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

• URL: http://arxiv.org/abs/2310.16982v1
• Date: Wed, 25 Oct 2023 20:33:59 GMT
• Title: Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries
• Authors: Guanyu Zhu, Shehryar Sikander, Elia Portnoy, Andrew W. Cross, and Benjamin J. Brown
• Abstract summary: We study fault-tolerant quantum computing for families of homological quantum low-density parity-check codes defined on 3-manifolds with constant or almost-constant encoding rate. We have developed a generic formalism to compute the triple intersection invariants for 3-manifolds.
• Score: 1.3194391758295114
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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. We have developed a generic formalism to compute the triple intersection invariants for 3-manifolds and also study the scaling of the Betti number and systoles with volume for various 3-manifolds, which translates to the encoding rate and distance. 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.

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