Gapped boundaries, group cohomology and fault-tolerant logical gates

Abstract

This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the $d$-dimensional quantum double model by using $d$-cocycles functions ($d\ge 2$). We point out that the system supports $m$-dimensional excitations ($m<d$), which we shall call fluctuating charges, that are superpositions of point-like electric charges characterized by $m$-dimensional bosonic SPT wavefunctions. There exist gapped boundaries where electric charges or magnetic fluxes may not condense by themselves, but may condense only when accompanied by fluctuating charges. Magnetic fluxes and codimension-$2$ fluctuating charges exhibit non-trivial multi-excitation braiding statistics, involving more than two excitations. The statistical angle can be computed by taking slant products of underlying cocycle functions sequentially. We find that excitations that may condense into a gapped boundary can be characterized by trivial multi-excitation braiding statistics, generalizing the notion of the Lagrangian subgroup. As an application, we construct fault-tolerantly implementable logical gates for the $d$-dimensional quantum double model by using $d$-cocycle functions. Namely, corresponding logical gates belong to the $d$th level of the Clifford hierarchy, but are outside of the $(d-1)$th level, if cocycle functions have non-trivial sequences of slant products.

29 pages, 26 figures, single column

References in corpus (14)

• Generalized Global Symmetries
• Local stabilizer codes in three dimensions without string logical operators
• Topological Quantum Distillation
• Restrictions on Transversal Encoded Quantum Gate Sets
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Condensate induced transitions between topologically ordered phases
• Topological boundary conditions in abelian Chern-Simons theory
• Unfolding the color code
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Universal transversal gates with color codes - a simplified approach
• Twisted Gauge Theory Model of Topological Phases in Three Dimensions
• Fault-tolerant logical gates in quantum error-correcting codes
• Generalized ADE Classification of Gapped Domain Walls
• Generalized Color Codes Supporting Non-Abelian Anyons

Cited by in corpus (63)

• Gapless topological phases and symmetry-enriched quantum criticality
• Hierarchy of topological order from finite-depth unitaries, measurement and feedforward
• Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter
• Topological Defect Networks for Fractons of all Types
• Cheshire charge in (3+1)-D topological phases
• Symmetry protected topological order at nonzero temperature
• Logical blocks for fault-tolerant topological quantum computation
• 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
• Pivot Hamiltonians as generators of symmetry and entanglement
• Non-Abelian Hybrid Fracton Orders
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries
• Symmetry-protected sign problem and magic in quantum phases of matter
• Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes
• Locality-Preserving Logical Operators in Topological Stabiliser Codes
• Higher-group symmetry in finite gauge theory and stabilizer codes
• Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states
• Topological Quantum Field Theory for Abelian Topological Phases and Loop Braiding Statistics in $(3+1)$-Dimensions
• From trivial to topological paramagnets: The case of ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_2^3$ symmetries in two dimensions
• Searching for Fracton Orders via Symmetry Defect Condensation
• Topological Order and Memory Time in Marginally Self-Correcting Quantum Memory
• Latent Computational Complexity of Symmetry-Protected Topological Order with Fractional Symmetry
• Quantum spin systems for measurement-based quantum computation
• Quantum computation from dynamic automorphism codes
• Building models of topological quantum criticality from pivot Hamiltonians
• Quantum Origami: Transversal Gates for Quantum Computation and Measurement of Topological Order
• Kennedy-Tasaki transformation and non-invertible symmetry in lattice models beyond one dimension
• Tensor Networks with a Twist: Anyon-permuting domain walls and defects in PEPS
• Non-invertible symmetry-protected topological order in a group-based cluster state
• Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases
• Universal quantum computing using $({\mathbb{Z}}_d{)}^3$ symmetry-protected topologically ordered states
• Symmetry-enriched topological order from partially gauging symmetry-protected topologically ordered states assisted by measurements
• Computing data for Levin-Wen with defects
• Robustness of Topological Order in the Toric Code with Open Boundaries
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Fusing Binary Interface Defects in Topological Phases: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ case
• Approximate symmetries and quantum error correction
• Multipole and fracton topological order via gauging foliated SPT phases
• Boundary states of Three Dimensional Topological Order and the Deconfined Quantum Critical Point
• Domain walls in topological phases and the Brauer-Picard ring for $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$
• 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
• Hierarchies of resources for measurement-based quantum computation
• Measurement-based quantum simulation of Abelian lattice gauge theories
• String operators for Cheshire strings in topological phases
• $Z_3$ and $(\times Z_3{)}^3$ symmetry protected topological paramagnets
• No Strings Attached: Boundaries and Defects in the Cubic Code
• Computing Defects Associated to Bounded Domain Wall Structures: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ Case
• Protocols for Creating Anyons and Defects via Gauging
• Entanglement Entropy of Topological Orders with Boundaries
• Low-overhead fault-tolerant quantum computation by gauging logical operators
• Gauging Defects in Quantum Spin Systems
• Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries
• Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate beyond the $n^{1/3}$ Distance Barrier
• Magic Boundaries of 3D Color Codes
• Topological Phase Transitions Induced by Varying Topology and Boundaries in the Toric Code
• Fermionic defects of topological phases and logical gates
• Solvable models for 2+1D quantum critical points: Loop soups of 1+1D conformal field theories
• Two-dimensional topological paramagnets protected by ${\mathbb{Z}}_3$ symmetry: Properties of the boundary Hamiltonian
• Universal fault tolerant quantum computation in 2D without getting tied in knots
• Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge