[论文解读] Ungauging quantum error-correcting codes

Aleksander Kubica, Beni Yoshida|arXiv (Cornell University)|May 4, 2018

Quantum many-body systems参考文献 1被引用 54

一句话总结

该论文将 Pauli CSS 量子码的 gauging 与 ungauging 过程形式化，应用于子系统码和 3D gauge color code，并展示这些构造如何产生新的对称结构、SPT 相以及 MBQC 模型。

ABSTRACT

We develop the procedures of gauging and ungauging, reveal their operational meaning and propose their generalization in a systematic manner within the framework of quantum error-correcting codes. We demonstrate with an example of the subsystem Bacon-Shor code that the ungauging procedure can result in models with unusual symmetry operators constrained to live on lower-dimensional structures. We apply our formalism to the three-dimensional gauge color code (GCC) and show that its codeword space is equivalent to the Hilbert space of six copies of $\mathbb{Z}_2$ lattice gauge theory with $1$-form symmetries. We find that three different stabilizer Hamiltonians associated with the GCC correspond to distinct thermal symmetry-protected topological (SPT) phases in the presence of the stabilizer symmetries of the GCC. One of the considered Hamiltonians describes the Raussendorf-Bravyi-Harrington model, which is universal for measurement-based quantum computation at non-zero temperature. We also propose a general procedure of creating $D$-dimensional SPT Hamiltonians from $(D+1)$-dimensional CSS stabilizer Hamiltonians by exploiting a relation between gapped domain walls and transversal logical gates. As a result, we find an explicit two-dimensional realization of a non-trivial fracton SPT phase protected by fractal-like symmetries.

研究动机与目标

• Develop a systematic framework for gauing and ungauging CSS stabilizer and subsystem codes.
• Characterize the codeword spaces of subsystem codes via ungauging.
• Apply the framework to the 3D gauge color code to reveal its phase structure and symmetries.
• Demonstrate the emergence of fracton frac-SPT phases from fractal-like symmetries.
• Propose a general method to construct D-dimensional SPT Hamiltonians from (D+1)-dimensional CSS stabilizer codes via gapped domain walls.

提出的方法

• Represent CSS stabilizer codes with chain complexes and boundary maps to connect stabilizers and syndromes.
• Define the ungauging map tilde Gamma as an isomorphism between Z-type symmetric subspaces and emergent X-type symmetric subspaces, with explicit action on Pauli operators (Z(c_Q) -> Z(partial_X c_Q), X(M_X c_X) -> X(c_X)).
• Use the ungauging chain complex (C_Z, C_Q, C_X, C_R) with boundaries partial_Z, partial_X, partial_R to determine final qubits and emergent symmetries.
• Illustrate with examples: ungauging the toric code and the 2D Bacon-Shor code to reveal lower-dimensional rigid symmetry operators.
• Apply the formalism to the 3D gauge color code to map its codeword space to six copies of Z2 lattice gauge theory with 1-form symmetries.
• Show how different GCC stabilizer Hamiltonians correspond to distinct fixed-point (thermal) SPT phases under stabilizer symmetries, including the Raussendorf–Bravyi–Harrington model.

实验结果

研究问题

• RQ1 What is a precise operational definition of gauging and ungauging for Pauli CSS symmetries within CSS subsystem codes?
• RQ2 How does ungauging transform the codeword space of subsystem codes into spaces defined by emergent X-type symmetries?
• RQ3 What is the structure of the GCC codeword space when ungauged, and how do GCC stabilizer Hamiltonians map to SPT phases under 1-form symmetries?
• RQ4 Can fracton codes be used to realize fractal symmetry-protected topological phases via ungauging and gapped domain walls?
• RQ5 Can one systematically construct D-dimensional SPT Hamiltonians from (D+1)-dimensional CSS stabilizer codes using gapped domain walls and transversal logical gates?

主要发现

• Ungauging 2D Bacon-Shor maps its codeword space to a subspace governed by X-type symmetries with unusual, lower-dimensional constrained symmetry operators.
• The GCC codeword space is equivalent to six copies of Z2 lattice gauge theory with 1-form symmetries under Z-type ungauging.
• Different stabilizer Hamiltonians derived from GCC correspond to distinct thermal SPT phases in the presence of GCC stabilizer symmetries.
• One GCC-related Hamiltonian corresponds to the RBH model, which is universal for measurement-based quantum computation at nonzero temperature due to its thermal order under 1-form symmetries.
• A general procedure is proposed to construct D-dimensional SPT Hamiltonians from (D+1)-D CSS stabilizer codes by exploiting a relation between gapped domain walls and transversal logical gates, enabling explicit fractal-like frac-SPT realizations in 2D.