Streaming Belief Propagation on Mixed-Alphabet Tanner Graphs for Practical Quantum Memory
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AbstractReliable quantum memory under circuit-level noise requires decoders that can process syndrome information continuously and at a rate comparable to its generation. In practical quantum error correction (QEC), repeated syndrome measurements cause the number of potential error locations to grow rapidly with both code size and time. In this paper, we propose a streaming mixed-alphabet belief propagation (SM-BP) decoder. We construct a space-time Tanner graph across multiple rounds of syndrome extraction with mixed-alphabet error variables, preserving correlations arising from multi-qubit faults. Additionally, we propose an adaptive sliding window procedure that captures long error events across window boundaries and adjusts the decoding in real time. To enhance SM-BP, we introduce a technique of probabilistic error consolidation to mitigate degeneracy effects and short cycles. Our simulations demonstrate high error thresholds of 0.4% to 0.87% and strong error-floor performance for topological code families, including rotated toric, toric color, and twisted XZZX toric codes. These results show that SM-BP provides a practical decoding framework for continuous QEC under circuit-level noise.Featured image: Space-time Tanner graph illustrating different decoding-window schemes. Fixed-size windows can truncate connected error chains, while the adaptive scheme shifts the window boundaries to capture entire connected error clusters.Popular summaryQuantum error correction is essential for protecting quantum information from noise, but it creates a demanding classical processing problem of its own. Error information is generated continuously during quantum computation, and a decoder must identify and correct faults quickly enough to keep up. In this work, we develop a streaming belief-propagation decoder for quantum error correction under realistic circuit-level noise. Our approach represents different physical faults using variables of different sizes, allowing important correlations from single-qubit, two-qubit, and measurement errors to be retained rather than immediately reducing everything to independent binary errors. We organize syndrome information over time in a sparse graphical model, consolidate redundant error descriptions, and use an adaptive sliding window so that error events extending across window boundaries are not artificially separated. The resulting decoder can process syndrome data continuously and efficiently, providing a practical approach to protecting quantum information over long periods of time.► BibTeX data@article{Kuo2026streamingbelief, doi = {10.22331/q-2026-09-10-2207}, url = {https://doi.org/10.22331/q-2026-09-10-2207}, title = {Streaming {B}elief {P}ropagation on {M}ixed-{A}lphabet {T}anner {G}raphs for {P}ractical {Q}uantum {M}emory}, author = {Kuo, Kao-Yueh and Lai, Ching-Yi}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2207}, month = sep, year = {2026} }► References [1] Peter W. Shor. ``Scheme for reducing decoherence in quantum computer memory''. Phys. Rev. A 52, 2493–2496 (1995). https://doi.org/10.1103/PhysRevA.52.R2493 [2] Andrew M. Steane. ``Error correcting codes in quantum theory''. Phys. Rev. Lett. 77, 793 (1996). https://doi.org/10.1103/PhysRevLett.77.793 [3] A. Robert Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Phys. Rev. A 54, 1098 (1996). https://doi.org/10.1103/PhysRevA.54.1098 [4] E. Knill and R. Laflamme. ``Theory of quantum error-correcting codes''. Phys. Rev. A 55, 900–911 (1997). https://doi.org/10.1103/PhysRevA.55.900 [5] Daniel Gottesman. ``Stabilizer codes and quantum error correction''. PhD thesis. California Institute of Technology. CA, USA (1997). [6] David P. DiVincenzo and Peter W. Shor. ``Fault-tolerant error correction with efficient quantum codes''. Phys. Rev. Lett. 77, 3260–3263 (1996). https://doi.org/10.1103/PhysRevLett.77.3260 [7] Daniel Gottesman. ``Theory of fault-tolerant quantum computation''. Phys. Rev. A 57, 127 (1998). https://doi.org/10.1103/PhysRevA.57.127 [8] Dorit Aharonov and Michael Ben-Or. ``Fault-tolerant quantum computation with constant error rate''. SIAM J. Comput. 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AbstractReliable quantum memory under circuit-level noise requires decoders that can process syndrome information continuously and at a rate comparable to its generation. In practical quantum error correction (QEC), repeated syndrome measurements cause the number of potential error locations to grow rapidly with both code size and time. In this paper, we propose a streaming mixed-alphabet belief propagation (SM-BP) decoder. We construct a space-time Tanner graph across multiple rounds of syndrome extraction with mixed-alphabet error variables, preserving correlations arising from multi-qubit faults. Additionally, we propose an adaptive sliding window procedure that captures long error events across window boundaries and adjusts the decoding in real time. To enhance SM-BP, we introduce a technique of probabilistic error consolidation to mitigate degeneracy effects and short cycles. Our simulations demonstrate high error thresholds of 0.4% to 0.87% and strong error-floor performance for topological code families, including rotated toric, toric color, and twisted XZZX toric codes. These results show that SM-BP provides a practical decoding framework for continuous QEC under circuit-level noise.Featured image: Space-time Tanner graph illustrating different decoding-window schemes. Fixed-size windows can truncate connected error chains, while the adaptive scheme shifts the window boundaries to capture entire connected error clusters.Popular summaryQuantum error correction is essential for protecting quantum information from noise, but it creates a demanding classical processing problem of its own. Error information is generated continuously during quantum computation, and a decoder must identify and correct faults quickly enough to keep up. In this work, we develop a streaming belief-propagation decoder for quantum error correction under realistic circuit-level noise. Our approach represents different physical faults using variables of different sizes, allowing important correlations from single-qubit, two-qubit, and measurement errors to be retained rather than immediately reducing everything to independent binary errors. We organize syndrome information over time in a sparse graphical model, consolidate redundant error descriptions, and use an adaptive sliding window so that error events extending across window boundaries are not artificially separated. The resulting decoder can process syndrome data continuously and efficiently, providing a practical approach to protecting quantum information over long periods of time.► BibTeX data@article{Kuo2026streamingbelief, doi = {10.22331/q-2026-09-10-2207}, url = {https://doi.org/10.22331/q-2026-09-10-2207}, title = {Streaming {B}elief {P}ropagation on {M}ixed-{A}lphabet {T}anner {G}raphs for {P}ractical {Q}uantum {M}emory}, author = {Kuo, Kao-Yueh and Lai, Ching-Yi}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2207}, month = sep, year = {2026} }► References [1] Peter W. Shor. ``Scheme for reducing decoherence in quantum computer memory''. Phys. Rev. A 52, 2493–2496 (1995). https://doi.org/10.1103/PhysRevA.52.R2493 [2] Andrew M. Steane. ``Error correcting codes in quantum theory''. Phys. Rev. Lett. 77, 793 (1996). https://doi.org/10.1103/PhysRevLett.77.793 [3] A. Robert Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Phys. Rev. A 54, 1098 (1996). https://doi.org/10.1103/PhysRevA.54.1098 [4] E. Knill and R. Laflamme. ``Theory of quantum error-correcting codes''. Phys. Rev. A 55, 900–911 (1997). https://doi.org/10.1103/PhysRevA.55.900 [5] Daniel Gottesman. ``Stabilizer codes and quantum error correction''. PhD thesis. California Institute of Technology. CA, USA (1997). [6] David P. DiVincenzo and Peter W. Shor. ``Fault-tolerant error correction with efficient quantum codes''. Phys. Rev. Lett. 77, 3260–3263 (1996). https://doi.org/10.1103/PhysRevLett.77.3260 [7] Daniel Gottesman. ``Theory of fault-tolerant quantum computation''. Phys. Rev. A 57, 127 (1998). https://doi.org/10.1103/PhysRevA.57.127 [8] Dorit Aharonov and Michael Ben-Or. ``Fault-tolerant quantum computation with constant error rate''. SIAM J. Comput. (2008). https://doi.org/10.1137/S0097539799359385 [9] Sergey Bravyi and Alexei Kitaev. ``Universal quantum computation with ideal Clifford gates and noisy ancillas''. Phys. Rev. A 71, 022316 (2005). https://doi.org/10.1103/PhysRevA.71.022316 [10] A. Yu. Kitaev. ``Fault-tolerant quantum computation by anyons''. Ann. Phys. 303, 2–30 (2003). https://doi.org/10.1016/S0003-4916(02)00018-0 [11] Hector Bombin and Miguel Angel Martin-Delgado. ``Topological quantum distillation''. Phys. Rev. Lett. 97, 180501 (2006). https://doi.org/10.1103/PhysRevLett.97.180501 [12] Héctor Bombin and Miguel A Martin-Delgado. ``Optimal resources for topological two-dimensional stabilizer codes: Comparative study''. Phys. Rev. A 76, 012305 (2007). https://doi.org/10.1103/PhysRevA.76.012305 [13] Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill. ``Topological quantum memory''. J. Math. 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