Locality-Preserving Logical Operators in Topological Stabiliser Codes
arXiv:1709.00020 · doi:10.1103/PhysRevA.97.012330
Abstract
Locality-preserving logical operators in topological codes are naturally fault-tolerant, since they preserve the correctability of local errors. Using a correspondence between such operators and gapped domain walls, we describe a procedure for finding all locality-preserving logical operators admitted by a large and important class of topological stabiliser codes. In particular, we focus on those equivalent to a stack of a finite number of surface codes of any spatial dimension, where our procedure fully specifies the group of locality-preserving logical operators. We also present examples of how our procedure applies to codes with different boundary conditions, including colour codes and toric codes, as well as more general codes such as abelian quantum double models and codes with fermionic excitations in more than two dimensions.
24 pages, 17 figures, comments welcome; v2 includes various minor improvements; v3 published version
References in corpus (32)
- Fault-tolerant quantum computation by anyons
- Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
- Surface codes: Towards practical large-scale quantum computation
- Quantum Error Correction for Quantum Memories
- Local stabilizer codes in three dimensions without string logical operators
- Restrictions on Transversal Encoded Quantum Gate Sets
- Models for gapped boundaries and domain walls
- Experimental Quantum Computations on a Topologically Encoded Qubit
- Topological Order with a Twist: Ising Anyons from an Abelian Model
- Quantum codes on a lattice with boundary
- Magic state distillation with low overhead
- Quantum Computers, Factoring, and Decoherence
- Fermions, strings, and gauge fields in lattice spin models
- An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation
- Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy
- Classification of topologically protected gates for local stabilizer codes
- Unfolding the color code
- Wigner function negativity and contextuality in quantum computation on rebits
- Poking holes and cutting corners to achieve Clifford gates with the surface code
- Fault-tolerant logical gates in quantum error-correcting codes
- Commuting Pauli Hamiltonians as maps between free modules
- Universal topological phase of 2D stabilizer codes
- Topological color code and symmetry-protected topological phases
- Feasibility of self-correcting quantum memory and thermal stability of topological order
- Topological phases and quantum computation
- Clifford Gates by Code Deformation
- Structure of 2D Topological Stabilizer Codes
- Symmetry protected topological order at nonzero temperature
- Gapped boundaries, group cohomology and fault-tolerant logical gates
- Protected gates for topological quantum field theories
- Tensor Networks with a Twist: Anyon-permuting domain walls and defects in PEPS
- A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)
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- Three-dimensional surface codes: Transversal gates and fault-tolerant architectures
- A fault-tolerant non-Clifford gate for the surface code in two dimensions
- The boundaries and twist defects of the color code and their applications to topological quantum computation
- Codimension-2 defects and higher symmetries in (3+1)D topological phases
- The disjointness of stabilizer codes and limitations on fault-tolerant logical gates
- Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries
- Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes
- Higher-group symmetry in finite gauge theory and stabilizer codes
- Symmetry-protected self-correcting quantum memories
- Cellular automaton decoders for topological quantum codes with noisy measurements and beyond
- Quantum Origami: Transversal Gates for Quantum Computation and Measurement of Topological Order
- Universal Fault-Tolerant Quantum Computing with Stabiliser Codes
- Higher-group symmetry of (3+1)D fermionic gauge theory: logical CCZ, CS, and T gates from higher symmetry
- Fault-tolerant logical gates in holographic stabilizer codes are severely restricted
- Symmetric Finite-Time Preparation of Cluster States via Quantum Pumps
- Lifting topological codes: Three-dimensional subsystem codes from two-dimensional anyon models
- Cross-cap defects and fault-tolerant logical gates in the surface code and the honeycomb Floquet code
- A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes
- Braiding defects in topological stabiliser codes of any dimension cannot be universal
- Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries
- Fermionic defects of topological phases and logical gates
- Magic Boundaries of 3D Color Codes