Analog Quantum Error Correction with Encoding a Qubit into an Oscillator

Abstract

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...

Fault-tolerant bosonic quantum error correction with the surface–Gottesman-Kitaev-Preskill code

“…Meanwhile, there have also been studies on scaling up the GKP code by concatenating it with a repetition code [22], the [ [4,2,2]] code [22,23], and the surface code [24][25][26], or by using cluster states and measurement-based quantum computation [25,27,28]. One of the recurring themes in these previous works is that the continuous error information gathered during the GKP code error correction protocol can boost the performance of the next layer of the concatenated error correction.…”

Quantum Computing with Rotation-Symmetric Bosonic Codes

“…A natural application is to use a bosonic code at the ground level in a concatenated error-correction scheme to suppress errors below the fault-tolerance threshold of a conventional qubit-based code [3,11], potentially reducing the total overhead. Decoders that exploit the continuousvariable nature of bosonic codes can improve the faulttolerance threshold [12][13][14] and reduce the number of physical qubits required [15]. From a hardware perspective, well controlled, low loss bosonic modes occur in many quantum technology platforms, such as electromagnetic modes in optical cavities [16,17] and free space [18], superconducting circuits and microwave cavities [19][20][21][22][23], and motional modes in ion traps [24][25][26].…”

Repetition Cat Qubits for Fault-Tolerant Quantum Computation

“…The concatenation of a CV code, such as the singlemode Gottesman-Kitaev-Preskill code, with a DV one, such as the surface code, has been recently investigated by various groups [30][31][32]. The main idea behind these proposals is that using CV codes as base qubits leads to important improvements in the accuracy threshold of the DV encoding.…”