To implement fault-tolerant quantum computation with continuous variables, the Gottesman-Kitaev-Preskill (GKP) qubit has been recognized as an important technological element. However, it is still challenging to experimentally generate the GKP qubit with the required squeezing level, 14.8 dB, of the existing fault-tolerant quantum computation. To reduce this requirement, we propose a high-threshold fault-tolerant quantum computation with GKP qubits using topologically protected measurement-based quantum computation with the surface code. By harnessing analog information contained in the GKP qubits, we apply analog quantum error correction to the surface code. Furthermore, we develop a method to prevent the squeezing level from decreasing during the construction of the large-scale cluster states for the topologically protected, measurement-based, quantum computation. We numerically show that the required squeezing level can be relaxed to less than 10 dB, which is within the reach of the current experimental technology. Hence, this work can considerably alleviate this experimental requirement and take a step closer to the realization of large-scale quantum computation.
To implement fault-tolerant quantum computation with continuous variables, the Gottesman-Kitaev-Preskill (GKP) qubit has been recognized as an important technological element. However, it is still challenging to experimentally generate the GKP qubit with the required squeezing level, 14.8 dB, of the existing fault-tolerant quantum computation. To reduce this requirement, we propose a high-threshold fault-tolerant quantum computation with GKP qubits using topologically protected measurement-based quantum computation with the surface code. By harnessing analog information contained in the GKP qubits, we apply analog quantum error correction to the surface code. Furthermore, we develop a method to prevent the squeezing level from decreasing during the construction of the large-scale cluster states for the topologically protected, measurement-based, quantum computation. We numerically show that the required squeezing level can be relaxed to less than 10 dB, which is within the reach of the current experimental technology. Hence, this work can considerably alleviate this experimental requirement and take a step closer to the realization of large-scale quantum computation.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance.In this work, we propose a highly effective use of the surface Gottesman-Kitaev-Preskill (GKP) code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-level qubits.In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum-likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates.Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12 dB, compared to the case where the simple closest-integer decoding is used.Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9 dB under the the assumption that finite squeezing of the GKP states is the dominant noise source.More importantly, we show that a low logical failure rate p L < 10 -7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12 dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p = 0.36%).Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface-code decoder.Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds.We also show that a highly squeezed GKP state of GKP squeezing 12 dB can be experimentally realized by using a dissipative stabilization method, namely, the big-small-big method, with fairly conservative experimental parameters.Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
To implement fault-tolerant quantum computation (FTQC) with continuous variables, continuous variables need to be digitized using an appropriate code such as the Gottesman-Kitaev-Preskill (GKP) qubit. The scheme introduced in [Fukui et al. Phys. Rev. X 8, 021054 (2018)] has reduced the threshold of the squeezing level required for continuous-variable FTQC to less than 10 dB, assuming noise derived from the GKP qubit itself. In this paper, we propose a scheme to improve noise tolerance during the construction of a large-scale cluster state used for FTQC with the GKP qubits. In our scheme, a small-scale cluster state is prepared by employing maximum-likelihood estimation, the entanglement generation via the Bell measurement, and probabilistic reliable measurement. Then, a large-scale cluster state is constructed from the small-scale cluster states via the reliable encoded Bell measurement. In the numerical calculations, we assume errors derived from the two-mode gate and loss in the homodyne measurement in addition to noise from the GKP qubit itself. The results show that the thresholds of a squeezing level are around 8.1, 9.6, and 12.4 dB for loss in the homodyne measurement 0, 5, and 10%, respectively. Hence, this paper provides a way toward continuous-variable FTQC with a feasible squeezing level.
Fault-tolerant quantum error correction is essential for implementing quantum algorithms of significant practical importance. In this work, we propose a highly effective use of the surface-GKP code, i.e., the surface code consisting of bosonic GKP qubits instead of bare two-dimensional qubits. In our proposal, we use error-corrected two-qubit gates between GKP qubits and introduce a maximum likelihood decoding strategy for correcting shift errors in the two-GKP-qubit gates. Our proposed decoding reduces the total CNOT failure rate of the GKP qubits, e.g., from 0.87% to 0.36% at a GKP squeezing of 12dB, compared to the case where the simple closest-integer decoding is used. Then, by concatenating the GKP code with the surface code, we find that the threshold GKP squeezing is given by 9.9dB under the the assumption that finite-squeezing of the GKP states is the dominant noise source. More importantly, we show that a low logical failure rate pL<10−7 can be achieved with moderate hardware requirements, e.g., 291 modes and 97 qubits at a GKP squeezing of 12dB as opposed to 1457 bare qubits for the standard rotated surface code at an equivalent noise level (i.e., p=0.36%). Such a low failure rate of our surface-GKP code is possible through the use of space-time correlated edges in the matching graphs of the surface code decoder. Further, all edge weights in the matching graphs are computed dynamically based on analog information from the GKP error correction using the full history of all syndrome measurement rounds. We also show that a highly-squeezed GKP state of GKP squeezing ≳12dB can be experimentally realized by using a dissipative stabilization method, namely, the Big-small-Big method, with fairly conservative experimental parameters. Lastly, we introduce a three-level ancilla scheme to mitigate ancilla decay errors during a GKP state preparation.
Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction.However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction.In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation.To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources.Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation.To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state.We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.
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 C4/C6 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.
Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction. However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction. In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation. To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources. Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation. To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state. We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.Received 11 January 2021Revised 18 May 2021Accepted 23 July 2021DOI:https://doi.org/10.1103/PRXQuantum.2.030325Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasMeasurement-based quantum computingOptical quantum information processingQuantum information processing with continuous variablesQuantum InformationAtomic, Molecular & Optical