With the significance of continuous-variable quantum computing increasing thanks to the achievements of lightbased quantum hardware, making it available to learner audiences outside physics has been an important yet seldomtackled challenge. Similarly, the rising focus on fault-tolerant quantum computing has shed light on quantum error correction schemes, turning it into the locus of attention for industry and academia alike. In this paper, we explore the widely adopted framework of quantum error correction based on continuous variable systems and suggest a guide on building a self-contained learning session targeting the famous Gottesman-Kitaev-Preskill (GKP) code through its geometric intuition.
With the significance of continuous-variable quantum computing increasing thanks to the achievements of lightbased quantum hardware, making it available to learner audiences outside physics has been an important yet seldomtackled challenge. Similarly, the rising focus on fault-tolerant quantum computing has shed light on quantum error correction schemes, turning it into the locus of attention for industry and academia alike. In this paper, we explore the widely adopted framework of quantum error correction based on continuous variable systems and suggest a guide on building a self-contained learning session targeting the famous Gottesman-Kitaev-Preskill (GKP) code through its geometric intuition.
Gottesman, Kitaev and Preskill have proposed a scheme to encode a qubit in a harmonic oscillator, which is called the GKP code. It is designed to be resistant to small shift errors contained in momentum and position quadratures. Thus there's some intrinsic fault tolerance of the GKP code. In this thesis,we propose a method to utilize all the information contained in the continuous shifts, not just simply map a GKP-encoded qubit to a normal qubit. This method enables us to do maximum-likelihood decisions and thus increase fault-tolerance of the GKP code. This thesis shows that the continuous nature of the GKP code is quite useful for quantum error correction.
The Gottesman-Kitaev-Preskill (gkp) quantum error-correcting code attracts much attention in continuous variable (CV) quantum computation and CV quantum communication due to the simplicity of error-correcting routines and the high tolerance against Gaussian errors. Since the gkp code state should be regarded as a limit of physically meaningful approximate ones, various approximations have been developed until today, but explicit relations among them are still unclear. In this paper, we rigorously prove the equivalence of these approximate gkp codes with an explicit correspondence of the parameters. We also propose a standard form of the approximate code states in the position representation, which enables us to derive closed-form expressions for the Wigner function, inner products, and the average photon number in terms of the theta functions. Our results serve as fundamental tools for further analyses of fault-tolerant quantum computation and channel coding using approximate gkp codes.
In order to solve practical problems on a quantum computer, it is necessary to use fault-tolerant quantum error correction schemes to overcome noise: the errors arising from imperfections in physical components. The Gottesman-Kitaev-Preskill (GKP) code aims at achieving this in a hardware efficient manner by encoding finite-dimensional logical subspaces in the Hilbert space of one or more continuous variable modes. In near term implementations, however, it is not feasible to eliminate errors entirely, so it is natural to also employ alternative error mitigation techniques together with error correction. In this work, we study a quantum error mitigation method known as probabilistic error cancellation in the context of the GKP code. We compare Steane-type and teleportation-based GKP error correction, and calculate the sampling overheads associated with the technique for square and hexagonal GKP codes. We employ the stabilizer subsystem decomposition for GKP codes to obtain an effective logical channel for noisy operations that is used to express the ideal one of a target logical unitary. We consider noise from finite squeezing on the data and the two ancilla modes, needed for error correction, and examine the relationship between the sampling overhead and the noise for different decoding methods. Our results are calculated for single- and two-qubit GKP Clifford gates and one round of error correction, and show the nontrivial combination of error correction and mitigation in continuous variable codes.
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 (CV) systems have shown remarkable potential for quantum computation, particularly excelling in scalability and holding great promise for error correction through bosonic encoding. Within this framework, the foundational notion of computational universality was introduced by Lloyd and Braunstein [], and has proven especially successful since it allows for the identification of finite sets of universal CV gates independent of the encoding scheme. However, realizing the full power of this computational model requires achieving fault tolerance, which in turn demands discretization via some form of encoding—and, to date, no proof exists that universal CV gates can give rise to encoded fault tolerance. We present compelling evidence in this direction by utilizing the Gottesman-Kitaev-Preskill (GKP) encoding. Specifically, we numerically optimize the generation of GKP states from vacua using circuits comprised solely of universal CV gates, and demonstrate that the resulting states are of sufficient quality to exhibit error probabilities below the threshold required for fault-tolerant memory via concatenated GKP-stabilizer codes.
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.