Qubit-oscillator concatenated codes: decoding formalism & code comparison

Abstract

Concatenating bosonic error-correcting codes with qubit codes can substantially boost the error-correcting power of the original qubit codes. It is not clear how to concatenate optimally, given there are several bosonic codes and concatenation schemes to choose from, including the recently discovered GKP-stabilizer codes [Phys. Rev. Lett. 125, 080503 (2020)}] that allow protection of a logical bosonic mode from fluctuations of the mode's conjugate variables. We develop efficient maximum-likelihood decoders for and analyze the performance of three different concatenations of codes taken from the following set: qubit stabilizer codes, analog/Gaussian stabilizer codes, GKP codes, and GKP-stabilizer codes. We benchmark decoder performance against additive Gaussian white noise, corroborating our numerics with analytical calculations. We observe that the concatenation involving GKP-stabilizer codes outperforms the more conventional concatenation of a qubit stabilizer code with a GKP code in some cases. We also propose a GKP-stabilizer code that suppresses fluctuations in both conjugate variables without extra quadrature squeezing, and formulate qudit versions of GKP-stabilizer codes.

20 pages, 8 figures

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• Linear-optical quantum computation with arbitrary error-correcting codes
• Optimal encoding of oscillators into more oscillators
• Clifford operations and homological codes for rotors and oscillators
• Bosonic Pauli+: Efficient Simulation of Concatenated Gottesman-Kitaev-Preskill Codes
• Safeguarding Oscillators and Qudits with Distributed Two-Mode Squeezing
• General Approach to Error Detection of Bosonic Codes via Phase Estimation
• Fault-Tolerant Non-Clifford GKP Gates using Polynomial Phase Gates and On-Demand Noise Biasing