We study mathematical, physical and computational aspects of the stabilizer formalism arising in quantum information and quantum computation. The measurement process of Pauli observables with its algorithm is given. It is shown that to detect genuine entanglement we need a full set of stabilizer generators and the stabilizer witness is coarser than the GHZ (Greenberger–Horne–Zeilinger) witness. We discuss stabilizer codes and construct a stabilizer code from a given linear code. We also discuss quantum error correction, error recovery criteria and syndrome extraction. The symplectic structure of the stabilizer formalism is established and it is shown that any stabilizer code is unitarily equivalent to a trivial code. The structure of graph codes as stabilizer codes is identified by obtaining the respective stabilizer generators. The distance of embeddable stabilizer codes in lattices is obtained. We discuss the Knill–Gottesman theorem, tableau representation and frame representation. The runtime of simulating stabilizer gates is obtained by applying stabilizer matrices. Furthermore, an algorithm for updating global phases is given. Resolution of quantum channels into stabilizer channels is shown. We discuss capacity achieving codes to obtain the capacity of the quantum erasure channel. Finally, we discuss the shadow tomography problem and an algorithm for constructing classical shadow is given.

我们研究量子信息和量子计算中出现的稳定器形式的数学、物理和计算方面。给出了泡利可观测量的测量过程及其算法。结果表明，为了检测真正的纠缠，我们需要一整套稳定器发生器，并且稳定器见证比 GHZ（Greenberger-Horne-Zeilinger）见证更粗糙。我们讨论稳定器代码并从给定的线性代码构造稳定器代码。我们还讨论了量子纠错、错误恢复标准和综合症提取。建立了稳定器形式主义的辛结构，并证明了任意稳定器码都统一等价于平凡码。通过获得各自的稳定器生成器来识别作为稳定器代码的图代码的结构。获得了格子中可嵌入稳定器代码的距离。我们讨论 Knill-Gottesman 定理、画面表示和框架表示。模拟稳定器门的运行时间是通过应用稳定器矩阵获得的。此外，还给出了更新全局相位的算法。显示了量子通道到稳定器通道的分辨率。我们讨论容量实现代码以获得量子擦除通道的容量。最后，我们讨论了阴影层析成像问题，并给出了构造经典阴影的算法。