Quantum error-correcting code in continuous-variable (CV) systems attracts much attention due to its flexibility and high resistance against specific noise. However, the theory of fault tolerance in CV systems is premature and lacks a general strategy to translate noise in CV systems into noise in logical qubits, leading to severe restrictions on correctable noise models. In this paper, we show that Markovian-type noise in CV systems is translated into Markovian-type noise in the logical qubits through the Gottesman-Kitaev-Preskill code. We analyze an upper bound on the resulting noise strength in terms of our newly introduced noise parameterization. Combined with the established threshold theorem of concatenated codes against Markovian-type noise, we show that CV quantum computation has a fault-tolerant threshold against general Markovian-type noise, closing the existing crucial gap in CV quantum computation. We also give a new insight into the fact that careful management of the energy of a state is required to achieve fault tolerance in CV systems.
Quantum error-correcting code in continuous-variable (CV) systems attracts much attention due to its flexibility and high resistance against specific noise. However, the theory of fault tolerance in CV systems is premature and lacks a general strategy to translate noise in CV systems into noise in logical qubits, leading to severe restrictions on correctable noise models. In this paper, we show that Markovian-type noise in CV systems is translated into Markovian-type noise in the logical qubits through the Gottesman-Kitaev-Preskill code. We analyze an upper bound on the resulting noise strength in terms of our newly introduced noise parameterization. Combined with the established threshold theorem of concatenated codes against Markovian-type noise, we show that CV quantum computation has a fault-tolerant threshold against general Markovian-type noise, closing the existing crucial gap in CV quantum computation. We also give a new insight into the fact that careful management of the energy of a state is required to achieve fault tolerance in CV systems.
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.
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 (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.
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.
Concatenation of a bosonic code with a qubit code is one of the promising ways to achieve fault-tolerant quantum computation. As one of the most important bosonic codes, the Gottesman-Kitaev-Preskill (GKP) code is proposed to correct small displacement errors in phase space. If the noise in phase space is biased, the square-lattice GKP code can be concatenated with the repetition code that promises a high fault-tolerant threshold to suppress the logical error. In this work, we study the performance of GKP repetition codes with finite-energy ancillary GKP qubits in correcting biased noise. We find that there exists a critical value of noise variance for the ancillary GKP qubit such that the logical Pauli error rate decreases when increasing the code size. Furthermore, one round of GKP error correction has to be performed before concatenating with the repetition code. Our study paves the way for practical implementation of error correction by concatenating the GKP code with low-level qubit codes.
Gottesman-Kitaev-Preskill (GKP) codes encode a qubit in displaced phase-space combs of a continuous-variable (CV) quantum system and are useful for correcting a variety of high-weight photonic errors. Here we propose atomic ensemble analogs of the single-mode CV GKP code by using the quantum central limit theorem to pull back the phase-space structure of a CV system to the compact phase space of a quantum spin system. We study the optimal recovery performance of these codes under error channels described by stochastic relaxation and isotropic ballistic dephasing processes using the diversity combining approach for calculating channel fidelity. We find that the spin GKP codes outperform other spin system codes such as cat codes or binomial codes. Our spin GKP codes based on the two-axis countertwisting interaction and superpositions of SU(2) coherent states are direct spin analogs of the finite-energy CV GKP codes, whereas our codes based on one-axis twisting do not yet have well-studied CV analogs. A state preparation scheme for the spin GKP codes is proposed which uses the linear-combination-of-unitaries method, applicable to both the CV and spin GKP settings. Finally, we discuss a fault-tolerant approximate gate set for quantum computing with spin-GKP-encoded qubits, obtained by translating gates from the CV GKP setting using the quantum central limit theorem.