We demonstrate that continuous-variable quantum error correction based on Gaussian ancilla states and Gaussian operations (for encoding, syndrome extraction, and recovery) can be very useful to suppress the effect of non-Gaussian error channels. For a certain class of stochastic error models, reminiscent of those typically considered in the qubit case, quantum error correction codes designed for single-channel errors may enhance the transfer fidelities even when errors occur in every channel employed for transmitting the encoded state. In fact, in this case, the error-correcting capability of the continuous-variable scheme turns out to be higher than that of its discrete-variable analogues.
We demonstrate that continuous-variable quantum error correction based on Gaussian ancilla states and Gaussian operations (for encoding, syndrome extraction, and recovery) can be very useful to suppress the effect of non-Gaussian error channels. For a certain class of stochastic error models, reminiscent of those typically considered in the qubit case, quantum error correction codes designed for single-channel errors may enhance the transfer fidelities even when errors occur in every channel employed for transmitting the encoded state. In fact, in this case, the error-correcting capability of the continuous-variable scheme turns out to be higher than that of its discrete-variable analogues.
Quantum error correction is essential for robust quantum-information processing with noisy devices. As bosonic quantum systems play a role in quantum sensing, communication, and computation, it is useful to design error-correction codes suitable for these systems against various different types of noises. While most efforts aim at protecting qubits encoded into the infinite-dimensional Hilbert space of a bosonic mode, Ref. [Phys. Rev. Lett. 125, 080503 (2020)] proposed an error-correction code to maintain the infinite-dimensional-Hilbert-space nature of bosonic systems by encoding a single bosonic mode into multiple bosonic modes. Enabled by Gottesman-Kitaev-Preskill states as ancilla, the code overcomes the no-go theorem of Gaussian error correction. In this work, we generalize the error-correction code to the scenario with general correlated and heterogeneous Gaussian noises, including memory effects. We introduce Gaussian preprocessing and postprocessing to convert the general noise model to an independent but heterogeneous collection of additive white Gaussian noise channels and then apply concatenated codes in an optimized manner. To evaluate the performance, we develop a theory framework to enable the efficient calculation of the noise SD after the error correction, despite the non-Gaussian nature of the codes. Our code provides the optimal scaling of the residue-noise SD with the number of modes and can be widely applied to distributed sensor networks, network communication, and composite quantum-memory systems.
1. IntroductionQuantum system is vulnerable to disturbance from the external environment, which leads todecoherence in the system. We have to overcome this difficulty to realize a working quanutumcomputer and a dependable quantum information processing. Quantum error correction (QEC)[1, 2, 3] is one of the most promising candidates for overcoming decoherence.QEC proposals to date are separated roughly into two classes, one employs extra ancilla qubitsfor error syndrome readout while the other, called operator quantum error correction (OQEC)employs a higher rank projection operator. However, the schemes have not been built in actualquantum computing or quantum information processing so far. This is dueto the following obstaclesinherent in these QECs; the syndrome must be read out by introducing ancilla qubits duringcomputing/information processing in the former case, while realization of a high-rank projectionoperator is physically challenging in the latter case.It was shown in [4] that for some quantum channels there exist different QECs in which nosyndrome measurements, no ancillas and no projection operators were required. Recovery operationand decoding operation are combined into a single unitary operation and the output state is a directproductof a decoded qubitstate and an ancilla state. Aqubit state isreproducedwithout recoveringthe codeword and, moreover, the projection operation is automatically built in our output state.The purpose of this paper is to extend the results in [4] to general quantum channels. We showthat for any quantum channel there is a unitary recovery operation, with which the output stateis a tensor product of decoded qubit state and an ancilla state. As a result, a decoding scheme canbe done by a unitary gate followed by a partial trace operation.The rest of the paper is organized as follows. We introduce the basic notions of OQEC andthen prove the main theorem in Section 2. We also give simple examples demonstrating our resultand a simplified proof of a theorem given in [2] illustrating that our recovery channel can be usedto do correction for many other channels related to ours. Section 3 is devoted to summary anddiscussions.
Robust continuous-variable (CV) quantum information processing requires correcting realistic errors in bosonic systems, but all existing schemes rely on auxiliary Gottesman-Kitaev-Preskill (GKP) states which the preparation and operation are demanding in many platforms. In this work, we propose a novel CV quantum error correction (QEC) scheme that utilizes a broadly accessible resource: discrete-variable (DV) ancilla. Our scheme extracts information about CV displacement to the DV ancilla, measuring that allows counteracting the unwanted displacement error. We show that a simple single-qubit ancilla can already suppress CV infidelity by more than 20%. By concatenating with DV QEC codes, our scheme is robust against the physical errors in hybrid CV-DV systems, and yields a new class of oscillator-in-oscillator code that does not involve GKP states. Our work facilitates the implementation of CV QEC on realistic platforms.
Quantum error correction protects the quantum state against noise and decoherence in quantum communication and quantum computation, which enables one to perform fault-torrent quantum information processing. We experimentally demonstrate a quantum error correction scheme with a five-wave-packet code against a single stochastic error, the original theoretical model of which was firstly proposed by S. L. Braunstein and T. A. Walker. Five submodes of a continuous variable cluster entangled state of light are used for five encoding channels. Especially, in our encoding scheme the information of the input state is only distributed on three of the five channels and thus any error appearing in the remained two channels never affects the output state, i.e. the output quantum state is immune from the error in the two channels. The stochastic error on a single channel is corrected for both vacuum and squeezed input states and the achieved fidelities of the output states are beyond the corresponding classical limit.
We reexamine a non-Gaussian quantum error-correction code designed to protect optical coherent-state qubits against errors due to an amplitude-damping channel. We improve on a previous result [R. Wickert, N. K. Bernardes, and P. van Loock, Phys. Rev. A 81, 062344 (2010)] by providing a tighter upper bound on the performance attained when considering realistic assumptions, which constrain the operation of the gates employed in the scheme. The quantitative characterization is performed through measures of fidelity and concurrence, the latter obtained by employing the code as an entanglement distillation protocol. We find that, when running the code in fully deterministic error-correction mode, direct transmission can only be beaten for certain combinations of channel and input state parameters. In contrast, in error-detection mode, the usage of higher repetition encodings remains beneficial throughout, however, at the expense of diminishing success probabilities.
Physical-level noise on traveling bosonic modes remains a critical bottleneck for scalable quantum information processing. We show that for any single-mode bosonic code (qumode) corrupted by thermal or Gaussian displacement noise at loss rate μ and amplification G, a hybrid continuous-discrete-variable (CV-DV) interferometer using a single qubit ancilla and two controlled-Fourier (CF) gates sandwiching the noise channel suppresses its effects from linear to quadratic scaling. This is achieved without active error correction or destructive measurements of the encoded state, maintaining high success probabilities ≥0.5 when μG≤0.5. When supplemented with multiple ancillas, the protocol converts photon loss into coherent Fock-damping, and thermal or displacement noise into a mixture of Fock-diagonal noise. The protocol is entirely code-agnostic. For the special case of 2K-fold rotation-symmetric bosonic codes, it simplifies to conventional error detection and projection with K ancillas. Suppression with simple gates and few ancillas demonstrates a clear hardware-efficient advantage over previously proposed ``bypass'' schemes, where quantum information transferred to the DV ancillas is readily corrupted by ancilla noise. Finally, we extend the protocol to a qutrit DV ancilla. This demonstrates resilience to both CV noise and composite DV damping noise, achieving a truly hybrid noise suppression scheme that operates effectively even on CV encodings lacking a well-defined photon-number parity syndrome.