Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries

Abstract

We study parallel fault-tolerant quantum computing for families of homological quantum low-density parity-check (LDPC) codes defined on 3-manifolds with constant or almost-constant encoding rate. We derive generic formula for a transversal $T$ gate of color codes on general 3-manifolds, which acts as collective non-Clifford logical CCZ gates on any triplet of logical qubits with their logical-$X$ membranes having a ${\mathbb{Z}}_2$ triple intersection at a single point. The triple intersection number is a topological invariant, which also arises in the path integral of the emergent higher symmetry operator in a topological quantum field theory: the ${\mathbb{Z}}_2^3$ gauge theory. Moreover, the transversal $S$ gate of the color code corresponds to a higher-form symmetry supported on a codimension-1 submanifold, giving rise to exponentially many addressable and parallelizable logical CZ gates. A construction of constant-depth circuits of the above logical gates via cup product cohomology operation is also presented for three copies of identical toric codes on arbitrary 3-manifolds. We have developed a generic formalism to compute the triple intersection invariants for 3-manifolds. We further develop three types of LDPC codes supporting such logical gates: (1) A quasi-hyperbolic code from the product of 2D hyperbolic surface and a circle, with almost-constant rate $k/n=O(1/\log (n))$ and $O(\log (n))$ distance; (2) A homological fibre bundle code with $O(1/{\log }^{\frac{1}{2}}(n))$ rate and $O({\log }^{\frac{1}{2}}(n))$ distance; (3) A specific family of 3D hyperbolic codes: the Torelli mapping torus code, constructed from mapping tori of a pseudo-Anosov element in the Torelli subgroup, which has constant rate while the distance scaling is currently unknown. We then show a generic constant-overhead scheme for applying a parallelizable universal gate set with the aid of logical-$X$ measurements.

arXiv comment40 pages, 32 figures. In the updated version v2, we have simplified the TQFT derivation of the logical gates via an operator-valued cochain formalism in Sec. III, which also gives rise to the explicit construction of constant-depth circuits corresponding to logical CCZ and CZ gates in three copies of identical toric codes defined on arbitrary 3-manifolds

References in corpus (42)

• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• String-net condensation: A physical mechanism for topological phases
• Restrictions on Transversal Encoded Quantum Gate Sets
• High-threshold and low-overhead fault-tolerant quantum memory
• Generalized Symmetries in Condensed Matter
• On 2-Group Global Symmetries and Their Anomalies
• Topological Computation without Braiding
• Fermionic SPT phases in higher dimensions and bosonization
• Quantum Hypergraph States
• Universal Non-Invertible Symmetries
• Criteria for virtual fibering
• Topological phases with generalized global symmetries
• Classification of topologically protected gates for local stabilizer codes
• Unfolding the color code
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Balanced Product Quantum Codes
• Universal transversal gates with color codes - a simplified approach
• Quantum computation with Turaev-Viro codes
• Low-overhead fault-tolerant quantum computing using long-range connectivity
• Triangular color codes on trivalent graphs with flag qubits
• Topological color code and symmetry-protected topological phases
• Hyperbolic and Semi-Hyperbolic Surface Codes for Quantum Storage
• Three-dimensional surface codes: Transversal gates and fault-tolerant architectures
• Symmetry-Enriched Quantum Spin Liquids in $(3+1)d$
• Gapped boundaries, group cohomology and fault-tolerant logical gates
• Codimension-2 defects and higher symmetries in (3+1)D topological phases
• Time-Efficient Constant-Space-Overhead Fault-Tolerant Quantum Computation
• Quantum error-correcting codes and 4-dimensional arithmetic hyperbolic manifolds
• Topological Order, Quantum Codes and Quantum Computation on Fractal Geometries
• Transversal Clifford gates on folded surface codes
• Locality-Preserving Logical Operators in Topological Stabiliser Codes
• Higher-group symmetry in finite gauge theory and stabilizer codes
• Higher cup products on hypercubic lattices: application to lattice models of topological phases
• Partitioning qubits in hypergraph product codes to implement logical gates
• Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes
• Fold-Transversal Clifford Gates for Quantum Codes
• Pseudo-Anosov stretch factors and homology of mapping tori
• Quantum Origami: Transversal Gates for Quantum Computation and Measurement of Topological Order
• Fermion decoration construction of symmetry protected trivial orders for fermion systems with any symmetries $G_f$ and in any dimensions
• Magic of quantum hypergraph states
• Low overhead Clifford gates from joint measurements in surface, color, and hyperbolic codes
• Quantum Pin Codes

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