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1. On maximum-likelihood decoding with circuit-level errors

   https://doi.org/10.22331/q-2020-08-06-304  

   Pryadko, Leonid P. (August 2020, Quantum)

   Error probability distribution associated with a given Clifford measurement circuit is described exactly in terms of the circuit error-equivalence group, or the circuit subsystem code previously introduced by Bacon, Flammia, Harrow, and Shi. This gives a prescription for maximum-likelihood decoding with a given measurement circuit. Marginal distributions for subsets of circuit errors are also analyzed; these generate a family of related asymmetric LDPC codes of varying degeneracy. More generally, such a family is associated with any quantum code. Implications for decoding highly-degenerate quantum codes are discussed. 

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2. Stabilizer Entanglement Distillation and Efficient Fault-Tolerant Encoders

   https://doi.org/10.1103/PRXQuantum.6.010339  

   Shi, Yu; Patil, Ashlesha; Guha, Saikat (March 2025, PRX Quantum)

   Entanglement is essential for quantum information processing, but is limited by noise. We address this by developing high-yield entanglement distillation protocols with several advancements. (1) We extend the 2-to-1 recurrence entanglement distillation protocol to higher-rate n-to-(n−1) protocols that can correct any single-qubit errors. These protocols are evaluated through numerical simulations focusing on fidelity and yield. We also outline a method to adapt any classical error-correcting code for entanglement distillation, where the code can correct both bit-flip and phase-flip errors by incorporating Hadamard gates. (2) We propose a constant-depth decoder for stabilizer codes that transforms logical states into physical ones using single-qubit measurements. This decoder is applied to entanglement distillation protocols, reducing circuit depth and enabling protocols derived from high-performance quantum error-correcting codes. We demonstrate this by evaluating the circuit complexity for entanglement distillation protocols based on surface codes and quantum convolutional codes. (3) Our stabilizer entanglement distillation techniques advance quantum computing. We propose a fault-tolerant protocol for constant-depth encoding and decoding of arbitrary states in surface codes, with potential extensions to more general quantum low-density parity-check codes. This protocol is feasible with state-of-the-art reconfigurable atom arrays and surpasses the limits of conventional logarithmic depth encoders. Overall, our study integrates stabilizer formalism, measurement-based quantum computing, and entanglement distillation, advancing both quantum communication and computing. 

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3. Fusion Blossom: Fast MWPM Decoders for QEC

   Wu, Yue; Zhong, Lin (September 2023, IEEE)

   The Minimum-Weight Perfect Matching (MWPM) decoder is widely used in Quantum Error Correction (QEC) decoding. Despite its high accuracy, existing implementations of the MWPM decoder cannot catch up with quantum hardware, e.g., 1 million measurements per second for superconducting qubits. They suffer from a backlog of measurements that grows exponentially and as a result, cannot realize the power of quantum computation. We design and implement a fast MWPM decoder, called Parity Blossom, which reaches a time complexity almost proportional to the number of defect measurements. We further design and implement a parallel version of Parity Blossom called Fusion Blossom. Given a practical circuit-level noise of 0.1%, Fusion Blossom can decode a million measurement rounds per second up to a code distance of 33. Fusion Blossom also supports stream decoding mode that reaches a 0.7 ms decoding latency at code distance 21 regardless of the measurement rounds. 

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4. Belief-Propagation with Quantum Messages for Polar Codes on Classical-Quantum Channels

   Mandal, Avijit; Brandsen; S. Pfister, Henry D. (January 2023, 2023 IEEE International Symposium on Information Theory (ISIT))

   This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results. 

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5. Belief-Propagation with Quantum Messages for Polar Codes on Classical-Quantum Channels, IEEE International Symposium on Information Theory, Taipei, Taiwan, June 25-30, 2023

   https://doi.org/10.1109/ISIT54713.2023.10206723  

   Mandal, Avijit; Brandsen, S; Pfister, Henry D. (June 2023, Abstracts of papers IEEE International Symposium on Information Theory)

   This paper considers the design and decoding of polar codes for general classical-quantum (CQ) channels. It focuses on decoding via belief-propagation with quantum messages (BPQM) and, in particular, the idea of paired-measurement BPQM (PM-BPQM) decoding. Since the PM-BPQM decoder admits a classical density evolution (DE) analysis, one can use DE to design a polar code for any CQ channel and then efficiently compute the trade-off between code rate and error probability. We have also implemented and tested a classical simulation of our PM-BPQM decoder for polar codes. While the decoder can be implemented efficiently on a quantum computer, simulating the decoder on a classical computer actually has exponential complexity. Thus, simulation results for the decoder are somewhat limited and are included primarily to validate our theoretical results. 

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