Local stabilizer codes in three dimensions without string logical operators

Abstract

We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have string-like logical operators, which make the codes non-self-correcting. We introduce a notion of "logical string segments" to avoid difficulties in defining one dimensional objects in discrete lattices. We prove that every string-like logical operator of our code can be deformed to a disjoint union of short segments, and each segment is in the stabilizer group. The code has surface-like logical operators whose partial implementation has unsatisfied stabilizers along its boundary.

18 pages, 12 figures; clarified intermidiate steps in the proof

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• Fractons on the edge
• Covariant field theory of 3D massive fractons
• Absence of topological order in the $U(1)$ checkerboard toric code
• Resonant dynamics of dipole-conserving Bose-Hubbard model with time-dependent tensor electric fields
• Homomorphism, substructure, and ideal: Elementary but rigorous aspects of renormalization group or hierarchical structure of topological orders
• There and Back Again: A Gauging Nexus between Topological and Fracton Phases
• Layer codes as partially self-correcting quantum memories
• Simulating Topological Order on Quantum Processors
• Unidirectional subsystem symmetry in a hole-doped honeycomb-lattice Ising magnet
• Entanglement renormalization of fractonic anisotropic ${\mathbb{Z}}_N$ Laplacian models
• Universal Freezing Transitions of Dipole-Conserving Chains
• A stabilizer code model with non-invertible symmetries: Strange fractons, confinement, and non-commutative and non-Abelian fusion rules
• Role of generic scale invariance in a Mott transition from a U(1) spin-liquid insulator to a Landau Fermi-liquid metal
• ${\mathbb{Z}}_N$ generalizations of three-dimensional stabilizer codes
• Emergent fracton strings from covariant bi-form gauge field theory
• Non-invertible translation from Lieb-Schultz-Mattis anomaly
• Exponential $U(1)$ Symmetry-Breaking Phase as a Disorder-Free Quantum Glass
• Hollow Lattice Tensor Gauge Theories with Bosonic Matter
• Error Resilience of Fracton Codes and Near Saturation of Code-Capacity Threshold in Three Dimensions
• Immobile and mobile excitations of three-spin interactions on the diamond chain
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains
• Symmetry-enriched topological order and quasifractonic behavior in ${\mathbb{Z}}_N$ stabilizer codes
• Continuum Fractons: Quantization and the Few Body Problem
• Fracton models from product codes
• Planon-modular fracton orders
• Collective dynamics in holographic fractonic solids
• Matrix product state classification of 1D multipole symmetry protected topological phases
• Non-Commutative weak measurements: Entanglement, Symmetry Breaking, and the Role of Readout
• Local symmetries and extensive ground-state degeneracy of a 1D supersymmetric fermionic chain
• Subsystem symmetries, critical Bose surface, and immobile excitations in an extended compass model
• Constrained motions and slow dynamics in one-dimensional bosons with double-well dispersion
• Covariant Fracton Electrodynamics in Six Dimensions
• Anisotropic Spin Ice on a Breathing Pyrochlore Lattice