High-threshold fault-tolerant quantum computation with the Gottesman-Kitaev-Preskill qubit under noise in an optical setup

Abstract

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.

Quantum error correction with the color-Gottesman-Kitaev-Preskill code

“…A simple analysis shows that replacing u 1 with −u 2 contributes nothing to the error correction, as u 1 and u 2 have equal variances. Actually, as noted in Section III, the Steane type error correction scheme with a maximum-likelihood estimation (ME-Steane scheme) [30] gives a better shift correction q (M E) cor = 1 2 [(u 1 + u 2 ) mod √ π] in this special case.…”

“…Before introducing ME-Steane scheme [30], we first simply review the conventional Steane error correction scheme. Our discussion is based on the result after Pauli twirling approximation, which means the noisy GKP state is regarded as a mixed state: du dvP σ (u)P σ (v)e −iup e iv q | ψ ψ|e −iv q e iup .…”

“…One can easily find the peaks of these Gaussian functions in u 1 = η(q out −k √ π) and the maximum point of f (u 1 ) is u 1 = η(q out mod √ π). Therefore the maximum-likelihood estimation of shift error u 1 is q (M E) cor = η(q out mod √ π) [30].…”

Generating Gottesman-Kitaev-Preskill qubit using a cross-Kerr interaction between a squeezed light and Fock states in optics

“…The probability of misidentifying the bit value of the GKP-like qubit is at least 1% [37], which corresponds to that of the GKP qubit with a squeezing level ∼6.2 dB. On the other hand, the threshold of the squeezing level is around 10 dB at least [10,33], which correspond to the probability of misidentifying the bit value ∼0.01%. Since the error probability for the GKP-like qubit is larger than that of the threshold value for FTQC with the GKP qubits, the GKP-like qubit generated by the conventional method is not sufficient to implement FTQC due to the probability of misidentifying the bit value.…”

“…where the squeezing level of generated GKP qubits is close to 10 dB, which is the required squeezing level for FTQC with CVs [10,24,[33][34][35][36]. On the other hand, a large-scale CV cluster state has so far not been demonstrated in such physical setups.…”