Concatenate codes, save qubits
arXiv:2402.09606 · doi:10.1038/s41534-025-01035-8
Abstract
The essential requirement for fault-tolerant quantum computation (FTQC) is the total protocol design to achieve a fair balance of all the critical factors relevant to its practical realization, such as the space overhead, the threshold, and the modularity. A major obstacle in realizing FTQC with conventional protocols, such as those based on the surface code and the concatenated Steane code, has been the space overhead, i.e., the required number of physical qubits per logical qubit. Protocols based on high-rate quantum low-density parity-check (LDPC) codes gather considerable attention as a way to reduce the space overhead, but problematically, the existing fault-tolerant protocols for such quantum LDPC codes sacrifice the other factors. Here we construct a new fault-tolerant protocol to meet these requirements simultaneously based on more recent progress on the techniques for concatenated codes rather than quantum LDPC codes, achieving a constant space overhead, a high threshold, and flexibility in modular architecture designs. In particular, under a physical error rate of , our protocol reduces the space overhead to achieve the logical CNOT error rates and by more than and , respectively, compared to the protocol for the surface code. Furthermore, our protocol achieves the threshold of under a conventional circuit-level error model, substantially outperforming that of the surface code. The use of concatenated codes also naturally introduces abstraction layers essential for the modularity of FTQC architectures. These results indicate that the code-concatenation approach opens a way to significantly save qubits in realizing FTQC while fulfilling the other essential requirements for the practical protocol design.
9+22 pages, 5+18 figures. The source codes for the simulation of the underlying codes (except for the surface code) and the quantum Hamming codes are available at https://github.com/sy3104/concatenated_code_threshold
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