallest-angle tests by exact $O(\log n)$ verification. Odd primes instead admit balanced statistical words with uniform mixed-weight imitation $2^{-(p-1)}$. For actual operations $R(s_i\alpha_j+\delta_{ij})$, an ideal rejecting position is lifted to $\sin^2\Delta_j$, where $\Delta_j=\sum_i\delta_{ij}$. This yields explicit robustness bounds with angular and binary-readout errors. Bell entanglement is not required for the additive algebra, but it keeps the traveler locally maximally mixed in every honest trial. We therefore present the result as a coded multiuser relation primitive with optional raw conference-key-candidate use, not as a composably secure conference-key protocol.
Quantum Physics (quant-ph); Cryptography and Security (cs.CR)
arXiv:2609.05645 [quant-ph]
(or arXiv:2609.05645v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.05645 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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