The surface code is a powerful quantum error correcting code that can be defined on a 2-D square lattice of qubits with only nearest neighbor interactions. Syndrome and data qubits form a checkerboard pattern. Information about errors is obtained by repeatedly measuring each syndrome qubit after appropriate interaction with its four nearest neighbor data qubits. Changes in the measurement value indicate the presence of chains of errors in space and time. The standard method of determining operations likely to return the code to its error-free state is to use the minimum weight matching algorithm to connect pairs of measurement changes with chains of corrections such that the minimum total number of corrections is used. Prior work has not taken into account the propagation of errors in space and time by the two-qubit interactions. We show that taking this into account leads to a quadratic improvement of the logical error rate.
The surface code is a powerful quantum error correcting code that can be defined on a 2-D square lattice of qubits with only nearest neighbor interactions. Syndrome and data qubits form a checkerboard pattern. Information about errors is obtained by repeatedly measuring each syndrome qubit after appropriate interaction with its four nearest neighbor data qubits. Changes in the measurement value indicate the presence of chains of errors in space and time. The standard method of determining operations likely to return the code to its error-free state is to use the minimum weight matching algorithm to connect pairs of measurement changes with chains of corrections such that the minimum total number of corrections is used. Prior work has not taken into account the propagation of errors in space and time by the two-qubit interactions. We show that taking this into account leads to a quadratic improvement of the logical error rate.
The surface code is a powerful quantum error correcting code that can be defined on a 2-D square lattice of qubits with only nearest neighbor interactions. Syndrome and data qubits form a checkerboard pattern. Information about errors is obtained by repeatedly measuring each syndrome qubit after appropriate interaction with its four nearest neighbor data qubits. Changes in the measurement value indicate the presence of chains of errors in space and time. The standard method of determining operations likely to return the code to its error-free state is to use the minimum weight matching algorithm to connect pairs of measurement changes with chains of corrections such that the minimum total number of corrections is used. Prior work has not taken into account the propagation of errors in space and time by the two-qubit interactions. We show that taking this into account leads to a quadratic improvement of the logical error rate.
Quantum error correction is essential to achieve fault-tolerant quantum computation in the presence of noisy qubits. Among the most promising approaches to quantum error correction is the surface code, thanks to a scalable two-dimensional architecture, only nearest-neighbor interactions, and a high error threshold. Decoding the surface code, i.e. finding the most likely error chain given a syndrome measurement outcome is a computationally complex task. Traditional decoders rely on classical algorithms, which, especially for larger systems, can be slow and may not always converge to the optimal solution. This thesis presents a novel approach to decoding the surface code using graph neural networks. By mapping the syndrome measurements to a graph and performing graph classification, we find that the graph neural networks can predict the most likely error configuration with high accuracy. Our results show that the GNN-based decoder outperforms the classic minimum weight perfect matching (MWPM) decoder in terms of accuracy. With a phenomenological noise model with depolarizing noise and perfect syndrome measurements, our networks beat MWPM up to code-size 15 across all relevant error rates. Furthermore, the GNN is capable of surpassing MWPM under circuit-level noise up to code size 7. We also show that training the network on repetition code data from a recent experiment [Google Quantum AI, Nature 614, 676 (2023)] produces per-step error rates comparable to those achieved with a matching decoder specifically adapted to the error rates of the physical qubits. This indicates that graph neural network decoders are capable of learning the underlying error distribution on the qubits. Our findings advance the field of quantum error correction and provide a promising new direction for the development of efficient decoding algorithms.
Abstract To date, the surface code has become a promising candidate for quantum error correcting codes because it achieves a high threshold and is composed of only the nearest gate operations and low-weight stabilizers. Here, we have exhibited that the logical failure rate can be enhanced by manipulating the lattice size of surface codes that they can show an enormous improvement in the number of physical qubits for a noise model where dephasing errors dominate over relaxation errors. We estimated the logical error rate in terms of the lattice size and physical error rate. When the physical error rate was high, the parameter estimation method was applied, and when it was low, the most frequently occurring logical error cases were considered. By using the minimum weight perfect matching decoding algorithm, we obtained the optimal lattice size by minimizing the number of qubits to achieve the required failure rates when physical error rates and bias are provided .
The field of quantum computation currently lacks a formal proof of experimental feasibility. Qubits are fragile and sophisticated quantum error correction is required to achieve reliable quantum computation. The surface code is a promising quantum error correction code, requiring only a physically reasonable 2D lattice of qubits with nearest neighbor interactions. However, existing proofs that reliable quantum computation is possible using this code assume the ability to measure four-body operators and, despite making this difficult to realize assumption, require that the error rate of these operator measurements is less than 10(-9), an unphysically low target. High error rates have been proved tolerable only when assuming tunable interactions of strength and error rate independent of distance, which is also unphysical. In this work, given a 2D lattice of qubits with only nearest neighbor two-qubit gates, and single-qubit measurement, initialization, and unitary gates, all of which have error rate p, we prove that arbitrarily reliable quantum computation is possible provided p < 7.4 × 10(-4), a target that many experiments have already achieved. This closes a long-standing open problem, formally proving the experimental feasibility of quantum computation under physically reasonable assumptions.
Large-scale quantum computation will only be achieved if experimentally implementable quantum error correction procedures are devised that can tolerate experimentally achievable error rates. We describe an improved decoding algorithm for the Kitaev surface code, which requires only a two-dimensional square lattice of qubits that can interact with their nearest neighbors, that raises the tolerable quantum gate error rate to over 1%. The precise maximum tolerable error rate depends on the error model, and we calculate values in the range 1.1--1.4% for various physically reasonable models. These values represent a very high threshold error rate calculated in a constrained setting.
Quantum error-correcting codes (QECCs) are necessary for fault-tolerant quantum computation. Surface codes are a class of topological QECCs that have attracted significant attention due to their exceptional error-correcting capabilities and easy implementation. In the decoding process of surface codes, the syndromes are crucial for error correction, however, they are not always correctly measured. Most of the existing decoding algorithms for surface codes need extra measurements to correct syndromes with errors, which implies a potential increase in inference complexity and decoding latency. In this paper, we propose a high-performance list decoding algorithm for surface codes with erroneous syndromes, where syndrome soft information is incorporated in the decoding, allowing qubits and syndrome to be recovered without needing extra measurements. Precisely, we first use belief propagation (BP) decoding for pre-processing with syndrome soft information, followed by ordered statistics decoding (OSD) for post-processing to list and recover both qubits and syndromes. Numerical results demonstrate that our proposed algorithm efficiently recovers erroneous syndromes and significantly improves the decoding performance of surface codes with erroneous syndromes compared to minimum-weight perfect matching (MWPM), BP and original BP-OSD algorithms.