design quantum error correction

Current quantum computing platforms are constrained by noise and limited qubit counts, hindering the realization of scalable systems. This work proposes a unified analytical framework that rapidly predicts the logical error rates of leading quantum error-correcting codes across mainstream hardware and distributed architectures by modeling two key factors: code structure and two-qubit gate overhead. For the first time, this framework analytically reproduces the qualitative trends observed in large-scale simulations and precisely identifies the dominant sources of logical errors—such as circuit volume, routing overhead, or asymmetric noise—across diverse platforms. Experimental validation confirms its cross-platform predictive accuracy and delineates the optimal design regime for distributed quantum error correction, offering critical guidance for the development of scalable distributed quantum computing systems.

This work addresses the compilation optimization of stabilizer codes for fault-tolerant quantum error correction (QEC) on 2×N quantum dot arrays under spin-qubit shuttling operations. We first prove that minimizing the number of shuttling operations is NP-hard. To overcome this, we propose a Shor-style syndrome extraction method enabling asymptotically optimal shuttling compilation for constant-column-weight qLDPC codes. Integrating heuristic circuit compilation, stabilizer synthesis, and realistic physical modeling, we validate the fault-tolerant feasibility of both surface codes and qLDPC codes under moderate noise. Key contributions include: (i) achieving constant—*O*(1)—shuttling overhead independent of code distance, breaking the conventional linear scaling bottleneck; and (ii) demonstrating high logical fidelity (>99.9%) in simulations, significantly enhancing the scalability and practicality of QEC on quantum-dot hardware.

This work addresses the challenge of fault-tolerant compilation on noisy quantum hardware by proposing an optimization method that integrates logical redundancy from quantum error-correcting codes with device connectivity constraints. It formalizes, for the first time, the selection of logically equivalent operations within the framework of the special unitary group and reformulates this problem as a tractable least-squares optimization. By synergistically combining the [[4,2,2]] code, analysis of logical operator equivalences, least-squares optimization, and compressed sensing techniques, the approach directly leverages native physical Hamiltonians to implement target logical operations—bypassing the need for high-overhead SWAP gates. This strategy significantly reduces compilation overhead while adhering to hardware connectivity limitations.

Logical state preparation circuits for CSS codes in fault-tolerant quantum computing are traditionally hand-designed, lacking automated synthesis methods that jointly optimize circuit depth and gate count—especially beyond distance-3 codes. Method: This paper introduces the first SAT-based fully automated synthesis framework for CSS code logical state preparation. It supports arbitrary code distance (d) (removing the conventional (d=3) restriction), jointly optimizes both preparation and verification subcircuits for depth and gate count, and incorporates scalable heuristics and non-deterministic construction strategies. Results: Experiments on distance-3, -5, and -7 CSS codes demonstrate that synthesized circuits achieve provable optimality in both depth and gate count; moreover, logical error rates exhibit exponential suppression with increasing code distance. The framework is open-sourced and integrated into the MQT toolchain.

Fault-tolerant implementation of Trotter circuits for quantum simulation incurs prohibitively high overhead under conventional quantum error correction. Method: This work introduces the first algorithm-specific fault-tolerant framework, departing from generic QEC paradigms. It features: (1) a novel “solve-and-stitch” synthesis algorithm enabling systematic compilation of Clifford Trotter circuits into the [[n,n−2,2]] code family; (2) a scalable, customized fault-tolerant design integrating the [[20,4,2]] hypergraph product code with flag gadgets; and (3) near-optimal circuit depth under realistic assumptions. Results: Experimental validation demonstrates successful execution of a 4-logical-qubit Clifford Trotter circuit encoded in 20 physical qubits. The approach achieves substantially reduced resource overhead—particularly in T-gate count and ancilla requirements—compared to standard fault-tolerant methods. This establishes a new pathway toward efficient, hardware-aware fault tolerance for domain-specific quantum algorithms.

This work addresses the challenge of deploying high-rate quantum error-correcting codes, which are often hindered by hardware constraints such as long-range couplings. On a single trapped-ion quantum computer and without any hardware reconfiguration, the authors demonstrate, for the first time, flexible implementation of nine distinct error-correcting codes—spanning qLDPC, topological, and concatenated families—with markedly different connectivity requirements. Leveraging an optical–metastable–ground (OMG) architecture, the system enables addressable mid-circuit measurement and reset without requiring ion shuttling or dedicated coolant ions. Notably, a qLDPC code encoding four logical qubits into eighteen physical qubits achieves break-even performance, exhibiting a logical error rate nine times lower than comparable superconducting-platform experiments; moreover, certain logical qubits surpass the coherence time of their constituent physical qubits, substantially enhancing resource efficiency.

Designing quantum error-correcting codes entails intricate trade-offs among code structure, hardware constraints, and decoding performance, making it challenging to achieve both efficiency and practicality. This work proposes OmniQEC, an AI-scientist-driven iterative discovery framework that uniquely integrates self-evolving reasoning with a fast-slow collaborative workflow: a fast loop employs low-cost code-level proxies to efficiently screen candidate codes, while a slow loop conducts physically realistic circuit-level simulations for fine-grained evaluation. Orchestrated by a large language model, the framework jointly optimizes code construction, syndrome extraction synthesis, and end-to-end decoder design. Under physical qubit budgets of 98 and 240, the discovered codes outperform canonical Bacon–Bravyi (BB) codes [72,12,6] and [144,12,12], respectively, demonstrating enhanced logical error suppression and hardware compatibility that scale favorably with available resources.

This work systematically examines the structures of classical and quantum error-correcting codes in storage and communication, elucidating their deep connections to mathematical and physical objects such as sphere packings, lattices, combinatorial designs, group theory, and quantum phases of matter. The project pioneers a handbook-style integration of the error-correction knowledge base, employing a taxonomic framework that unifies information theory, algebraic coding theory, and interdisciplinary mapping techniques. Codes are organized structurally according to symbol types, enabling coherent classification and cross-referencing. The resulting resource not only serves as a rigorous and comprehensive reference but also empowers researchers to trace interrelationships among codes and inspire novel discoveries, thereby addressing a critical gap in systematic synthesis and cross-domain linkage within the field.