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Quantum Error Correction in Continuous-Variable System

Warit Asavanant, Akira Furusawa

Published Jan 25, 2022• 1 citation
Publisher

Abstract

On the physical level, digital computers are not error free. However, by adding the redundancy and using the error-correction technique, the errors on the computational level can be mitigated to the point that the probability of error occurring is negligible for most tasks. In quantum computation, a similar strategy can be employed. Here, a main difference between classical computer and quantum computer is that the quantum error correction needs to mitigate the errors, while maintaining the quantum coherence. In this chapter, we will first discuss the conceptual ideas of how to implement quantum error correction and achieve fault tolerance in the continuous-variable system. In the case of the continuous-variable system, we need to first encode the information using a logical qubit which, in a sense, digitize the broad Hilbert space. We will give some examples of the logical qubits and then discuss a particular type of encoding called Gottesman-Kitaev-Preskill (GKP) qubit, which is one of the currently most promising encoding.

DOI
10.1063/9780735424074_006

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On the physical level, digital computers are not error free. However, by adding the redundancy and using the error-correction technique, the errors on the computational level can be mitigated to the point that the probability of error occurring is negligible for most tasks. In quantum computation, a similar strategy can be employed. Here, a main difference between classical computer and quantum computer is that the quantum error correction needs to mitigate the errors, while maintaining the quantum coherence. In this chapter, we will first discuss the conceptual ideas of how to implement quantum error correction and achieve fault tolerance in the continuous-variable system. In the case of the continuous-variable system, we need to first encode the information using a logical qubit which, in a sense, digitize the broad Hilbert space. We will give some examples of the logical qubits and then discuss a particular type of encoding called Gottesman-Kitaev-Preskill (GKP) qubit, which is one of the currently most promising encoding.
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 bit-flip 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.
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 nnn-cycles of the quantum error-correction process using Knill's C4/C6{C}_{4}/{C}_{6}C4​/C6​ code with the concatenation level lll. 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.
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.
Quantum computing is entering a transitional regime between noisy intermediate-scale quantum (NISQ) processing and early fault-tolerant quantum computation (FTQC), in which increasingly capable hardware is beginning to support repeated syndrome measurements, partial error correction, and logical-qubit operations, while residual physical and logical errors remain non-negligible. In this regime, error suppression, error mitigation, and quantum error correction are increasingly better viewed as complementary layers of a unified error-reduction strategy rather than as separate approaches, with each acting at a different stage of the quantum computation to improve simulation reliability. Thus, in this review, we provide a practical and forward-looking overview of the principal hardware error sources and the corresponding error suppression and mitigation methods for reducing their impact across the current NISQ-FTQC transition. We discuss hardware-aware circuit design, coherent-error suppression, readout mitigation, noise extrapolation, classical inference, and software-supported workflows, with particular emphasis on their implementation on actual quantum processors. We further examine how error mitigation techniques can be adapted to encoded and logical-qubit settings so that they can operate alongside quantum error correction to suppress residual logical errors and improve the accuracy of computation in the early fault-tolerant regime.