哈密顿公式中的量子纠错码
Quantum Error Correction Code in the Hamiltonian Formulation
摘要：
文献中的量子纠错码的哈密顿模型往往是借助其稳定子形式构造的。但是已经有许多已知的非加性码的例子，它们超出了使用稳定器形式主义的标准量子纠错理论。本文提出了量子纠错码的另一种不涉及稳定子形式的哈密顿形式，并通过研究Shor九比特码及其推广给出了解释.在该哈密顿公式中，在特定时刻的幺正演化算子是从乘积基到量子纠错码的幺正基变换矩阵。该基变换矩阵作为一个纠缠量子算符，将一个分离态变换为纠缠态，从而明确地揭示了量子纠错码的纠缠特性。此外，由于它形成了Artin辫子群的幺正表示，所以量子纠错码可以用辫子算子来描述。此外，由于幺正演化算符是量子杨-巴克斯特方程的解，相应的哈密顿模型可以解释为杨-巴克斯特理论中的可积模型。另一方面，我们推广了Shor九量子比特码，并提出了一个称为量子纠错码的主题，使用Greenberger-Horne-Zeilinger状态产生新的非加性码和信道自适应码。
英文摘要：
The Hamiltonian model of quantum error correction code in the literature is often constructed with the help of its stabilizer formalism. But there have been many known examples of nonadditive codes which are beyond the standard quantum error correction theory using the stabilizer formalism. In this paper, we suggest the other type of Hamiltonian formalism for quantum error correction code without involving the stabilizer formalism, and explain it by studying the Shor nine-qubit code and its generalization. In this Hamiltonian formulation, the unitary evolution operator at a specific time is a unitary basis transformation matrix from the product basis to the quantum error correction code. This basis transformation matrix acts as an entangling quantum operator transforming a separate state to an entangled one, and hence the entanglement nature of the quantum error correction code can be explicitly shown up. Furthermore, as it forms a unitary representation of the Artin braid group, the quantum error correction code can be described by a braiding operator. Moreover, as the unitary evolution operator is a solution of the quantum Yang--Baxter equation, the corresponding Hamiltonian model can be explained as an integrable model in the Yang--Baxter theory. On the other hand, we generalize the Shor nine-qubit code and articulate a topic called quantum error correction codes using Greenberger-Horne-Zeilinger states to yield new nonadditive codes and channel-adapted codes.
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