Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate beyond the $n^{1/3}$ Distance Barrier

Abstract

We construct a family of infinitely many new candidate non-Abelian self-correcting topological quantum memories in $D\ge 5+1$ spacetime dimensions without particle excitations using local commuting non-Pauli stabilizer lattice models and field theories of ${\mathbb{Z}}_2^3$ higher-form gauge fields with nontrivial topological action. We call such non-Pauli stabilizer models magic stabilizer codes. The family of topological orders have Abelian electric excitations and non-Abelian magnetic excitations that obey Ising-like fusion rules and non-Abelian braiding, including Borromean ring type braiding which is a signature of non-Abelian topological order, generalizing the dihedral group ${\mathbb{D}}_8$ gauge theory in (2+1)D. The simplest example includes a new non-Abelian self-correcting memory in (5+1)D with Abelian loop excitations and non-Abelian membrane excitations. We prove the self-correction property and the thermal stability, and devise a probabilistic local cellular-automaton decoder. We also construct fault-tolerant non-Clifford CCZ logical gate using constant depth circuit from higher cup products in the 5D non-Abelian code. The use of higher-cup products and non-Pauli stabilizers allows us to get an $O(n^{2/5})$ distance overcoming the $O(n^{1/3})$ distance barrier in conventional topological stabilizer codes, including the 3D color code and the 6D self-correcting color code.

28 pages, 9 figures. Title revised. Close to the published version

References in corpus (61)

• Fault-tolerant quantum computation by anyons
• Topological quantum memory
• Generalized Global Symmetries
• String-net condensation: A physical mechanism for topological phases
• Symmetry protected topological orders and the group cohomology of their symmetry group
• Topological Quantum Distillation
• Roads towards fault-tolerant universal quantum computation
• Braiding statistics approach to symmetry-protected topological phases
• Symmetry Fractionalization, Defects, and Gauging of Topological Phases
• Noise tailoring for scalable quantum computation via randomized compiling
• Coupling a QFT to a TQFT and Duality
• Restrictions on Transversal Encoded Quantum Gate Sets
• High-threshold and low-overhead fault-tolerant quantum memory
• Five-Brane Effective Action In M-Theory
• A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes
• On 2-Group Global Symmetries and Their Anomalies
• Quantum memories at finite temperature
• Comments on One-Form Global Symmetries and Their Gauging in 3d and 4d
• D-brane Charges in Five-brane backgrounds
• Autocorrelations and Thermal Fragility of Anyonic Loops in Topologically Quantum Ordered Systems
• Non-Abelian Topological Order and Anyons on a Trapped-Ion Processor
• AdS/CFT Correspondence And Topological Field Theory
• Foliated fracton order from gauging subsystem symmetries
• Topological Order at Non-zero Temperature
• Single-shot fault-tolerant quantum error correction
• Finite Group Modular Data
• Topological phases with generalized global symmetries
• Classification of topologically protected gates for local stabilizer codes
• Unfolding the color code
• State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter
• Symmetry fractionalization and twist defects
• Balanced Product Quantum Codes
• Universal transversal gates with color codes - a simplified approach
• Theory of Twist Liquids: Gauging an Anyonic Symmetry
• Quantum computation with Turaev-Viro codes
• Braiding Statistics and Link Invariants of Bosonic/Fermionic Topological Quantum Matter in 2+1 and 3+1 dimensions
• A classification of 3+1D bosonic topological orders (I): the case when point-like excitations are all bosons
• Low-overhead fault-tolerant quantum computing using long-range connectivity
• Topological color code and symmetry-protected topological phases
• Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage
• Feasibility of self-correcting quantum memory and thermal stability of topological order
• Fiber Bundle Codes: Breaking the $N^{1/2}\mathrm{polylog}(N)$ Barrier for Quantum LDPC Codes
• Three-dimensional surface codes: Transversal gates and fault-tolerant architectures
• Symmetry-Enriched Quantum Spin Liquids in $(3+1)d$
• A fault-tolerant non-Clifford gate for the surface code in two dimensions
• Gapped boundaries, group cohomology and fault-tolerant logical gates
• 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 $n$
• Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds
• Higher-group symmetry in finite gauge theory and stabilizer codes
• Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies
• Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes
• Fusion 2-categories with no line operators are grouplike
• Higher-group symmetry of (3+1)D fermionic ${\mathbb{Z}}_2$ gauge theory: logical CCZ, CS, and T gates from higher symmetry
• Instantaneous braids and Dehn twists in topologically ordered states
• Quantum error correction with fractal topological codes
• Topological Orders in (4+1)-Dimensions
• Experiments with the 4D Surface Code on a QCCD Quantum Computer
• Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits
• Confinement in partially broken abelian Chern-Simons theories
• Non-invertible symmetries in finite-group gauge theory