黑盒阿贝尔群分解的改进量子算法
Improved Quantum Algorithms for Black-Box Abelian Group Decomposition
- School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
- School of Artificial Intelligence, Wuhan University（武汉大学人工智能学院）
- Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对黑盒阿贝尔群分解问题，提出改进量子算法，通过改编Regev采样与格归约，将电路门数、时间和空间复杂度分别降至O~(n^(3/2)T_op)、O~(n^(5/2)T_op)和O(n)量子比特。
AI中文摘要：
将有限阿贝尔黑盒群分解为循环因子是量子计算中的一个基本问题。我们通过改编Regev因子分解算法中的量子采样和经典格归约方法，给出了一种分解有限阿贝尔黑盒群的量子算法。该算法以高概率计算不变因子分解及相应的循环生成元。对于阶数至多为2^n、具有唯一编码且可逆群操作代价T_op = Ω(√n)的群G，我们的算法使用O(√n)个量子电路，每个电路包含O~(n T_op)个门，且每个电路最多执行O(n)次。作为比较，我们考虑了Cheung-Mosca算法，该算法分别分解Sylow p子群。我们还考虑了扩展Cheung-Mosca算法的公共模数实现。我们的算法将电路门计数总和从O~(n^2 T_op)降低到O~(n^(3/2) T_op)。总量子时间界从O~(n^3 T_op)降低到O~(n^(5/2) T_op)，量子空间界从O(n^2)量子比特降低到O(n)量子比特。经典计算使用多项式级别的比特操作和群操作查询。这些改进依赖于两个技术要素。我们证明了我们的算法中使用的整数关系格以高概率具有整数基，其向量的欧几里得范数至多为exp(O(√n))。我们通过纳入所有O(n)个采样生成元，同时保持格归约维度为O(√n)，保留了Regev算法在群分解中的电路规模优势。
英文摘要：
Decomposing finite Abelian black-box groups into cyclic factors is a basic problem in quantum computation. We give a quantum algorithm for decomposing finite Abelian black-box groups by adapting the quantum sampling and classical lattice-reduction method of Regev's factoring algorithm. The algorithm computes an invariant-factor decomposition and corresponding cyclic generators with high probability. For a group G of order at most 2^n with unique encodings and reversible group-operation cost T_op = Omega(sqrt(n)), our algorithm uses O(sqrt(n)) quantum circuits, each with O~(n T_op) gates and executed at most O(n) times. For comparison, we consider the Cheung-Mosca algorithm, which decomposes the Sylow p-subgroups separately. We also consider the common-modulus implementation of the extended Cheung-Mosca algorithm. Our algorithm reduces the sum of the circuit gate counts from O~(n^2 T_op) to O~(n^(3/2) T_op). The total quantum time bound reduces from O~(n^3 T_op) to O~(n^(5/2) T_op), and the quantum space bound reduces from O(n^2) to O(n) qubits. The classical computation uses polynomially many bit operations and group-operation queries. These improvements rely on two technical ingredients. We prove that the integer relation lattices used in our algorithm admit, with high probability, integral bases whose vectors have Euclidean norm at most exp(O(sqrt(n))). We preserve the circuit-size advantage of Regev's algorithm for group decomposition by incorporating all O(n) sampled generators while keeping the lattice-reduction dimension at O(sqrt(n)).
