Optimal encoding of oscillators into more oscillators

Jing Wu1, Anthony J. Brady2, and Quntao Zhuang3,1,2

1James C. Wyant College of Optical Sciences, University of Arizona, Tucson, AZ 85721, USA
2Department of Electrical and Computer Engineering, University of Arizona, Tucson, Arizona 85721, USA
3Ming Hsieh Department of Electrical and Computer Engineering & Department of Physics and Astronomy, University of Southern California, Los Angeles, California 90089, USA

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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.

Quantum error correction is important for robust quantum information processing in presence of noise. Bosonic encoding of quantum information into harmonic oscillators is a hardware efficient approach for quantum error correction, as exemplified by the Gottesman–Kitaev–Preskill (GKP) code and cat codes in the case of encoding a qubit. Beyond qubits, Noh, Girvin and Jiang recently provided a route to encode an oscillator into many oscillators—via GKP-stabilizer codes—in their seminal paper [Phys. Rev. Lett. 125, 080503 (2020)]. 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. To benefit from those codes maximally, an important open problem is the performance limits of such GKP-stabilizer codes, especially their optimal forms in terms of noise suppression.

In this work, we solve this important open problem for oscillator-to-oscillator encoding, by proving that the generalized GKP-two-mode-squeezing code is optimal. For single-mode data and ancilla, we further show that hexagonal lattice is the optimal GKP lattice; while for multi-mode case, we find that multimode GKP states with high dimensional lattice can perform better than single-mode low-dimensional GKP states, therefore highlighting the need of considering high dimensional lattices of GKP states. We also obtain a much simpler proof of a no-threshold theorem of such codes with finite squeezing.

The proposed optimal codes can be readily implemented in various physical platforms, promising improvement in the suppression of different types of noises.

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Cited by

[1] Anthony J. Brady, Alec Eickbusch, Shraddha Singh, Jing Wu, and Quntao Zhuang, "Advances in Bosonic Quantum Error Correction with Gottesman-Kitaev-Preskill Codes: Theory, Engineering and Applications", arXiv:2308.02913, (2023).

[2] Zheshen Zhang, Chenglong You, Omar S. Magaña-Loaiza, Robert Fickler, Roberto de J. León-Montiel, Juan P. Torres, Travis Humble, Shuai Liu, Yi Xia, and Quntao Zhuang, "Entanglement-Based Quantum Information Technology", arXiv:2308.01416, (2023).

[3] Yijia Xu, Yixu Wang, En-Jui Kuo, and Victor V. Albert, "Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison", PRX Quantum 4 2, 020342 (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-09-22 20:03:03). The list may be incomplete as not all publishers provide suitable and complete citation data.

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