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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Updated version: The authors have uploaded version v4 of this work to the arXiv which may contain updates or corrections not contained in the published version v3. The authors left the following comment on the arXiv:
30 pages, 13 figures. Final version accepted by Quantum, with typo corrected

Abstract

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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► References

[1] A. R. Calderbank and Peter W. Shor. ``Good quantum error-correcting codes exist''. Phys. Rev. A 54, 1098–1105 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.1098

[2] Andrew Steane. ``Multiple-particle interference and quantum error correction''. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 452, 2551–2577 (1996).
https:/​/​doi.org/​10.1098/​rspa.1996.0136

[3] Daniel Gottesman, Alexei Kitaev, and John Preskill. ``Encoding a qubit in an oscillator''. Phys. Rev. A 64, 012310 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.012310

[4] A. Romanenko, R. Pilipenko, S. Zorzetti, D. Frolov, M. Awida, S. Belomestnykh, S. Posen, and A. Grassellino. ``Three-dimensional superconducting resonators at $t<20$ mk with photon lifetimes up to ${\tau}=2$ s''. Phys. Rev. Applied 13, 034032 (2020).
https:/​/​doi.org/​10.1103/​PhysRevApplied.13.034032

[5] Nissim Ofek, Andrei Petrenko, Reinier Heeres, Philip Reinhold, Zaki Leghtas, Brian Vlastakis, Yehan Liu, Luigi Frunzio, SM Girvin, Liang Jiang, et al. ``Extending the lifetime of a quantum bit with error correction in superconducting circuits''. Nature 536, 441–445 (2016).
https:/​/​doi.org/​10.1038/​nature18949

[6] VV Sivak, A Eickbusch, B Royer, S Singh, I Tsioutsios, S Ganjam, A Miano, BL Brock, AZ Ding, L Frunzio, et al. ``Real-time quantum error correction beyond break-even'' (2022).
https:/​/​doi.org/​10.1038/​s41586-023-05782-6
arXiv:2211.09116

[7] Nithin Raveendran, Narayanan Rengaswamy, Filip Rozpędek, Ankur Raina, Liang Jiang, and Bane Vasić. ``Finite rate QLDPC-GKP coding scheme that surpasses the CSS Hamming bound''. Quantum 6, 767 (2022).
https:/​/​doi.org/​10.22331/​q-2022-07-20-767

[8] Filip Rozpędek, Kyungjoo Noh, Qian Xu, Saikat Guha, and Liang Jiang. ``Quantum repeaters based on concatenated bosonic and discrete-variable quantum codes''. npj Quantum Inf. 7, 1–12 (2021).
https:/​/​doi.org/​10.1038/​s41534-021-00438-7

[9] Christopher Chamberland, Kyungjoo Noh, Patricio Arrangoiz-Arriola, Earl T Campbell, Connor T Hann, Joseph Iverson, Harald Putterman, Thomas C Bohdanowicz, Steven T Flammia, Andrew Keller, et al. ``Building a fault-tolerant quantum computer using concatenated cat codes''. PRX Quantum 3, 010329 (2022).
https:/​/​doi.org/​10.1103/​PRXQuantum.3.010329

[10] Kyungjoo Noh, SM Girvin, and Liang Jiang. ``Encoding an oscillator into many oscillators'' (2019). arXiv:1903.12615.
https:/​/​doi.org/​10.1103/​PhysRevLett.125.080503
arXiv:1903.12615

[11] Kyungjoo Noh, S. M. Girvin, and Liang Jiang. ``Encoding an Oscillator into Many Oscillators''. Phys. Rev. Lett. 125, 080503 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.080503

[12] Lisa Hänggli and Robert König. ``Oscillator-to-oscillator codes do not have a threshold''. IEEE Trans. Inf. Theory 68, 1068–1084 (2021).
https:/​/​doi.org/​10.1109/​TIT.2021.3126881

[13] Yijia Xu, Yixu Wang, En-Jui Kuo, and Victor V Albert. ``Qubit-Oscillator Concatenated Codes: Decoding Formalism and Code Comparison''. PRX Quantum 4, 020342 (2023).
https:/​/​doi.org/​10.1103/​PRXQuantum.4.020342

[14] Quntao Zhuang, John Preskill, and Liang Jiang. ``Distributed quantum sensing enhanced by continuous-variable error correction''. New Journal of Physics 22, 022001 (2020).
https:/​/​doi.org/​10.1088/​1367-2630/​ab7257

[15] Boyu Zhou, Anthony J. Brady, and Quntao Zhuang. ``Enhancing distributed sensing with imperfect error correction''. Phys. Rev. A 106, 012404 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.106.012404

[16] Bo-Han Wu, Zheshen Zhang, and Quntao Zhuang. ``Continuous-variable quantum repeaters based on bosonic error-correction and teleportation: architecture and applications''. Quantum Science and Technology 7, 025018 (2022).
https:/​/​doi.org/​10.1088/​2058-9565/​ac4f6b

[17] Baptiste Royer, Shraddha Singh, and S.M. Girvin. ``Encoding Qubits in Multimode Grid States''. PRX Quantum 3, 010335 (2022).
https:/​/​doi.org/​10.1103/​PRXQuantum.3.010335

[18] Jonathan Conrad, Jens Eisert, and Francesco Arzani. ``Gottesman-Kitaev-Preskill codes: A lattice perspective''. Quantum 6, 648 (2022).
https:/​/​doi.org/​10.22331/​q-2022-02-10-648

[19] Julien Niset, Jaromír Fiurášek, and Nicolas J. Cerf. ``No-Go Theorem for Gaussian Quantum Error Correction''. Phys. Rev. Lett. 102, 120501 (2009).
https:/​/​doi.org/​10.1103/​PhysRevLett.102.120501

[20] Jing Wu and Quntao Zhuang. ``Continuous-variable error correction for general gaussian noises''. Phys. Rev. Applied 15, 034073 (2021).
https:/​/​doi.org/​10.1103/​PhysRevApplied.15.034073

[21] Alonso Botero and Benni Reznik. ``Modewise entanglement of Gaussian states''. Phys. Rev. A 67, 052311 (2003).
https:/​/​doi.org/​10.1103/​PhysRevA.67.052311

[22] Ben Q. Baragiola, Giacomo Pantaleoni, Rafael N. Alexander, Angela Karanjai, and Nicolas C. Menicucci. ``All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code''. Phys. Rev. Lett. 123, 200502 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.123.200502

[23] Thomas M. Cover and Joy A. Thomas. ``Elements of information theory''. John Wiley & Sons. (2006). 2 edition.

[24] Kasper Duivenvoorden, Barbara M. Terhal, and Daniel Weigand. ``Single-mode displacement sensor''. Phys. Rev. A 95, 012305 (2017).
https:/​/​doi.org/​10.1103/​PhysRevA.95.012305

[25] Kyungjoo Noh, Victor V Albert, and Liang Jiang. ``Quantum capacity bounds of Gaussian thermal loss channels and achievable rates with Gottesman-Kitaev-Preskill codes''. IEEE Transactions on Information Theory 65, 2563–2582 (2018).
https:/​/​doi.org/​10.1109/​TIT.2018.2873764

[26] Michael M Wolf. ``Not-so-normal mode decomposition''. Phys. Rev. Lett. 100, 070505 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.100.070505

[27] Filippo Caruso, Jens Eisert, Vittorio Giovannetti, and Alexander S Holevo. ``Multi-mode bosonic Gaussian channels''. New J. Phys. 10, 083030 (2008).
https:/​/​doi.org/​10.1088/​1367-2630/​10/​8/​083030

[28] Kyungjoo Noh and Christopher Chamberland. ``Fault-tolerant bosonic quantum error correction with the surface–gottesman-kitaev-preskill code''. Phys. Rev. A 101, 012316 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.101.012316

[29] Baptiste Royer, Shraddha Singh, and S. M. Girvin. ``Stabilization of Finite-Energy Gottesman-Kitaev-Preskill States''. Phys. Rev. Lett. 125, 260509 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.125.260509

[30] Samuel L Braunstein. ``Squeezing as an irreducible resource''. Phys. Rev. A 71, 055801 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.055801

[31] Michael Reck, Anton Zeilinger, Herbert J Bernstein, and Philip Bertani. ``Experimental realization of any discrete unitary operator''. Phys. Rev. Lett. 73, 58 (1994).
https:/​/​doi.org/​10.1103/​PhysRevLett.73.58

[32] Alessio Serafini. ``Quantum Continuous Variables: A Primer of Theoretical Methods''. CRC press. (2017).

[33] Christian Weedbrook, Stefano Pirandola, Raúl García-Patrón, Nicolas J. Cerf, Timothy C. Ralph, Jeffrey H. Shapiro, and Seth Lloyd. ``Gaussian quantum information''. Rev. Mod. Phys. 84, 621–669 (2012).
https:/​/​doi.org/​10.1103/​RevModPhys.84.621

[34] Alexander S Holevo. ``One-mode quantum Gaussian channels: Structure and quantum capacity''. Probl. Inf. Transm. 43, 1–11 (2007).
https:/​/​doi.org/​10.1134/​S0032946007010012

[35] Gerardo Adesso. ``Entanglement of Gaussian states'' (2007). arXiv:quant-ph/​0702069.
arXiv:quant-ph/0702069

[36] Alessio Serafini, Gerardo Adesso, and Fabrizio Illuminati. ``Unitarily localizable entanglement of Gaussian states''. Phys. Rev. A 71, 032349 (2005).
https:/​/​doi.org/​10.1103/​PhysRevA.71.032349

[37] Jim Harrington and John Preskill. ``Achievable rates for the Gaussian quantum channel''. Phys. Rev. A 64, 062301 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.64.062301

[38] Lisa Hänggli, Margret Heinze, and Robert König. ``Enhanced noise resilience of the surface–Gottesman-Kitaev-Preskill code via designed bias''. Phys. Rev. A 102, 052408 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.052408

[39] Blayney W. Walshe, Ben Q. Baragiola, Rafael N. Alexander, and Nicolas C. Menicucci. ``Continuous-variable gate teleportation and bosonic-code error correction''. Phys. Rev. A 102, 062411 (2020).
https:/​/​doi.org/​10.1103/​PhysRevA.102.062411

[40] Frank Schmidt and Peter van Loock. ``Quantum error correction with higher Gottesman-Kitaev-Preskill codes: Minimal measurements and linear optics''. Phys. Rev. A 105, 042427 (2022).
https:/​/​doi.org/​10.1103/​PhysRevA.105.042427

[41] Benjamin Schumacher and M. A. Nielsen. ``Quantum data processing and error correction''. Phys. Rev. A 54, 2629–2635 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.2629

[42] Seth Lloyd. ``Capacity of the noisy quantum channel''. Phys. Rev. A 55, 1613–1622 (1997).
https:/​/​doi.org/​10.1103/​PhysRevA.55.1613

[43] Igor Devetak. ``The private classical capacity and quantum capacity of a quantum channel''. IEEE Transactions on Information Theory 51, 44–55 (2005).
https:/​/​doi.org/​10.1109/​TIT.2004.839515

[44] Michael M. Wolf, Geza Giedke, and J. Ignacio Cirac. ``Extremality of Gaussian Quantum States''. Phys. Rev. Lett. 96, 080502 (2006).
https:/​/​doi.org/​10.1103/​PhysRevLett.96.080502

[45] A. S. Holevo and R. F. Werner. ``Evaluating capacities of bosonic Gaussian channels''. Phys. Rev. A 63, 032312 (2001).
https:/​/​doi.org/​10.1103/​PhysRevA.63.032312

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).

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