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, Gottesman-Kitaev-Preskill (GKP) qubits have been recognized as an important technological element. However, the analog outcome of GKP qubits, which includes beneficial information to improve the error tolerance, has been wasted, because the GKP qubits have been treated as only discrete variables. In this Letter, we propose a hybrid quantum error correction approach that combines digital information with the analog information of the GKP qubits using a maximum-likelihood method. As an example, we demonstrate that the three-qubit bitflip code can correct double errors, whereas the conventional method based on majority voting on the binary measurement outcome can correct only a single error. As another example, we show that a concatenated code known as Knill's C 4 =C 6 code can achieve the hashing bound for the quantum capacity of the Gaussian quantum channel (GQC). To the best of our knowledge, this approach is the first attempt to draw both digital and analog information to improve quantum error correction performance and achieve the hashing bound for the quantum capacity of the GQC. DOI: 10.1103/PhysRevLett.119.180507 Quantum computation (QC) has a great deal of potential [1,2]. Although small-scale quantum circuits with various qubits have been demonstrated [3,4], a large-scale quantum circuit that requires scalable entangled states is still a significant experimental challenge for most candidates of qubits. In continuous variable (CV) QC, squeezed vacuum (SV) states with the optical setting have shown great potential to generate scalable entangled states because the entanglement is generated by only beam splitter (BS) coupling between two SV states [5]. However, scalable computation with SV states has been shown to be difficult to achieve because of the accumulation of errors during the QC process, even though the states are created with perfect experimental apparatus [6]. Therefore, fault-tolerant (FT) protection from noise is required that uses the quantum error correcting code. Because noise accumulation originates from the "continuous" nature of the CVQC, it can be circumvented by encoding CVs into digitized variables using an appropriate code, such as Gottesman-KitaevPreskill (GKP) code [7], which are referred to as GKP qubits in this Letter. Menicucci showed that CV-FTQC is possible within the framework of measurement-based QC using SV states with GKP qubits [6]. Moreover, GKP qubits keep the advantage of SV states on optical implementation that they can be entangled by only BS coupling. Hence, GKP qubits offer a promising element for the implementation of CV-FTQC.To be practical, the squeezing level required for FTQC should be experimentally achievable. Unfortunately, Menicucci's scheme still requires a 14.8 dB squeezing level to achieve the FT threshold 2 × 10 −2 [8][9][10]. Thus, another twist is necessary to reduce the required squeezing level. It is analog information contained in the GKP qubit that has been overlooked. Th...

To implement fault-tolerant quantum computation with continuous variables, continuous variables need to be digitized using an appropriate code such as the Gottesman-Kitaev-Preskill (GKP) qubit. We have developed a method to alleviate the required squeezing level to realize fault-tolerant quantum computation with the GKP qubit [K. Fukui, A. Tomita, A. Okamoto, and K. Fujii, Phys. Rev. X 8, 021054 (2018)]. In the previous work, the required squeezing level can be reduced to less than 10 dB, assuming a noise derived from only the deviation of the GKP qubit itself. Considering realistic devices such as the two-qubit gate and the homodyne measurement, however, there is some additional noise, which leads to the degradation of the squeezing level. Specifically, the required squeezing level degrades to 15.0 dB, assuming the transmission loss in the homodyne measurement 5 % and the imperfection of the two-qubit gate. In this work, we propose a scheme to improve the required squeezing level, combining the previous schemes with the proposed maximum-likelihood methods and encoded measurements. The numerical calculations show that the required squeezing levels are 8.3, 9.6, and 11.7 dB for the transmission loss in the homodyne measurement 0, 5, and 10 %, respectively, considering the imperfection of the two-qubit gate. Hence, we believe this work will open up a new way to implement continuous-variable fault-tolerant quantum computation with a moderate squeezing level and realistically noisy devices.

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