Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry

Abstract

It has recently been understood that the complete global symmetry of finite group topological gauge theories contains the structure of a higher-group. Here we study the higher-group structure in (3+1)D ${\mathbb{Z}}_2$ gauge theory with an emergent fermion, and point out that pumping chiral $p+ip$ topological states gives rise to a ${\mathbb{Z}}_8$ 0-form symmetry with mixed gravitational anomaly. This ordinary symmetry mixes with the other higher symmetries to form a 3-group structure, which we examine in detail. We then show that in the context of stabilizer quantum codes, one can obtain logical CCZ and CS gates by placing the code on a discretization of $T^3$ (3-torus) and $T^2{⋊}_{C_2}{S}^1$ (2-torus bundle over the circle) respectively, and pumping $p+ip$ states. Our considerations also imply the possibility of a logical $T$ gate by placing the code on ${\mathbb{RP}}^3$ and pumping a $p+ip$ topological state.

arXiv comment34 pages, 6 figures. minor revisions

References in corpus (78)

• Fault-tolerant quantum computation by anyons
• Non-Abelian Anyons and Topological Quantum Computation
• Anyons in an exactly solved model and beyond
• Unpaired Majorana fermions in quantum wires
• Paired states of fermions in two dimensions with breaking of parity and time-reversal symmetries, and the fractional quantum Hall effect
• Topological quantum memory
• Generalized Global Symmetries
• Entanglement Spectrum as a Generalization of Entanglement Entropy: Identification of Topological Order in Non-Abelian Fractional Quantum Hall Effect States
• Quantum Error Correction for Quantum Memories
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Topological phases of fermions in one dimension
• Roads towards fault-tolerant universal quantum computation
• Symmetry Protected Topological phases of Quantum Matter
• The effects of interactions on the topological classification of free fermion systems
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Models for gapped boundaries and domain walls
• A Duality Web in 2+1 Dimensions and Condensed Matter Physics
• General Relationship Between the Entanglement Spectrum and the Edge State Spectrum of Topological Quantum States
• Exotic non-Abelian anyons from conventional fractional quantum Hall states
• Topological Order with a Twist: Ising Anyons from an Abelian Model
• Fermionic Symmetry Protected Topological Phases and Cobordisms
• Generalized Symmetries in Condensed Matter
• Exploring 2-Group Global Symmetries
• Superconductors are topologically ordered
• Level/rank Duality and Chern-Simons-Matter Theories
• On gauging finite subgroups
• Genons, twist defects, and projective non-Abelian braiding statistics
• Topological Nematic States and Non-Abelian Lattice Dislocations
• Fractionalizing Majorana fermions: non-abelian statistics on the edges of abelian quantum Hall states
• On 2-Group Global Symmetries and Their Anomalies
• Superconducting Proximity Effect on the Edge of Fractional Topological Insulators
• Fermionic SPT phases in higher dimensions and bosonization
• Exactly Soluble Model of a 3D Symmetry Protected Topological Phase of Bosons with Surface Topological Order
• Theory of defects in Abelian topological states
• Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology
• Gauging spatial symmetries and the classification of topological crystalline phases
• Fermionic Modular Categories and the 16-fold Way
• Classification of Topological Defects in Abelian Topological States
• Projective non-Abelian Statistics of Dislocation Defects in a Z_N Rotor Model
• Emergent (anomalous) higher symmetries from topological orders and from dynamical electromagnetic field in condensed matter systems
• Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond
• Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups
• The Global Form of Flavor Symmetries and 2-Group Symmetries in 5d SCFTs
• Fusion 2-categories and a state-sum invariant for 4-manifolds
• Topological color code and symmetry-protected topological phases
• Discrete Theta Angles, Symmetries and Anomalies
• Symmetry-Enriched Quantum Spin Liquids in $(3+1)d$
• Many-body topological invariants in fermionic symmetry protected topological phases: Cases of point group symmetries
• Quantum ether: photons and electrons from a rotor model
• Construction and classification of symmetry protected topological phases in interacting fermion systems
• $U(1)\times U(1)⋊Z_2$ Chern-Simons Theory and Z_4 Parafermion Fractional Quantum Hall States
• Gapped boundaries, group cohomology and fault-tolerant logical gates
• Classification of (2+1)D invertible fermionic topological phases with symmetry
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Lattice models that realize ${\mathbb{Z}}_n$-1-symmetry protected topological states for even $n$
• Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism
• Time-reversal and spatial reflection symmetry localization anomalies in (2+1)D topological phases of matter
• Tunneling Topological Vacua via Extended Operators: (Spin-)TQFT Spectra and Boundary Deconfinement in Various Dimensions
• Decorated ${\mathbb{Z}}_2$ Symmetry Defects and Their Time-Reversal Anomalies
• Fault-Tolerant Quantum Gates with Defects in Topological Stabiliser Codes
• Higher-group symmetry in finite gauge theory and stabilizer codes
• Locality-Preserving Logical Operators in Topological Stabiliser Codes
• Lorentz Symmetry Fractionalization and Dualities in (2+1)d
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• Fractional disclination charge and discrete shift in the Hofstadter butterfly
• Higher representations for extended operators
• Anomalies of $(1+1)D$ categorical symmetries
• Quantized charge polarization as a many-body invariant in (2+1)D crystalline topological states and Hofstadter butterflies
• Gravitational anomaly of 3+1 dimensional Z_2 toric code with fermionic charges and fermionic loop self-statistics
• Pin TQFT and Grassmann integral
• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Complete crystalline topological invariants from partial rotations in (2+1)D invertible fermionic states and Hofstadter's butterfly
• Surface Topological Order and a new 't Hooft Anomaly of Interaction Enabled 3+1D Fermion SPTs
• Domain walls in topological phases and the Brauer-Picard ring for $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$
• (3+1)D topological orders with only a ${\mathbb{Z}}_2$-charged particle
• Extracting higher central charge from a single wave function
• A 3-categorical perspective on G-crossed braided categories
• Pumping Chirality in Three Dimensions

Cited by in corpus (11)

• Non-invertible symmetry-protected topological order in a group-based cluster state
• LDPC-cat codes for low-overhead quantum computing in 2D
• Fractionalization of Coset Non-Invertible Symmetry and Exotic Hall Conductance
• Generalized toric codes on twisted tori for quantum error correction
• Higher-Matter and Landau-Ginzburg Theory of Higher-Group Symmetries
• Subsystem Symmetry Fractionalization and Foliated Field Theory
• Non-invertible symmetries in finite-group gauge theory
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate beyond the $n^{1/3}$ Distance Barrier
• Generalized Statistics on Lattices
• Exploring two dimensional ${\mathbb{Z}}_2$ invariant phases with time reversal symmetry and their transitions with topological operations