Quantum Error Correction Code in the Hamiltonian Formulation

TL;DR

This paper introduces a novel Hamiltonian formalism for quantum error correction codes that does not rely on stabilizer formalism, highlighting entanglement, braiding, and integrability aspects, and generalizes to nonadditive codes.

Contribution

It proposes a new Hamiltonian approach for quantum error correction codes beyond stabilizer formalism, linking to braid groups and Yang-Baxter equations, and extends to nonadditive codes.

Findings

Unitary evolution operator acts as a basis transformation to the code.

Code can be described by a braiding operator as a representation of the Artin braid group.

Hamiltonian model is an integrable system satisfying the Yang-Baxter equation.

Abstract

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.