The stabiliser formalism plays a central role in quantum computing, error correction, and fault-tolerance. Stabiliser states are used to encode computational basis states. Clifford gates are those which can be easily performed fault-tolerantly in the most common error correction schemes. Their mathematical properties are the subject of significant research interest. Conversions between and verifications of different specifications of stabiliser states and Clifford gates are important components of many classical algorithms in quantum information, e.g. for gate synthesis, circuit optimisation, and for simulating quantum circuits. These core functions are also used in the numerical experiments critical to formulating and testing mathematical conjectures on the stabiliser formalism. We develop novel mathematical insights concerning stabiliser states and Clifford gates that significantly clarify their descriptions. We then utilise these to provide ten new fast algorithms which offer asymptotic advantages over any existing implementations. We show how to rapidly verify that a vector is a stabiliser state, and interconvert between its specification as amplitudes, a quadratic form, and a check matrix. These methods are leveraged to rapidly check if a given unitary matrix is a Clifford gate and to interconvert between the matrix of a Clifford gate and its compact specification as a stabiliser tableau. For example, we extract the stabiliser tableau of a Clifford gate matrix with $N^2$ entries in $O(N \log N)$ time. Remarkably, it is not necessary to read all the elements of a Clifford matrix to extract its stabiliser tableau. This is an asymptotic speedup over the best-known method that is superexponential in the number of qubits. We provide example implementations of our algorithms in Python.

翻译：稳定子形式体系在量子计算、纠错与容错中占据核心地位。稳定子态被用于编码计算基态。Clifford门则是在最常见纠错方案中能够以容错方式轻松执行的量子门。它们的数学性质是当前重要研究课题。稳定子态与Clifford门不同规范之间的相互转换及验证，是量子信息领域中许多经典算法（如门综合、电路优化和量子电路模拟）的关键组成部分。这些核心功能也广泛应用于数值实验，对构建和检验稳定子形式体系的数学猜想至关重要。本文提出了关于稳定子态与Clifford门的新数学见解，显著澄清了其描述方式。基于这些见解，我们开发了十个新型快速算法，其渐近复杂度优于现有所有实现方案。我们展示了如何快速验证向量是否为稳定子态，并在振幅表示、二次型表示和校验矩阵表示之间进行高效转换。这些方法被进一步应用于快速检测给定酉矩阵是否为Clifford门，并在Clifford门的矩阵表示与其紧凑的稳定子表示形式之间实现转换。例如，我们仅需$O(N \log N)$时间即可从包含$N^2$个元素的Clifford门矩阵中提取其稳定子表示。值得注意的是，提取稳定子表示无需读取Clifford矩阵的所有元素。这相较于当前已知最佳方法实现了超越指数级的渐近加速。我们在Python中提供了算法的示例实现。