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.
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.
Two-dimensional color codes are a promising candidate for fault-tolerant quantum computing, as they have high encoding rates, transversal implementation of logical Clifford gates, and resource-efficient magic state preparation schemes. However, decoding color codes presents a significant challenge due to their structure, where elementary errors violate three checks instead of just two (a key feature in surface code decoding), and the complexity of extracting syndrome is greater. We introduce an efficient color-code decoder that tackles these issues by combining two matching decoders for each color, generalized to handle circuit-level noise by employing detector error models. We provide comprehensive analyses of the decoder, covering its threshold and sub-threshold scaling both for bit-flip noise with ideal measurements and for circuit-level noise. Our simulations reveal that this decoding strategy nearly reaches the best possible scaling of logical failure ( pfail∼pd/2 ) for both noise models, where p is the noise strength, in the regime of interest for fault-tolerant quantum computing. While its noise thresholds are comparable with other matching-based decoders for color codes ( 8.2% for bit-flip noise and 0.46% for circuit-level noise), the scaling of logical failure rates below threshold significantly outperforms the best matching-based decoders.
Practical large-scale quantum computation requires both efficient error correction and robust implementation of logical operations. Three-dimensional (3D) color codes are a promising candidate for fault-tolerant quantum computation due to their transversal non-Clifford gates, but efficient decoding remains challenging. In this work, we extend previous decoders for two-dimensional color codes [1], which are based on the restriction of the decoding problem to a subset of the qubit lattice, to three dimensions. Including boundaries of 3D color codes, we demonstrate that the 3D restriction decoder achieves optimal scaling of the logical error rate and a threshold value of 1.55(6)% for code-capacity bit- and phase-flip noise, which is almost a factor of two higher than previously reported for this family of codes [2, 3]. We furthermore present qCodePlot3D, a Python package for visualizing 2D and 3D color codes, error configurations, and decoding paths, which supports the development and analysis of such decoders. These advancements contribute to making 3D color codes a more practical option for exploring fault-tolerant quantum computation.
Fault-tolerant quantum computing will require error rates far below those achievable with physical qubits. Quantum error correction (QEC) bridges this gap, but depends on decoders being simultaneously fast, accurate, and scalable. This combination of requirements remains unmet by a machine-learning decoder, nor by any decoder for promising resource-efficient codes such as the color code. Here we introduce AlphaQubit 2, a neural-network decoder that achieves near-optimal logical error rates for both surface and color codes at scale under realistic noise. For the color code, it is orders of magnitude faster than other high-accuracy decoders. We demonstrate real-time decoding faster than 1μs per cycle on commercial accelerators: for the surface code to distance 11, with better accuracy than leading real-time decoders; and the first real-time decoding of the color code to distance 9. These results support the practical application of a wider class of promising QEC codes, and establish a credible path towards high-accuracy, real-time neural decoding at the scales required for fault-tolerant quantum computation.
Color codes are a promising topological code for fault-tolerant quantum computing. Insufficient research on color codes has delayed their practical application. In this work, we address several key issues to facilitate practical fault-tolerant quantum computing based on color codes. First, by introducing decoding graphs with error-rate-related weights, we obtain the threshold of 0.47 % of the 6.6.6 triangular color code under the standard circuit-level noise model, narrowing the gap to that of the surface code. Second, our work first investigates the circuit-level decoding of color code lattice surgery, then gives an efficient decoding algorithm, which is crucial to perform logical operations in a quantum computer with two-dimensional architectures. Last, a state injection protocol of the triangular color code is proposed, reducing the output magic state error rate in one round of 15 to 1 distillation by two orders of magnitude compared to a previous rough protocol. We also prove that our protocol offers the lowest logical error rates for state injection among all possible codes. Published by the American Physical Society 2024
Efficient decoding is essential for the practical realization of fault-tolerant quantum computers. We study the computational complexity of minimum-weight decoding for topological quantum codes. For surface codes under the depolarizing channel, we consider Minimum-Weight decoding, which seeks a minimum-weight Pauli error consistent with both the X- and Z-syndromes. For color codes under independent X- and Z-error models, we consider Separate Minimum-Weight decoding.
Assuming P=NP, we establish polynomial additive inapproximability gaps for these problems. Specifically, for the toric code and the 4.8.8 color code on the torus, there exists a constant c>0 such that no polynomial-time algorithm can always produce a solution whose weight is within cN1/14 of the optimum, where N is the number of qubits, unless P=NP. For the planar surface code, we obtain an Ω(N1/18) gap. Our inapproximability results use Håstad's hardness of approximation for MAX-3SAT.
Our reduction develops a general, modular framework for embedding logical constraints into coupled primal--dual join problems on a lattice. A key ingredient is a localization argument that controls unintended interactions between different parts of the construction.
Quantum error correction is an essential ingredient for reliable quantum computation for theoretically provable quantum speedup. Topological color codes, one of the quantum error correction codes, have an advantage against the surface codes in that all Clifford gates can be implemented transversally. However, the hardness of decoding makes the color codes not suitable as the best candidate for experimentally feasible implementation of quantum error correction. Here we propose an Ising model formulation that enables highly accurate decoding of the color codes. In this formulation, we map stabilizer operators to classical spin variables to represent an error satisfying the syndrome. Then we construct an Ising Hamiltonian that counts the number of errors and formulate the decoding problem as an energy minimization problem of an Ising Hamiltonian, which is solved by simulated annealing. In numerical simulations on the (4.8.8) lattice, we find an error threshold of 10.36(5)% for bit-flip noise model, 18.47(5)% for depolarizing noise model, and 2.90(4)% for phenomenological noise model (bit-flip error is located on each of data and measurement qubits), all of which are higher than the thresholds of existing efficient decoding algorithms. Furthermore, we verify that the achieved logical error rates are almost optimal in the sense that they are almost the same as those obtained by exact optimizations by CPLEX with smaller decoding time in many cases. Since the decoding process has been a bottleneck for performance analysis, the proposed decoding method is useful for further exploration of the possibility of the topological color codes. Published by the American Physical Society 2024