Quantum computing promises to solve problems in factorization, quantum chemistry, and dynamics simulation faster than classical computers, but it requires fault tolerance, meaning error suppression to a level suitable for algorithms. Traditional surface-code architectures are reliable but inefficient: the ratio of physical to logical qubits is high. This work sits within a broader field of quantum error correction and quantum algorithms: neighboring studies have already demonstrated quantum speedups by factors, silicon chips with 18 qubits, self-healing quantum computers, and iceberg codes where errors boost performance. The bicycle architecture joins this line but bets on modularity and long-range connectivity between qubits, opening the path to low-density parity-check codes.
The foundation is two-dimensional bivariate bicycle codes, which generalize the toric code: qubits and checks are placed on a torus, and each check is additionally connected to two distant qubits via long-range links. For the gross code (144 physical qubits, 12 logical qubits, distance 12) and the two-gross code (288 physical qubits, 12 logical qubits, distance 18), we construct explicit syndrome measurement circuits, logical operations via code deformation (quantum surgery), and injection of T-states from factories. Fault tolerance is verified by simulating circuit-level noise with error probability p, including preparation, measurement, and two-qubit gate errors. To estimate rare logical errors, we use the spectral fitting method, and phenomenological distances are computed via integer programming.
For the gross code at p=10⁻³, the logical idle error is 10⁻⁸·⁸, and for two-gross it is 10⁻²⁰·¹. At p=10⁻⁴, two-gross reaches 10⁻³⁸·³, many orders of magnitude lower than a surface code of comparable distance. Intra-module and inter-module measurements have error rates on the order of 10⁻⁵–10⁻⁷ for gross and 10⁻⁹–10⁻¹⁸ for two-gross. End-to-end estimates for random circuits and transverse-field Ising model simulations show that with 5,000 physical qubits and p=10⁻³, the bicycle architecture performs 10 times more logical operations than the surface code; for the Ising model, 8,138 qubits suffice at p≈7×10⁻⁴, compared to 58–73 thousand for the surface code.
This work shows that low-density parity-check codes can be not only theoretically more efficient but also practically realizable in a modular architecture with long-range links. Unlike neighboring works that often consider isolated components, such as error correction, compilation, or state factories, here we assemble a full stack, from physical connectivity to algorithm compilation and resource estimation. This is the first end-to-end demonstration that modularity and long-range links yield an order-of-magnitude gain in the number of logical qubits without compromising reliability.
Future prospects involve improving inter-module measurements, whose logical errors currently limit overall performance. Possible new adapter designs, decoder optimization, and parallel measurements are on the horizon. Another direction is increasing code distances: the family of bivariate bicycle codes is scalable, and other low-density parity-check codes, up to constant-rate codes, could be used. Finally, magic state factories could be improved by leveraging long-range links for cultivation or distillation.
The architecture will influence the design of quantum processors, especially superconducting ones, where long-range links are already being demonstrated. It also sets new standards for compilers and error correction systems.
Immediate next steps include optimizing inter-module operations and the decoder, searching for syndrome measurement circuits with larger circuit-level distance, and implementing a prototype module on real hardware.
In quantum error correction, the practical implementation of low-density parity-check codes remains an open question: how to ensure long-range connectivity without loss of quality, how to decode quickly and accurately, and how to scale modules. This work directly connects these problems by proposing concrete circuits and estimates, but does not fully solve them, for instance, the circuit-level distance for the gross code is smaller than the code distance, and inter-module measurements require improvement.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
The paper proposes a modular bicycle architecture for quantum computing based on two-dimensional bivariate bicycle codes. For the gross and two-gross codes, it constructs fault-tolerant sets of logical instructions, estimates error probabilities under circuit-level noise, and develops a compilation strategy. End-to-end resource estimates show that, with the same number of physical qubits, the bicycle architecture runs logical circuits an order of magnitude larger than surface-code architectures.
Where its strength is
The strength of the work lies in its full stack: from physical connectivity and error correction to algorithm compilation and resource estimation, giving a holistic picture of the practical feasibility of low-density parity-check codes.
Where it could go next
Worth trying
Why this may matter
If modular quantum computers with low-density parity-check codes can be realized on existing platforms, this could substantially reduce the resources needed for practical quantum algorithms and bring their execution closer.
Nearby works in our archive
The archive is sparse around this work: the closest review is only 0.56, against a usual 0.64. It looks like few have drilled nearby.
The code is named 'gross' (from the English 'gross', meaning 12 dozen) because it encodes exactly 12 logical qubits, the same number as items in a dozen dozen.