Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes

Abstract

Braiding defects in topological stabiliser codes has been widely studied as a promising approach to fault-tolerant quantum computing. Here, we explore the potential and limitations of such schemes in codes of all spatial dimensions. We prove that a universal gate set for quantum computing cannot be realised by supplementing locality-preserving logical operators with defect braiding, even in more than two dimensions. However, notwithstanding this no-go theorem, we demonstrate that higher dimensional defect-braiding schemes have the potential to play an important role in realising fault-tolerant quantum computing. Specifically, we present an approach to implement the full Clifford group via braiding in any code possessing twist defects on which a fermion can condense. We explore three such examples in higher dimensional codes, specifically: in self-dual surface codes; the three dimensional Levin-Wen fermion mode; and the checkerboard model. Finally, we show how our no-go theorems can be circumvented to provide a universal scheme in three-dimensional surface codes without magic state distillation. Specifically, our scheme employs adaptive implementation of logical operators conditional on logical measurement outcomes to lift a combination of locality-preserving and braiding logical operators to universality.

arXiv comment22 pages + appendices, 25 figures; v2 improvements to presentation; v3 published version

References in corpus (58)

• Fault-tolerant quantum computation by anyons
• Anyons in an exactly solved model and beyond
• Surface codes: Towards practical large-scale quantum computation
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Quantum Error Correction for Quantum Memories
• Fault-tolerant quantum computation with high threshold in two dimensions
• Local stabilizer codes in three dimensions without string logical operators
• Topological Quantum Distillation
• Topological fault-tolerance in cluster state quantum computation
• Roads towards fault-tolerant universal quantum computation
• Surface code quantum computing by lattice surgery
• Fracton Topological Order, Generalized Lattice Gauge Theory and Duality
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Quantum Glassiness
• High threshold universal quantum computation on the surface code
• Restrictions on Transversal Encoded Quantum Gate Sets
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Genons, twist defects, and projective non-Abelian braiding statistics
• Magic State Distillation: Not as Costly as You Think
• Fermions, strings, and gauge fields in lattice spin models
• Topological Computation without Braiding
• Braiding statistics of loop excitations in three dimensions
• Theory of defects in Abelian topological states
• Single-shot fault-tolerant quantum error correction
• Topological phases with generalized global symmetries
• Classification of topologically protected gates for local stabilizer codes
• Unfolding the color code
• Classification of Topological Defects in Abelian Topological States
• Universal transversal gates with color codes - a simplified approach
• Poking holes and cutting corners to achieve Clifford gates with the surface code
• Quantum Measurements and Gates by Code Deformation
• Fault-tolerant logical gates in quantum error-correcting codes
• Fractional excitations in foliated fracton phases
• Commuting Pauli Hamiltonians as maps between free modules
• The surface code with a twist
• Topological color code and symmetry-protected topological phases
• Three-dimensional surface codes: Transversal gates and fault-tolerant architectures
• A fault-tolerant non-Clifford gate for the surface code in two dimensions
• Clifford Gates by Code Deformation
• Cheshire charge in (3+1)-D topological phases
• Symmetry protected topological order at nonzero temperature
• Gapped boundaries, group cohomology and fault-tolerant logical gates
• 2-D color code quantum computation
• The boundaries and twist defects of the color code and their applications to topological quantum computation
• Foliated fracton order in the checkerboard model
• Protected gates for topological quantum field theories
• Sorting topological stabilizer models in three dimensions
• Universal Quantum Computation with Gapped Boundaries
• Stacked codes: universal fault-tolerant quantum computation in a two-dimensional layout
• Locality-Preserving Logical Operators in Topological Stabiliser Codes
• Changing topology by topological defects in three-dimensional topologically ordered phases
• Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes
• Generalized Cluster States Based on Finite Groups
• 2D quantum computation with 3D topological codes
• Universal fault-tolerant quantum computation with Bacon-Shor codes
• An Interferometry-Free Protocol for Demonstrating Topological Order
• Transversal gates and error propagation in 3D topological codes
• A Hierarchy of Anyon Models Realised by Twists in Stacked Surface Codes

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• Higher-group symmetry in finite gauge theory and stabilizer codes
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• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• A Compact Fermion to Qubit Mapping Part 2: Alternative Lattice Geometries
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• Braiding defects in topological stabiliser codes of any dimension cannot be universal
• Topological quantum compilation of metaplectic anyons based on the genetic optimized algorithms
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