Ungauging quantum error-correcting codes

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.

arXiv comment32 pages, 8 figures

References in corpus (1)

• Symmetry-protected self-correcting quantum memories

Cited by in corpus (34)

• The Fracton Gauge Principle
• Foliated fracton order from gauging subsystem symmetries
• Long-range entanglement from measuring symmetry-protected topological phases
• Twisted Fracton Models in Three Dimensions
• Spurious topological entanglement entropy from subsystem symmetries
• Pinch Point Singularities of Tensor Spin Liquids
• Twisted foliated fracton phases
• Pauli stabilizer models of twisted quantum doubles
• Foliated fracton order in the checkerboard model
• Pivot Hamiltonians as generators of symmetry and entanglement
• Non-Abelian Hybrid Fracton Orders
• Gauging permutation symmetries as a route to non-Abelian fractons
• Symmetry-protected self-correcting quantum memories
• Compactifying fracton stabilizer models
• Hybrid Fracton Phases: Parent Orders for Liquid and Non-Liquid Quantum Phases
• Entanglement Spectra of Stabilizer Codes: A Window into Gapped Quantum Phases of Matter
• Searching for Fracton Orders via Symmetry Defect Condensation
• Type-II fractons from coupled spin chains and layers
• Classifying local fractal subsystem symmetry protected topological phases
• Efficient color code decoders in $d\ge 2$ dimensions from toric code decoders
• Building models of topological quantum criticality from pivot Hamiltonians
• Hyperbolic Fracton Model, Subsystem Symmetry, and Holography II: The Dual Eight-Vertex Model
• Boson-fermion duality with subsystem symmetry
• Detecting subsystem symmetry protected topological order via entanglement entropy
• Planar p-String Condensation: Chiral Fracton Phases from Fractional Quantum Hall Layers and Beyond
• Distilling Fractons from Layered Subsystem-Symmetry Protected Phases
• Symmetry protected self correcting quantum memory in three space dimensions
• Fractalizing quantum codes
• Measurement-based quantum simulation of Abelian lattice gauge theories
• Dynamical Signatures of Rank-2 $U(1)$ Spin Liquids
• Anomaly inflow for CSS and fractonic lattice models and dualities via cluster state measurement
• Non-Abelian fracton order from gauging a mixture of subsystem and global symmetries
• Magic Boundaries of 3D Color Codes
• Thermal Stability of Dynamical Phase Transitions in Higher Dimensional Stabilizer Codes