Quantum spherical codes

Abstract: As with classical computers, quantum computers require error-correction schemes to reliably perform useful large-scale calculations. The nature and frequency of errors depends on the quantum computing platform, and although there is a large literature on qubit-based coding, these are often not directly applicable to devices that store information in bosonic systems such as photonic resonators. Here, we introduce a framework for constructing quantum codes defined on spheres by recasting such codes as quantum analogues of the classical spherical codes. We apply this framework to bosonic coding, and we obtain multimode extensions of the cat codes that can outperform previous constructions but require a similar type of overhead. Our polytope-based cat codes consist of sets of points with large separation that, at the same time, form averaging sets known as spherical designs. We also recast concatenations of Calderbank–Shor–Steane codes with cat codes as quantum spherical codes, which establishes a method to autonomously protect against dephasing noise.

Conclusion: We introduce a framework for constructing quantum analogues of the classical spherical codes, encapsulating several physically relevant quantum coding schemes for bosonic, spin and molecular systems. We apply our framework to obtain multimode coherent-state codes based on polytopes, CSS codes and classical codes. These QSCs outperform previous cat-code constructions29^,30^,36 both in terms of code parameters and a numerical performance comparison of qudit encodings. We show how passive protection of several instances of these QSCs can be realized in microwave cavities.There are many other ways of constructing spherical codes, for example, as group-orbit codes69^,70^,71, as spherical embeddings of association schemes32, through computer searches72^,73 and many others31^,32^,74^,75, as well as ways of constructing spherical designs76^,77^,78. As such, we anticipate that this work will pave the way for many new, well-protected and experimentally feasible logical qubits.

文章信息：https://doi.org/10.1038/s41567-024-02496-y