Pauli topological subsystem codes from Abelian anyon theories

Abstract

We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories--this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the ${\mathbb{Z}}_4^{(1)}$ anyon theory with degenerate braiding relations and the chiral semion theory--both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory containing a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.

arXiv comment67 + 35 pages, single column format, v2 published version

References in corpus (70)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Surface codes: Towards practical large-scale quantum computation
• Quantum Teleportation is a Universal Computational Primitive
• Topological quantum memory
• String-net condensation: A physical mechanism for topological phases
• Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order
• Fractons
• Models for gapped boundaries and domain walls
• A fault-tolerant one-way quantum computer
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Stabilizer Formalism for Operator Quantum Error Correction
• Quantum tensor product structures are observable-induced
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• Higher Gauging and Non-invertible Condensation Defects
• Universal fault-tolerant quantum computation with only transversal gates and error correction
• Topological boundary conditions in abelian Chern-Simons theory
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Hilbert Space Fragmentation and Commutant Algebras
• Ultrahigh Error Threshold for Surface Codes with Biased Noise
• Fermions, strings, and gauge fields in lattice spin models
• Fault-tolerant conversion between the Steane and Reed-Muller quantum codes
• Dynamically Generated Logical Qubits
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Poking holes and cutting corners to achieve Clifford gates with the surface code
• Topological Subsystem Codes
• Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups
• Fault-tolerant logical gates in quantum error-correcting codes
• Topological invariants for gauge theories and symmetry-protected topological phases
• Fault-tolerant error correction with the gauge color code
• Universal topological phase of 2D stabilizer codes
• Statistical mechanical models for quantum codes with correlated noise
• Higher central charges and topological boundaries in 2+1-dimensional TQFTs
• A Fault-Tolerant Honeycomb Memory
• A fault-tolerant non-Clifford gate for the surface code in two dimensions
• Structure of 2D Topological Stabilizer Codes
• In and around Abelian anyon models
• Floquet codes without parent subsystem codes
• Standard Form of Qudit Stabilizer Groups
• Nontrivial Quantum Cellular Automata in Higher Dimensions
• Fracton phases via exotic higher-form symmetry-breaking
• Pauli stabilizer models of twisted quantum doubles
• From gauge to higher gauge models of topological phases
• Code Deformation and Lattice Surgery Are Gauge Fixing
• Topology, criticality, and dynamically generated qubits in a stochastic measurement-only Kitaev model
• Microscopic definitions of anyon data
• Boundaries for the Honeycomb Code
• Adiabatic paths of Hamiltonians, symmetries of topological order, and automorphism codes
• Subsystem codes with spatially local generators
• Thermal Hall conductance and a relative topological invariant of gapped two-dimensional systems
• Performance of planar Floquet codes with Majorana-based qubits
• Constructions and Noise Threshold of Topological Subsystem Codes
• Algebraic and information-theoretic conditions for operator quantum error-correction
• Clifford Quantum Cellular Automata: Trivial group in 2D and Witt group in 3D
• Minimal nondegenerate extensions
• Generalized Kitaev Models and Slave Genons
• Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead
• Benchmarking the Planar Honeycomb Code
• Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes
• Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond
• Logical operator tradeoff for local quantum codes
• Quantum spin systems on infinite lattices
• From Symmetries to Commutant Algebras in Standard Hamiltonians
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices
• Interacting Anyonic Fermions in a Two-Body Color Code Model
• Gravitational anomaly of 3+1 dimensional Z_2 toric code with fermionic charges and fermionic loop self-statistics
• On the stability of charges in infinite quantum spin systems
• Disentangling modular Walker-Wang models via fermionic invertible boundaries

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• Intrinsic Mixed-state Topological Order
• A Noisy Approach to Intrinsically Mixed-State Topological Order
• Towards a classification of mixed-state topological orders in two dimensions
• Quantum computation from dynamic automorphism codes
• Engineering 3D Floquet codes by rewinding
• Topological error correcting processes from fixed-point path integrals
• Symmetries and anomalies of Kitaev spin-$S$ models: Identifying symmetry-enforced exotic quantum matter
• Multipole groups and fracton phenomena on arbitrary crystalline lattices
• Lifting topological codes: Three-dimensional subsystem codes from two-dimensional anyon models
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime
• Homological invariants of Pauli stabilizer codes
• Intrinsic mixed state topological order in a stabilizer system under stochastic decoherence: Strong-to-weak spontaneous symmetry breaking from percolation point of view
• Generalizations of Kitaev's honeycomb model from braided fusion categories
• The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems
• Single-Shot Quantum Error Correction in Intertwined Toric Codes
• Finite-temperature quantum topological order in three dimensions
• Low-overhead non-Clifford fault-tolerant circuits for all non-chiral abelian topological phases
• Chiral spin liquid in a ${\mathbb{Z}}_3$ Kitaev model
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