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.

40 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 (8)

• Restrictions on Transversal Encoded Quantum Gate Sets
• High-threshold and low-overhead fault-tolerant quantum memory
• Topological Computation without Braiding
• Universal Non-Invertible Symmetries
• Exact Topological Quantum Order in D=3 and Beyond: Branyons and Brane-Net Condensates
• Universal transversal gates with color codes - a simplified approach
• Locality-Preserving Logical Operators in Topological Stabiliser Codes
• Magic of quantum hypergraph states

Cited by in corpus (3)

• Low-overhead fault-tolerant quantum computation by gauging logical operators
• Parallel Logical Measurements via Quantum Code Surgery
• Universal fault tolerant quantum computation in 2D without getting tied in knots