发表机构
Peking University; Beijing Normal University; South China Normal University（北京大学; 北京师范大学; 华南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出多变量多项式变换的完整构造性合成理论，实现高效电路与量子信道变换，为多算子量子算法提供统一框架。
AI 中文摘要
多项式变换是量子算法中的基本原语：量子信号处理和奇异值变换将单变量多项式编译为电路，其查询复杂度主要由次数决定。然而，非交换矩阵的多变量变换缺乏类似的合成理论。我们在联合块访问下，为所有在指定矩阵域上压缩的多项式发展了一套完整的构造性理论。给定一个次数为$D$的多项式$P$的紧凑有限状态描述以及$0<\tau<1$，我们使用$O(D/\sqrt{\tau})$次查询合成具有$(1+\tau)$最优归一化的$P$，并在行块访问下精确使用$D$次查询，匹配次数下界。完整的电路实现在经典上可在多项式时间内计算，对精度的依赖为多对数级。我们构造的关键是一个有限的算法Schur--Agler定理：系数递推定义了一个多项式维度的延拓空间，支持在指定域上压缩性的完整半定证书。分解该证书产生查询算法背后的压缩实现。超越矩阵变换，我们的框架将多变量多项式合成提升到量子信道变换。给定相干Kraus访问$K=\{K_a\}_a$，我们合成任何有限的联合压缩非交换多项式映射族$K\mapsto\{F_b(K)\}_b$作为完全正操作，允许Kraus历史之间的相干干涉。此外，由因果Choi数据指定的信道级变换可以显式合成为固定阶量子梳。总的来说，这些结果指向一个更广泛的纲领：多变量逼近作为多算子量子算法和高阶量子信息处理的语言。
英文摘要
Polynomial transformations are basic primitives in quantum algorithms: quantum signal processing and singular value transformation compile univariate polynomials into circuits with query complexity essentially set by degree. Multivariable transformations of noncommuting matrices, however, lack a comparable synthesis theory. We develop a complete constructive theory under joint block access for all polynomials contractive on the prescribed matrix domain. Given a compact finite-state description of a degree-$D$ polynomial $P$ and $0<τ<1$, we synthesize $P$ with $(1+τ)$-optimal normalization using $O(D/\sqrtτ)$ queries, and exactly $D$ under row-block access, matching a degree lower bound. The complete circuit realization is classically computable in polynomial time with polylogarithmic dependence on accuracy. The key to our construction is a finite algorithmic Schur--Agler theorem: the coefficient recurrence defines a polynomial-dimensional continuation space supporting a complete semidefinite certificate for contractivity on the prescribed domain. Factoring this certificate yields the contractive realization underlying the query algorithm. Beyond matrix transformations, our framework lifts multivariable polynomial synthesis to quantum channel transformations. Given coherent Kraus access $K=\{K_a\}_a$, we synthesize any finite jointly contractive family of noncommutative polynomial maps $K\mapsto\{F_b(K)\}_b$ as completely positive operations, allowing coherent interference among Kraus histories. Furthermore, channel-level transformations specified by causal Choi data can be synthesized explicitly as fixed-order quantum combs. Collectively, these results point to a broader program: multivariable approximation as a language for multi-operator quantum algorithms and higher-order quantum information processing.
Comments51+37 pages
