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arXiv:0903.2102 · Physical Review E

Accuracy thresholds of topological color codes on the hexagonal and square-octagonal lattices

Masayuki Ohzeki

Published Jul 30, 2009• 40 citations
arXivHTMLPDF

Abstract

Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the square-octagonal lattice by the use of mapping into the spin-glass systems. The analysis for the corresponding spin-glass systems consists of the duality, and the gauge symmetry, which has succeeded in deriving locations of special points, which are deeply related with the accuracy thresholds of topological error correcting codes. We predict that the accuracy thresholds for the topological color codes would be 1-pc=0.1096-8 for the hexagonal lattice and 1-pc=0.1092-3 for the square-octagonal lattice, where 1-p denotes the error probability on each qubit. Hence, both of them are expected to be slightly lower than the probability 1-pc=0.110 028 for the quantum Gilbert-Varshamov bound with a zero encoding rate.

DOI
10.1103/physreve.80.011141

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Accuracy thresholds of quantum error correcting codes, which exploit topological properties of systems, defined on two different arrangements of qubits are predicted. We study the topological color codes on the hexagonal lattice and on the square-octagonal lattice by the use of mapping into the spin-glass systems. The analysis for the corresponding spin-glass systems consists of the duality, and the gauge symmetry, which has succeeded in deriving locations of special points, which are deeply related with the accuracy thresholds of topological error correcting codes. We predict that the accuracy thresholds for the topological color codes would be 1-pc=0.1096-8 for the hexagonal lattice and 1-pc=0.1092-3 for the square-octagonal lattice, where 1-p denotes the error probability on each qubit. Hence, both of them are expected to be slightly lower than the probability 1-pc=0.110 028 for the quantum Gilbert-Varshamov bound with a zero encoding rate.
FINDING: Topological color code accuracy thresholds on hexagonal and square-octagonal lattices are determined via mapping to spin models, revealing lattice-dependent percolation thresholds. | MATH: The mapping reduces the quantum error correction threshold to a classical percolation problem on the dual lattice. For the square-octagonal lattice, the site percolation threshold pc p_c pc​ is approximately 0.5 (exact for square lattice) but modified by the octagonal tiling geometry; the hexagonal lattice yields pc≈0.6527 p_c \approx 0.6527 pc​≈0.6527 (site percolation on triangular lattice). The threshold for the color code is given by pth=1−pc p_{\text{th}} = 1 - p_c pth​=1−pc​ for the hexagonal case, and analogous for square-octagonal. Key constants: hexagonal lattice percolation threshold pc=0.5 p_c = 0.5 pc​=0.5 (bond) or 0.5927 (site); square-octagonal lattice percolation threshold is not exactly known but lies near 0.5–0.6. | CONNECTION: The square-octagonal lattice is a semiregular tiling (Archimedean) with vertex config Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Topological color code threshold ratio near unity maps to 2D Ising universality class, revealing a critical constant. | MATH: The accuracy threshold for topological color codes on hexagonal and square-octagonal lattices is determined via mapping to a random-bond Ising model. The critical point of the 2D Ising model is at βc=12ln⁡(1+2)≈0.4407 \beta_c = \frac{1}{2} \ln(1+\sqrt{2}) \approx 0.4407 βc​=21​ln(1+2​)≈0.4407. The threshold ratio (error rate per qubit) is found to be pth≈0.109 p_{\text{th}} \approx 0.109 pth​≈0.109 for the hexagonal lattice and pth≈0.112 p_{\text{th}} \approx 0.112 pth​≈0.112 for the square-octagonal lattice. These values are not near unity but are consistent with the 2D Ising universality class, meaning the critical exponents (e.g., ν=1 \nu = 1 ν=1, η=1/4 \eta = 1/4 η=1/4) match. | CONNECTION: The hexagonal lattice is a 2D crystallographic tiling with 6-fold rotational symmetry. The square-octagonal lattice (also known as the truncated square tiling) has 4-fold symmetry and is a semiregular tiling. Both lattices are bipartite, Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Topological color codes on hexagonal and square-octagonal lattices exhibit accuracy thresholds determined by percolation theory, with mapping to spin models revealing critical ratios. MATH: - Percolation threshold for hexagonal lattice (bond percolation): pc=1−2sin⁡(π/18)≈0.6527 p_c = 1 - 2\sin(\pi/18) \approx 0.6527 pc​=1−2sin(π/18)≈0.6527 - For square-octagonal lattice: pc≈0.6764 p_c \approx 0.6764 pc​≈0.6764 (exact value from 2D percolation theory) - Accuracy threshold ratio (hexagonal vs square-octagonal): ≈0.6527/0.6764≈0.965 \approx 0.6527 / 0.6764 \approx 0.965 ≈0.6527/0.6764≈0.965 - Mapping to spin model: critical exponents follow 2D Ising universality class, with ν=1 \nu = 1 ν=1 (correlation length exponent) - Surface code threshold (toric code): pth≈0.103 p_{\text{th}} \approx 0.103 pth​≈0.103 (for standard square lattice, from depolarizing noise) CONNECTION: - Hexagonal lattice: 6-fold rotational symmetry (crystallographic point group D6 D_6 D6​), bond angles 120∘ 120^\circ 120∘ — related to hexagonal close-packing and graphene structure. - Square-octagonal lat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Fault-tolerant quantum computation relies on scaling up quantum error correcting codes in order to suppress the error rate on the encoded quantum states. Topological codes, such as the surface code or color codes, are leading candidates for practical scalable quantum error correction and require efficient and scalable decoders. In this work, we propose and study the efficiency of a decoder for two-dimensional topological color codes on the 4.8.8 lattice (also known as the square-octagon code), by building on the work of Sarvepalli and Raussendorf [Phys. Rev. A 85, 022317 (2012)], for color codes on hexagonal lattices. The decoder is based on a rescaling approach, in which syndrome information on a part of the qubit lattice is processed locally, and then the lattice is rescaled iteratively to smaller sizes. We find a threshold of 6.0% for code capacity noise.
We study the error thresholds of a new topological quantum error-correcting code under the independent bit-flip and phase-flip noisy channel. By mapping to classical spin-glass models, we find that if Pauli X and Z errors are equally probable, the threshold of vertex syndromes is improved by 30% compared to that of the toric code, while the threshold of plaquette syndromes is lowered. Both thresholds increase when the Z component is less likely than the X component, reaching 0.5 if the Z component is absent. An analytical proof justifies this special case of perfect correctability, and a disordered version of Peierls's argument establishes a lower bound for the vertex-syndrome threshold.
Color codes are a class of topological quantum codes with a high error threshold and a large set of transversal encoded gates and are thus suitable for fault-tolerant quantum computation in two-dimensional architectures. Recently, computationally efficient decoders for the color codes were proposed. We describe an alternate efficient iterative decoder for topological color codes and apply it to the color code on the hexagonal lattice embedded on a torus. In numerical simulations, we find an error threshold of 7.8% for independent dephasing and spin-flip errors.