精确量子分裂与有限代数的结构

中文摘要 AI 辅助

Berlekamp算法通过确定性线性代数对有限域𝔽_q[x]中的无平方多项式f进行因式分解，将问题简化为将显式交换代数B≅𝔽_q^r分裂为其r个单因子。对于大奇数q，标准高效分裂步骤是随机化的，而已知的去随机化方法依赖于广义黎曼假设。我们在允许通过高效计算角度进行单量子比特旋转的电路模型中，给出了一种无条件的精确量子实现，该构造使用无条件计数论证。对于包含s≥2个不可约因子的块，奇特征下的二次特征测试和特征2下的绝对迹测试会以概率p_{q,s}≥1/2产生非恒定测试元素，该概率预先精确已知，仅取决于q和s，与未知因式分解无关。因此，精确幅度放大将每个随机测试转化为在一次放大迭代后以确定性成功的过程。所得算法恰好使用r-1次量子分裂轮次，以及O(n³log q)次量子𝔽_q运算和O(n³)次经典运算，不需要本原根、二次非剩余或不同次数的预处理。该方法还可分裂由结构常数给出的任意有限维可分交换𝔽_q代数。结合Rónyai的经典结构理论（其确定性计算根基并将剩余任务确定性归约为多项式因式分解），对于由结构常数给出的任意n维结合𝔽_q代数，它可确定性地得到根基和A/Rad(A)的Wedderburn分解为极小双边理想，使用O(n⁴log q)次量子𝔽_q运算。

英文摘要

Berlekamp's algorithm factors a squarefree polynomial $f\in\mathbb{F}_q[x]$ by deterministic linear algebra, reducing the problem to splitting an explicit commutative algebra $B\cong\mathbb{F}_q^r$ into its $r$ simple factors. For large odd $q$, the standard efficient splitting step is randomized, while known derandomizations are conditional on the Extended Riemann Hypothesis. We give an unconditional exact quantum implementation in a circuit model permitting single-qubit rotations through efficiently computable angles. The construction uses an unconditional counting argument. For a block containing $s\ge2$ irreducible factors, a quadratic-character test in odd characteristic and an absolute-trace test in characteristic $2$ yield a nonconstant test element with probability $p_{q,s}\ge\tfrac12$, known exactly in advance and depending only on $q$ and $s$, not on the unknown factorization. Exact amplitude amplification therefore converts each randomized test into a procedure succeeding with certainty after one amplification iteration. The resulting algorithm uses exactly $r-1$ quantum splitting rounds and $O(n^3\log q)$ quantum $\mathbb{F}_q$-operations and $O(n^3)$ classical operations, requiring no primitive root, quadratic non-residue, or distinct-degree preprocessing. The method also splits arbitrary finite-dimensional separable commutative $\\mathbb{F}_q$-algebras given by structure constants. Combined with R'onyai's classical structure theory, which computes the radical deterministically and reduces the remaining tasks deterministically to polynomial factorization, it yields the radical and the Wedderburn decomposition of $A/\mathrm{Rad}(A)$ into minimal two-sided ideals, with certainty, for any $n$-dimensional associative $\mathbb{F}_q$-algebra given by structure constants, using $O(n^4\log q)$ quantum $\mathbb{F}_q$-operations.

发表机构

• School of Computer Science, University of Birmingham（伯明翰大学计算机学院）
• CISITS Lab, Department of Mathematics, Universitas Indonesia（印度尼西亚大学数学系）

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