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Quantum Information and Computation

Golden codes: quantum LDPC codes from regular tessellations of hyperbolic 4-manifolds

Vivien Londe, Anthony Leverrier

Published May 1, 2019• 13 citations
Publisher

Abstract

We adapt a construction of Guth and Lubotzky \cite{GL14} to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like n^{0.1} where n is the number of physical qubits. Similarly as in Ref.~\cite{GL14}, our homological code family stems from hyperbolic 4-manifolds equipped with tessellations. The main novelty of this work is that we consider a regular tessellation consisting of hypercubes. We exploit this strong local structure to design and analyze an efficient decoding algorithm.

DOI
10.26421/qic19.5-6-1

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We adapt a construction of Guth and Lubotzky \cite{GL14} to obtain a family of quantum LDPC codes with non-vanishing rate and minimum distance scaling like n^{0.1} where n is the number of physical qubits. Similarly as in Ref.~\cite{GL14}, our homological code family stems from hyperbolic 4-manifolds equipped with tessellations. The main novelty of this work is that we consider a regular tessellation consisting of hypercubes. We exploit this strong local structure to design and analyze an efficient decoding algorithm.
We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, distance scaling as$n^\epsilon forforfor\epsilon > 0andefficientdecodingschemes.Thecodefamilyisbasedontessellationsofclosed,four−dimensional,hyperbolicmanifolds,asfirstsuggestedbyGuthandLubotzky.ThemaincontributionofthisworkistheconstructionofsuitablemanifoldsviafinitepresentationsofCoxetergroups,theirlinearrepresentationsoverGaloisfieldsandtopologicalcoverings.Weestablishalowerboundontheencodingrateand efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rateandefficientdecodingschemes.Thecodefamilyisbasedontessellationsofclosed,four−dimensional,hyperbolicmanifolds,asfirstsuggestedbyGuthandLubotzky.ThemaincontributionofthisworkistheconstructionofsuitablemanifoldsviafinitepresentationsofCoxetergroups,theirlinearrepresentationsoverGaloisfieldsandtopologicalcoverings.Weestablishalowerboundontheencodingratek/n$of$13/72 = 0.180\ldots $and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime.
Recent advances in quantum hardware relax the strict geometric-locality constraints traditionally imposed on quantum error-correcting codes, motivating interest in high-rate quantum low-density parity-check (qLDPC) codes. At the same time, color codes provide a particularly rich setting for fault-tolerant quantum computation, underlying protocols such as single-shot error correction and self-correcting quantum computation. Hyperbolic color codes provide a class of high-rate qLDPC codes that also retain the structural features of color codes relevant to fault-tolerant quantum computation. However, previous constructions of hyperbolic color codes have achieved at most logarithmic code distance. In this work, we construct hyperbolic color codes with both constant rate and polynomial distance by building on arithmetic hyperbolic manifolds that support polynomial-distance hyperbolic toric codes. Our construction applies in arbitrary dimension D≥4D\geq 4D≥4. In even dimensions, the resulting type-D/2D/2D/2 color codes have constant encoding rate and polynomial distance, while in other cases the number of logical qubits and the code distance both exhibit polynomial scaling. We further derive explicit exponents for polynomial lower bounds as functions of the dimension and code type. These results establish a family of hyperbolic color codes simultaneously achieving constant rate and polynomial distance and providing a testbed to explore fault-tolerant quantum computation protocols that combine high-rate quantum codes with the structural advantages of color codes.
We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, distance scaling as nϵ for ϵ>0 and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate k/n of 13/72=0.180… and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime.
We construct toric codes on various high-dimensional manifolds. Assuming a conjecture in geometry we find families of quantum CSS stabilizer codes on NNN qubits with logarithmic weight stabilizers and distance N1−εN^{1-ε}N1−ε for any ε>0ε>0ε>0. The conjecture is that there is a constant C>0C>0C>0 such that for any nnn-dimensional torus Tn=Rn/Λ{\mathbb T}^n={\mathbb R}^n/ΛTn=Rn/Λ, where ΛΛΛ is a lattice, the least volume unoriented n/2n/2n/2-dimensional surface (using the Euclidean metric) representing nontrivial homology has volume at least CnC^nCn times the volume of the least volume n/2n/2n/2-dimensional hyperplane representing nontrivial homology; in fact, it would suffice to have this result for ΛΛΛ an integral lattice with the surface restricted to faces of a cubulation by unit hypercubes. The main technical result is an estimate of Rankin invariants\cite{rankin} for certain random lattices, showing that in a certain sense they are optimal. Additionally, we construct codes with square-root distance, logarithmic weight stabilizers, and inverse polylogarithmic soundness factor (considered as quantum locally testable codes\cite{qltc}). We also provide an short, alternative proof that the shortest vector in the exterior power of a lattice may be non-split\cite{coulangeon}.