稳定子电路模拟可归约为布尔线性代数的一个精简证明
A streamlined demonstration that stabilizer circuits simulation reduces to Boolean linear algebra
AI总结：
本文通过将稳定子 tableau 形式体系扩展至任意 Clifford 信道，把稳定子电路模拟归约为图收缩与一系列高斯消元，从而为稳定子模拟提供了新视角并开辟了张量收缩技术的探索空间。
摘要：
Gottesman-Knill 定理指出，稳定子电路上的计算可以在经典计算机上模拟，传统模拟算法广泛使用比特串上的线性代数。例如，给定一个非自适应稳定子电路，计算给定结果的概率（强模拟）问题已知可对数空间归约为求解布尔变量上的线性方程组，通常通过高斯消元法完成。本文旨在使稳定子电路与布尔线性代数之间的联系更加明确。为此，我们扩展了稳定子 tableau 形式体系，使其包含任意稳定子操作（Clifford 信道）的稳定子 tableau 描述。寻找两个信道复合所对应的 tableau 成为一个线性代数问题。任何稳定子电路都可以重写为一个顶点上带有稳定子 tableau 的图，收缩一条边意味着对信道进行复合，计算电路的结果意味着完全收缩该图。因此，模拟稳定子电路归约为一系列高斯消元。该方法为解释稳定子 tableau 方法的工作提供了新视角（重现了渐近复杂度），并为在稳定子模拟中探索各种张量收缩技术创造了机会。
英文摘要：
Gottesman-Knill theorem states that computations on stabilizer circuits can be simulated on a classical computer, conventional simulation algorithms extensively use linear algebra over bit strings. For instance, given a non-adaptive stabilizer circuit, the problem of computing the probability of a given outcome (strong simulation) is known to be log-space reducible to solving the system of linear equations over Boolean variables, which is commonly done by Gaussian elimination. This note aims to make the connection between stabilizer circuits and Boolean linear algebra even more explicit. To do this, we extend the stabilizer tableau formalism to include stabilizer tableau descriptions of arbitrary stabilizer operations (Clifford channels). Finding the tableau corresponding to the composition of two channels becomes a linear algebra problem. Any stabilizer circuit rewrites to a diagram with stabilizer tableaux on vertices, contracting an edge means to take the composition of channels, to compute the result of the circuit means to fully contract the diagram. Thus, simulating stabilizer circuits reduces to a sequence of Gaussian eliminations. This approach gives a new perspective on explaining the work of stabilizer tableau methods (reproducing the asymptotics) and creates opportunity for exploring various tensor-contraction techniques in stabilizer simulation.
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