It is known that continuous variable quantum information cannot be protected against naturally occurring noise using Gaussian states and operations only. Nohet al.proposed bosonic oscillator-to-oscillator codes relying on non-Gaussian resource states as an alternative, and showed that these encodings can lead to a reduction of the effective error strength at the logical level as measured by the variance of the classical displacement noise channel. An oscillator-to-oscillator code embeds K logical bosonic modes (in an arbitrary state) into N physical modes by means of a Gaussian N-mode unitary and N-K auxiliary one-mode Gottesman-Kitaev-Preskill-states. Here we ask if – in analogy to qubit error-correcting codes – there are families of oscillator-to-oscillator codes with the following threshold property: They allow to convert physical displacement noise with variance below some threshold value to logical noise with variance upper bounded by any (arbitrary) constant. We find that this is not the case if encoding unitaries involving a constant amount of squeezing and maximum likelihood error decoding are used. We show a general lower bound on the logical error probability which is only a function of the amount of squeezing and independent of the number of modes. As a consequence, any physically implementable family of oscillator-to-oscillator codes combined with maximum likelihood error decoding does not admit a threshold.
It is known that continuous variable quantum information cannot be protected against naturally occurring noise using Gaussian states and operations only. Nohet al.proposed bosonic oscillator-to-oscillator codes relying on non-Gaussian resource states as an alternative, and showed that these encodings can lead to a reduction of the effective error strength at the logical level as measured by the variance of the classical displacement noise channel. An oscillator-to-oscillator code embeds K logical bosonic modes (in an arbitrary state) into N physical modes by means of a Gaussian N-mode unitary and N-K auxiliary one-mode Gottesman-Kitaev-Preskill-states. Here we ask if – in analogy to qubit error-correcting codes – there are families of oscillator-to-oscillator codes with the following threshold property: They allow to convert physical displacement noise with variance below some threshold value to logical noise with variance upper bounded by any (arbitrary) constant. We find that this is not the case if encoding unitaries involving a constant amount of squeezing and maximum likelihood error decoding are used. We show a general lower bound on the logical error probability which is only a function of the amount of squeezing and independent of the number of modes. As a consequence, any physically implementable family of oscillator-to-oscillator codes combined with maximum likelihood error decoding does not admit a threshold.
An outstanding challenge for quantum information processing using bosonic systems is Gaussian errors such as excitation loss and added thermal noise errors. Thus, bosonic quantum error correction is essential. Most bosonic quantum error correction schemes encode a finite-dimensional logical qubit or qudit into noisy bosonic oscillator modes. In this case, however, the infinite-dimensional bosonic nature of the physical system is lost at the error-corrected logical level. On the other hand, there are several proposals for encoding an oscillator mode into many noisy oscillator modes. However, these oscillator-into-oscillators encoding schemes are in the class of Gaussian quantum error correction. Therefore, these codes cannot correct practically relevant Gaussian errors due to the established no-go theorems that state that Gaussian errors cannot be corrected by using only Gaussian resources. Here, we circumvent these no-go results and show that it is possible to correct Gaussian errors by using Gottesman-Kitaev-Preskill (GKP) states as non-Gaussian resources. In particular, we propose a non-Gaussian oscillator-into-oscillators code, namely the GKP two-mode squeezing code, and demonstrate that it can quadratically suppress additive Gaussian noise errors in both the position and momentum quadratures except for a small sublogarithmic correction. Furthermore, we demonstrate that our GKP two-mode squeezing code is near optimal in the weak noise limit by proving via quantum information theoretic tools that quadratic noise suppression is optimal when we use two physical oscillator modes. Lastly, we show that our non-Gaussian oscillator encoding scheme can also be used to correct excitation loss and thermal noise errors, which are dominant error sources in many realistic bosonic systems.
Bosonic encoding of quantum information into harmonic oscillators is a hardware efficient approach to battle noise. In this regard, oscillator-to-oscillator codes not only provide an additional opportunity in bosonic encoding, but also extend the applicability of error correction to continuous-variable states ubiquitous in quantum sensing and communication. In this work, we derive the optimal oscillator-to-oscillator codes among the general family of Gottesman-Kitaev-Preskill (GKP)-stablizer codes for homogeneous noise. We prove that an arbitrary GKP-stabilizer code can be reduced to a generalized GKP two-mode-squeezing (TMS) code. The optimal encoding to minimize the geometric mean error can be constructed from GKP-TMS codes with an optimized GKP lattice and TMS gains. For single-mode data and ancilla, this optimal code design problem can be efficiently solved, and we further provide numerical evidence that a hexagonal GKP lattice is optimal and strictly better than the previously adopted square lattice. For the multimode case, general GKP lattice optimization is challenging. In the two-mode data and ancilla case, we identify the D4 lattice—a 4-dimensional dense-packing lattice—to be superior to a product of lower dimensional lattices. As a by-product, the code reduction allows us to prove a universal no-threshold-theorem for arbitrary oscillators-to-oscillators codes based on Gaussian encoding, even when the ancilla are not GKP states.
We examine the performance of the single-mode Gottesman-Kitaev-Preskill (GKP) code and its concatenation with the toric code for a noise model of Gaussian shifts, or displacement errors. We show how one can optimize the tracking of errors in repeated noisy error correction for the GKP code. We do this by examining the maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean path-integral modeling a particle in a random cosine potential. We demonstrate the efficiency of a minimum-energy decoding strategy as a proxy for the path integral evaluation. In the second part of this paper, we analyze and numerically assess the concatenation of the GKP code with the toric code. When toric code measurements and GKP error correction measurements are perfect, we find that by using GKP error information the toric code threshold improves from 10 to 14. When only the GKP error correction measurements are perfect we observe a threshold at 6. In the more realistic setting when all error information is noisy, we show how to represent the maximum likelihood decoding problem for the toric-GKP code as a 3D compact QED model in the presence of a quenched random gauge field, an extension of the random-plaquette gauge model for the toric code. We present a decoder for this problem which shows the existence of a noise threshold at shift-error standard deviation ${\ensuremath{\sigma}}_{0}\ensuremath{\approx}0.243$ for toric code measurements, data errors and GKP ancilla errors. If the errors only come from having imperfect GKP states, then this corresponds to states with just four photons or more. Our last result is a no-go result for linear oscillator codes, encoding oscillators into oscillators. For the Gaussian displacement error model, we prove that encoding corresponds to squeezing the shift errors. This shows that linear oscillator codes are useless for quantum information protection against Gaussian shift errors.
The continuous-variable (CV) Gaussian no-go theorem fundamentally limits the suppression of Gaussian displacement errors using only Gaussian gates and states. Prior studies have employed Gottesman-Kitaev-Preskill (GKP) states as ancillary qumodes to suppress small Gaussian displacement errors, but when the displacement magnitude becomes large, lattice-crossing events arise beyond the correctable range of the GKP state. To address this issue, we concatenate a Gaussian-noise-suppression circuit with an outer analog Steane code that corrects such occasional lattice-crossing events as well as other abrupt displacement errors. Unlike conventional concatenation, which primarily aims to reduce logical error rates, the Steane-GKP duality in encoding provides complementary protection against both large and small displacement errors, enabling CV error correction within the continuous encoding space and contrasting with earlier approaches that concatenate GKP states with repetition codes for discrete qubit or qudit encodings. Analytical results show that, under infinite squeezing, the concatenated code suppresses the variance of Gaussian displacement errors across all qumodes by up to 50 percent while enabling unbiased correction of lattice-crossing events, with a success probability determined by the ratio between the residual Gaussian error standard deviation and the lattice-crossing magnitude. Even with finite squeezing, the proposed architecture continues to provide Gaussian-error suppression together with lattice-crossing correction, and the presence of the outer analog Steane code relaxes the squeezing requirement of the inner GKP states, indicating near-term experimental feasibility. This work establishes a viable route toward fault-tolerant continuous-variable quantum computation and provides new insight into the design of concatenated CV error-correcting architectures.
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.
Bosonic quantum error-correcting codes encode logical information in a harmonic oscillator, with the Gottesman-Kitaev-Preskill (GKP) and number-phase (NP) codes representing two fundamentally different encoding paradigms. Although both have been extensively studied, it remains unclear under what physical noise conditions (including photon loss and dephasing) one encoding intrinsically outperforms the other. Here we estimate a quantitative performance boundary between GKP and NP codes under general photon loss-dephasing noise. By optimizing code parameters within each encoding family, we identify the noise regimes in which each code exhibits a fundamental advantage. In particular, we find that the crossover occurs when the dephasing strength is approximately two orders of magnitude smaller than the loss strength, revealing a sharp separation between operational regimes. Beyond this specific comparison, our work establishes a practical and extensible methodology for benchmarking bosonic codes and optimizing their parameters, providing concrete guidance for the experimental selection and deployment of bosonic encodings in realistic noise environments.