Protected gates for topological quantum field theories
arXiv:1409.3898 · doi:10.1063/1.4939783
Abstract
We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators --- for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically-local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons; in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.
50 pages, many figures, v3: updated to match published version
References in corpus (16)
- Surface codes: Towards practical large-scale quantum computation
- Local stabilizer codes in three dimensions without string logical operators
- Topological Quantum Distillation
- Lieb-Robinson bounds and the generation of correlations and topological quantum order
- Restrictions on Transversal Encoded Quantum Gate Sets
- Topological Order with a Twist: Ising Anyons from an Abelian Model
- A Family of Non-Abelian Kitaev Models on a Lattice: Topological Confinement and Condensation
- Topological Computation without Braiding
- Fault-tolerant logical gates in quantum error-correcting codes
- Braided fusion categories, gravitational anomalies, and the mathematical framework for topological orders in any dimensions
- Majorana braiding with thermal noise
- An invariant of topologically ordered states under local unitary transformations
- 3-d topological quantum memory with a power-law energy barrier
- Classical Simulation of Quantum Error Correction in a Fibonacci Anyon Code
- Continuous error correction for Ising anyons
- 3-d quantum stabilizer codes with a power law energy barrier
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