Tracking quantum error correction

Abstract

To implement fault-tolerant quantum computation with continuous variables, the Gottesman--Kitaev--Preskill (GKP) qubit has been recognized as an important technological element. We have proposed a method to reduce the required squeezing level to realize large-scale quantum computation with the GKP qubit [Phys. Rev. X 8, 021054 (2018)], harnessing the virtue of analog information in the GKP qubits. In the present work, to reduce the number of qubits required for large-scale quantum computation, we propose the tracking quantum error correction, where the logical-qubit-level quantum error correction is partially substituted by the single-qubit-level quantum error correction. In the proposed method, the analog quantum error correction is utilized to make the performances of the single-qubit-level quantum error correction almost identical to those of the logical-qubit-level quantum error correction in a practical noise level. The numerical results show that the proposed tracking quantum error correction reduces the number of qubits during a quantum error-correction process by the reduction rate ${2(n\ensuremath{-}1){4}^{l\ensuremath{-}1}\ensuremath{-}n+1}/(2n\ifmmode\times\else\texttimes\fi{}{4}^{l\ensuremath{-}1})$ for $n$-cycles of the quantum error-correction process using Knill's ${C}_{4}/{C}_{6}$ code with the concatenation level $l$. Hence, the proposed tracking quantum error correction has great advantage in reducing the required number of physical qubits, and will open a new way to expoloit the advantages of the GKP qubits in practical quantum computation.