Encoding an oscillator into many oscillators (1903.12615v3)

Abstract: 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 (QEC) is essential. Most bosonic QEC 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 which 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, the two-mode GKP-repetition code, and demonstrate that it can correct additive Gaussian noise errors. In addition, we generalize the two-mode GKP-repetition code to an even broader class of non-Gaussian oscillator codes, namely, GKP-stabilizer codes. Specifically, we show that there exists a highly hardware-efficient GKP-stabilizer code, the GKP-two-mode-squeezing code, that can quadratically suppress additive Gaussian noise errors in both the position and momentum quadratures up to a small logarithmic correction. Moreover, for any GKP-stabilizer code, we show that logical Gaussian operations can be readily implemented by using only physical Gaussian operations. We also show that our oscillator codes can correct practically relevant excitation loss and thermal noise errors.

Non-Gaussian Oscillator-into-Oscillator Codes for Quantum Error Correction

This paper presents a notable advancement in bosonic quantum error correction (QEC) by proposing a non-Gaussian framework that circumvents the limitations imposed by established no-go theorems on Gaussian QEC. These no-go results, which preclude the correction of Gaussian errors using Gaussian resources alone, have historically constrained the development of robust encoding schemes for continuous-variable (CV) quantum systems. Using Gottesman-Kitaev-Preskill (GKP) states as a non-Gaussian resource, the authors introduce several innovative encoding schemes that maintain the infinite-dimensional nature of the logical Hilbert space and are capable of correcting Gaussian errors—a feat previously deemed infeasible.

Two-Mode GKP-Repetition Code

One of the principal contributions is the introduction of the two-mode GKP-repetition code. This scheme encodes a single oscillator mode into two oscillator modes using a SUM gate and a canonical GKP state. This encoding can correct additive Gaussian noise errors, effectively preserving the bosonic characteristics at the logical level. The paper demonstrates that this approach reduces the variance of the data position quadrature noise by a factor of two without increasing the variance of the momentum quadrature noise.

GKP-Two-Mode-Squeezing Code

The authors further refine their approach with the GKP-two-mode-squeezing code, which significantly enhances noise suppression. This encoding strategy employs a two-mode squeezing operation to achieve quadratic suppression of Gaussian noise in both quadrature modes. Such substantial noise reduction is achieved despite using only two bosonic modes, highlighting the hardware efficiency of this approach compared to qubit-based QEC methods, which typically require larger overheads.

GKP-Stabilizer Codes

The paper extends these insights by proposing a broader class of codes termed GKP-stabilizer codes. These codes generalize the two-mode schemes and have the potential to achieve higher-order noise suppression by leveraging different Gaussian operations in the encoding circuits. The framework also supports straightforward implementation of logical Gaussian operations using physical Gaussian resources, facilitating a variety of CV quantum information processing tasks.

Practical Implications and Future Directions

These non-Gaussian oscillator-into-oscillator codes offer several practical implications across quantum computing and communications. They promise improved scalability and fault tolerance for CV quantum information processing, potentially enhancing protocols such as boson sampling, vibrational quantum dynamics simulation, and even quantum metrology. The paper suggests that the ability to perform logical Gaussian operations with physical Gaussian resources could lead to a more seamless integration of these codes into existing quantum architectures.

Furthermore, the reliance on GKP states, now realizable in platforms such as trapped ions and circuit QED systems, underscores the feasibility of these codes. However, practical implementation will need to contend with imperfections in GKP states, notably finite squeezing, which can limit performance gains. The authors propose strategies such as offline preparation with post-selection to mitigate these challenges.

Conclusion

The exploration of GKP-stabilizer codes marks a meaningful stride toward overcoming the traditional barriers in bosonic QEC. By devising these non-Gaussian encoding schemes that are both theoretically robust and experimentally feasible, the paper sets the stage for future research aimed at refining and extending these methods. This work signifies a substantial contribution to quantum error correction methodologies, offering both a new theoretical framework and practical pathways for enhancing the resilience of bosonic quantum systems.