Watch QNumerics 2026 4-3 — Quantum Error Correction: Toric Decoding & QLDPC Codes Video Tutorial

QNumerics Summer School 2026 — Session 4-3 Quantum Error Correction: Syndromes, Toric-Code Decoding, and QLDPC Constructions Instructor: Stefan Krastanov This session develops quantum error correction from the algebra of classical linear codes to modern stabilizer-code constructions. A three-bit repetition code introduces codewords, parity-check matrices, syndromes, and the key fact that a syndrome depends on the error rather than the encoded message. Lookup-table decoding illustrates maximum-likelihood ambiguity, logical failures, and Monte Carlo evaluation of logical versus physical error probability, including pseudo-thresholds. The same language is then transferred to the seven-qubit Steane CSS code, where commuting X- and Z-type stabilizers detect phase and bit flips without revealing the logical state. The toric code supplies a geometric picture: errors form chains, nonzero syndrome checks mark their endpoints, and minimum-weight perfect matching proposes likely corrections. Contractible loops act trivially on the code space, while noncontractible loops implement logical errors and determine code distance. The final section gives a high-level route to quantum LDPC codes through chain complexes, the boundary-of-a-boundary condition, sparse commuting parity checks, product constructions, and compact polynomial representations of circulant matrices. Learning objectives: - Formulate linear codes using codewords, parity-check matrices, and error-dependent syndromes. - Build a lookup-table decoder and evaluate logical error probability with Monte Carlo sampling. - Read CSS stabilizer checks and determine a code's physical qubits, logical qubits, and distance. - Interpret toric-code syndromes as error-chain endpoints and decode them with minimum-weight matching. - Relate commuting X/Z checks to chain complexes and motivate product-based quantum LDPC constructions. Learn more about QNumerics: https://qnumerics.org/2026/ QuantumSavory, the umbrella organization: https://quantumsavory.org/