The problem of discriminating between many quantum channels with certainty is analyzed under the assumption of prior knowledge of algebraic relations among possible channels. It is shown, by explicit construction of a novel family of quantum algorithms, that when the set of possible channels faithfully represents a finite subgroup of SU(2) (e.g., $C_n, D_{2n}, A_4, S_4, A_5$) the recently developed techniques of quantum signal processing can be modified to constitute subroutines for quantum hypothesis testing. These algorithms, for group quantum hypothesis testing, intuitively encode discrete properties of the channel set in SU(2) and improve query complexity at least quadratically in $n$, the size of the channel set and group, compared to naïve repetition of binary hypothesis testing. Intriguingly, performance is completely defined by explicit group homomorphisms; these in turn inform simple constraints on polynomials embedded in unitary matrices. These constructions demonstrate a flexible technique for mapping questions in quantum inference to the well-understood subfields of functional approximation and discrete algebra. Extensions to larger groups and noisy settings are discussed, as well as paths by which improved protocols for quantum hypothesis testing against structured channel sets have application in the transmission of reference frames, proofs of security in quantum cryptography, and algorithms for property testing.

中文翻译：

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具有群结构的量子假设检验

在可能信道之间的代数关系的先验知识的假设下，分析了确定区分许多量子信道的问题。通过显式构造一个新的量子算法族，表明当可能的信道集忠实地表示 SU(2) 的一个有限子群时（例如，$C_n, D_{2n}, {\mathrm{一种}}_4, {秒}_4, {\mathrm{一种}}_5$) 可以修改最近开发的量子信号处理技术，以构成用于量子假设检验的子程序。这些算法，用于群量子假设检验，直观地编码 SU(2) 中信道集的离散属性，并至少以二次方的方式提高查询复杂度$n$，通道集和组的大小，与二元假设检验的简单重复相比。有趣的是，性能完全由显式群同态定义；这些反过来又为嵌入在酉矩阵中的多项式提供了简单的约束。这些结构展示了一种灵活的技术，可以将量子推理中的问题映射到泛函近似和离散代数的广为人知的子领域。讨论了对更大群体和嘈杂设置的扩展，以及用于针对结构化通道集的量子假设测试的改进协议在参考帧的传输、量子密码学中的安全性证明和用于属性测试的算法中的应用的路径。