The $2T$-qutrit, a two-mode bosonic qutrit

Aurélie Denys and Anthony Leverrier

Inria Paris, France

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

Quantum computers often manipulate physical qubits encoded on two-level quantum systems. Bosonic qubit codes depart from this idea by encoding information in a well-chosen subspace of an infinite-dimensional Fock space. This larger physical space provides a natural protection against experimental imperfections and allows bosonic codes to circumvent no-go results that apply to states constrained by a 2-dimensional Hilbert space. A bosonic qubit is usually defined in a single bosonic mode but it makes sense to look for multimode versions that could exhibit better performance.
In this work, building on the observation that the cat code lives in the span of coherent states indexed by a finite subgroup of the complex numbers, we consider a two-mode generalisation living in the span of 24 coherent states indexed by the binary tetrahedral group $2T$ of the quaternions. The resulting $2T$-qutrit naturally inherits the algebraic properties of the group $2T$ and appears to be quite robust in the low-loss regime. We initiate its study and identify stabilisers as well as some logical operators for this bosonic code.

► BibTeX data

► References

[1] Victor V. Albert, Kyungjoo Noh, Kasper Duivenvoorden, Dylan J. Young, R. T. Brierley, Philip Reinhold, Christophe Vuillot, Linshu Li, Chao Shen, S. M. Girvin, Barbara M. Terhal, and Liang Jiang. Performance and structure of single-mode bosonic codes. Phys. Rev. A, 97: 032346, Mar 2018. 10.1103/​PhysRevA.97.032346. URL https:/​/​doi.org/​10.1103/​PhysRevA.97.032346.
https:/​/​doi.org/​10.1103/​PhysRevA.97.032346

[2] Victor V Albert, Shantanu O Mundhada, Alexander Grimm, Steven Touzard, Michel H Devoret, and Liang Jiang. Pair-cat codes: autonomous error-correction with low-order nonlinearity. Quantum Science and Technology, 4 (3): 035007, jun 2019. 10.1088/​2058-9565/​ab1e69. URL https:/​/​dx.doi.org/​10.1088/​2058-9565/​ab1e69.
https:/​/​doi.org/​10.1088/​2058-9565/​ab1e69

[3] Marcel Bergmann and Peter van Loock. Quantum error correction against photon loss using noon states. Phys. Rev. A, 94: 012311, Jul 2016a. 10.1103/​PhysRevA.94.012311. URL https:/​/​doi.org/​10.1103/​PhysRevA.94.012311.
https:/​/​doi.org/​10.1103/​PhysRevA.94.012311

[4] Marcel Bergmann and Peter van Loock. Quantum error correction against photon loss using multicomponent cat states. Phys. Rev. A, 94: 042332, Oct 2016b. 10.1103/​PhysRevA.94.042332. URL https:/​/​doi.org/​10.1103/​PhysRevA.94.042332.
https:/​/​doi.org/​10.1103/​PhysRevA.94.042332

[5] Mario Berta, Francesco Borderi, Omar Fawzi, and Volkher B Scholz. Semidefinite programming hierarchies for constrained bilinear optimization. Mathematical Programming, 194 (1): 781–829, 2022. 10.1007/​s10107-021-01650-1.
https:/​/​doi.org/​10.1007/​s10107-021-01650-1

[6] Samuel L. Braunstein and Peter van Loock. Quantum information with continuous variables. Rev. Mod. Phys., 77: 513–577, Jun 2005. 10.1103/​RevModPhys.77.513. URL https:/​/​doi.org/​10.1103/​RevModPhys.77.513.
https:/​/​doi.org/​10.1103/​RevModPhys.77.513

[7] Earl T. Campbell. Enhanced fault-tolerant quantum computing in $d$-level systems. Phys. Rev. Lett., 113: 230501, Dec 2014. 10.1103/​PhysRevLett.113.230501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.113.230501.
https:/​/​doi.org/​10.1103/​PhysRevLett.113.230501

[8] Earl T. Campbell, Hussain Anwar, and Dan E. Browne. Magic-state distillation in all prime dimensions using quantum reed-muller codes. Phys. Rev. X, 2: 041021, Dec 2012. 10.1103/​PhysRevX.2.041021. URL https:/​/​doi.org/​10.1103/​PhysRevX.2.041021.
https:/​/​doi.org/​10.1103/​PhysRevX.2.041021

[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, Gil Refael, John Preskill, Liang Jiang, Amir H. Safavi-Naeini, Oskar Painter, and Fernando G.S.L. Brandão. Building a fault-tolerant quantum computer using concatenated cat codes. PRX Quantum, 3: 010329, Feb 2022. 10.1103/​PRXQuantum.3.010329. URL https:/​/​doi.org/​10.1103/​PRXQuantum.3.010329.
https:/​/​doi.org/​10.1103/​PRXQuantum.3.010329

[10] Isaac L. Chuang and Yoshihisa Yamamoto. Simple quantum computer. Phys. Rev. A, 52: 3489–3496, Nov 1995. 10.1103/​PhysRevA.52.3489. URL https:/​/​doi.org/​10.1103/​PhysRevA.52.3489.
https:/​/​doi.org/​10.1103/​PhysRevA.52.3489

[11] Isaac L. Chuang, Debbie W. Leung, and Yoshihisa Yamamoto. Bosonic quantum codes for amplitude damping. Phys. Rev. A, 56: 1114–1125, Aug 1997. 10.1103/​PhysRevA.56.1114. URL https:/​/​doi.org/​10.1103/​PhysRevA.56.1114.
https:/​/​doi.org/​10.1103/​PhysRevA.56.1114

[12] P. T. Cochrane, G. J. Milburn, and W. J. Munro. Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping. Phys. Rev. A, 59: 2631–2634, Apr 1999. 10.1103/​PhysRevA.59.2631. URL https:/​/​doi.org/​10.1103/​PhysRevA.59.2631.
https:/​/​doi.org/​10.1103/​PhysRevA.59.2631

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

[14] HSM Coxeter. Regular Complex Polytopes. Cambridge University Press, Cambridge, 1991.

[15] Andrew S. Fletcher, Peter W. Shor, and Moe Z. Win. Optimum quantum error recovery using semidefinite programming. Phys. Rev. A, 75: 012338, Jan 2007. 10.1103/​PhysRevA.75.012338. URL https:/​/​doi.org/​10.1103/​PhysRevA.75.012338.
https:/​/​doi.org/​10.1103/​PhysRevA.75.012338

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

[17] Arne L. Grimsmo and Shruti Puri. Quantum Error Correction with the Gottesman-Kitaev-Preskill Code. PRX Quantum, 2: 020101, Jun 2021. 10.1103/​PRXQuantum.2.020101. URL https:/​/​doi.org/​10.1103/​PRXQuantum.2.020101.
https:/​/​doi.org/​10.1103/​PRXQuantum.2.020101

[18] Arne L. Grimsmo, Joshua Combes, and Ben Q. Baragiola. Quantum computing with rotation-symmetric bosonic codes. Phys. Rev. X, 10: 011058, Mar 2020. 10.1103/​PhysRevX.10.011058. URL https:/​/​doi.org/​10.1103/​PhysRevX.10.011058.
https:/​/​doi.org/​10.1103/​PhysRevX.10.011058

[19] Jérémie Guillaud and Mazyar Mirrahimi. Repetition cat qubits for fault-tolerant quantum computation. Phys. Rev. X, 9: 041053, Dec 2019. 10.1103/​PhysRevX.9.041053. URL https:/​/​doi.org/​10.1103/​PhysRevX.9.041053.
https:/​/​doi.org/​10.1103/​PhysRevX.9.041053

[20] Jim Harrington and John Preskill. Achievable rates for the Gaussian quantum channel. Phys. Rev. A, 64: 062301, Nov 2001. 10.1103/​PhysRevA.64.062301. URL https:/​/​doi.org/​10.1103/​PhysRevA.64.062301.
https:/​/​doi.org/​10.1103/​PhysRevA.64.062301

[21] Shubham P Jain, Joseph T Iosue, Alexander Barg, and Victor V Albert. Quantum spherical codes. arXiv preprint arXiv:2302.11593, 2023.
arXiv:2302.11593

[22] Emanuel Knill, Raymond Laflamme, and Gerald J Milburn. A scheme for efficient quantum computation with linear optics. Nature, 409 (6816): 46–52, 2001. 10.1038/​35051009.
https:/​/​doi.org/​10.1038/​35051009

[23] Anirudh Krishna and Jean-Pierre Tillich. Towards low overhead magic state distillation. Phys. Rev. Lett., 123: 070507, Aug 2019. 10.1103/​PhysRevLett.123.070507. URL https:/​/​doi.org/​10.1103/​PhysRevLett.123.070507.
https:/​/​doi.org/​10.1103/​PhysRevLett.123.070507

[24] Felipe Lacerda, Joseph M. Renes, and Volkher B. Scholz. Coherent-state constellations and polar codes for thermal gaussian channels. Phys. Rev. A, 95: 062343, Jun 2017. 10.1103/​PhysRevA.95.062343. URL https:/​/​doi.org/​10.1103/​PhysRevA.95.062343.
https:/​/​doi.org/​10.1103/​PhysRevA.95.062343

[25] Ludovico Lami and Mark M Wilde. Exact solution for the quantum and private capacities of bosonic dephasing channels. Nature Photonics, 2023. 10.1038/​s41566-023-01190-4.
https:/​/​doi.org/​10.1038/​s41566-023-01190-4

[26] Ulf Leonhardt. Quantum physics of simple optical instruments. Reports on Progress in Physics, 66 (7): 1207, 2003. 10.1088/​0034-4885/​66/​7/​203.
https:/​/​doi.org/​10.1088/​0034-4885/​66/​7/​203

[27] Peter Leviant, Qian Xu, Liang Jiang, and Serge Rosenblum. Quantum capacity and codes for the bosonic loss-dephasing channel. Quantum, 6: 821, September 2022. ISSN 2521-327X. 10.22331/​q-2022-09-29-821. URL https:/​/​doi.org/​10.22331/​q-2022-09-29-821.
https:/​/​doi.org/​10.22331/​q-2022-09-29-821

[28] H.-A. Loeliger. Signal sets matched to groups. IEEE Transactions on Information Theory, 37 (6): 1675–1682, 1991. 10.1109/​18.104333.
https:/​/​doi.org/​10.1109/​18.104333

[29] Marios H. Michael, Matti Silveri, R. T. Brierley, Victor V. Albert, Juha Salmilehto, Liang Jiang, and S. M. Girvin. New class of quantum error-correcting codes for a bosonic mode. Phys. Rev. X, 6: 031006, Jul 2016. 10.1103/​PhysRevX.6.031006. URL https:/​/​doi.org/​10.1103/​PhysRevX.6.031006.
https:/​/​doi.org/​10.1103/​PhysRevX.6.031006

[30] Mazyar Mirrahimi, Zaki Leghtas, Victor V Albert, Steven Touzard, Robert J Schoelkopf, Liang Jiang, and Michel H Devoret. Dynamically protected cat-qubits: a new paradigm for universal quantum computation. New Journal of Physics, 16 (4): 045014, apr 2014. 10.1088/​1367-2630/​16/​4/​045014. URL https:/​/​doi.org/​10.1088/​1367-2630/​16/​4/​045014.
https:/​/​doi.org/​10.1088/​1367-2630/​16/​4/​045014

[31] J. Niset, U. L. Andersen, and N. J. Cerf. Experimentally feasible quantum erasure-correcting code for continuous variables. Phys. Rev. Lett., 101: 130503, Sep 2008. 10.1103/​PhysRevLett.101.130503. URL https:/​/​doi.org/​10.1103/​PhysRevLett.101.130503.
https:/​/​doi.org/​10.1103/​PhysRevLett.101.130503

[32] Murphy Yuezhen Niu, Isaac L. Chuang, and Jeffrey H. Shapiro. Hardware-efficient bosonic quantum error-correcting codes based on symmetry operators. Phys. Rev. A, 97: 032323, Mar 2018. 10.1103/​PhysRevA.97.032323. URL https:/​/​doi.org/​10.1103/​PhysRevA.97.032323.
https:/​/​doi.org/​10.1103/​PhysRevA.97.032323

[33] 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 (4): 2563–2582, 2019. 10.1109/​TIT.2018.2873764.
https:/​/​doi.org/​10.1109/​TIT.2018.2873764

[34] B. O'Donoghue, E. Chu, N. Parikh, and S. Boyd. Conic optimization via operator splitting and homogeneous self-dual embedding. Journal of Optimization Theory and Applications, 169 (3): 1042–1068, June 2016. URL http:/​/​stanford.edu/​ boyd/​papers/​scs.html.
http:/​/​stanford.edu/​~boyd/​papers/​scs.html

[35] B. O'Donoghue, E. Chu, N. Parikh, and S. Boyd. SCS: Splitting conic solver, version 2.0.2. https:/​/​github.com/​cvxgrp/​scs, November 2017.
https:/​/​github.com/​cvxgrp/​scs

[36] Yingkai Ouyang and Rui Chao. Permutation-invariant constant-excitation quantum codes for amplitude damping. IEEE Transactions on Information Theory, 66 (5): 2921–2933, 2020. 10.1109/​TIT.2019.2956142.
https:/​/​doi.org/​10.1109/​TIT.2019.2956142

[37] Shruti Puri, Lucas St-Jean, Jonathan A. Gross, Alexander Grimm, Nicholas E. Frattini, Pavithran S. Iyer, Anirudh Krishna, Steven Touzard, Liang Jiang, Alexandre Blais, Steven T. Flammia, and S. M. Girvin. Bias-preserving gates with stabilized cat qubits. Science Advances, 6 (34): eaay5901, 2020. 10.1126/​sciadv.aay5901. URL https:/​/​www.science.org/​doi/​abs/​10.1126/​sciadv.aay5901.
https:/​/​doi.org/​10.1126/​sciadv.aay5901

[38] T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy. Quantum computation with optical coherent states. Phys. Rev. A, 68: 042319, Oct 2003. 10.1103/​PhysRevA.68.042319. URL https:/​/​doi.org/​10.1103/​PhysRevA.68.042319.
https:/​/​doi.org/​10.1103/​PhysRevA.68.042319

[39] T. C. Ralph, A. J. F. Hayes, and Alexei Gilchrist. Loss-tolerant optical qubits. Phys. Rev. Lett., 95: 100501, Aug 2005. 10.1103/​PhysRevLett.95.100501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.95.100501.
https:/​/​doi.org/​10.1103/​PhysRevLett.95.100501

[40] M. Reimpell and R. F. Werner. Iterative optimization of quantum error correcting codes. Phys. Rev. Lett., 94: 080501, Mar 2005. 10.1103/​PhysRevLett.94.080501. URL https:/​/​doi.org/​10.1103/​PhysRevLett.94.080501.
https:/​/​doi.org/​10.1103/​PhysRevLett.94.080501

[41] Alessio Serafini. Quantum continuous variables: a primer of theoretical methods. CRC press, 2017.

[42] David Slepian. Group codes for the gaussian channel. Bell System Technical Journal, 47 (4): 575–602, 1968. https:/​/​doi.org/​10.1002/​j.1538-7305.1968.tb02486.x. URL https:/​/​onlinelibrary.wiley.com/​doi/​abs/​10.1002/​j.1538-7305.1968.tb02486.x.
https:/​/​doi.org/​10.1002/​j.1538-7305.1968.tb02486.x

[43] B M Terhal, J Conrad, and C Vuillot. Towards scalable bosonic quantum error correction. Quantum Science and Technology, 5 (4): 043001, jul 2020. 10.1088/​2058-9565/​ab98a5. URL https:/​/​doi.org/​10.1088/​2058-9565/​ab98a5.
https:/​/​doi.org/​10.1088/​2058-9565/​ab98a5

[44] Allan DC Tosta, Thiago O Maciel, and Leandro Aolita. Grand unification of continuous-variable codes. arXiv preprint arXiv:2206.01751, 2022.
arXiv:2206.01751

[45] Christophe Vuillot, Hamed Asasi, Yang Wang, Leonid P. Pryadko, and Barbara M. Terhal. Quantum error correction with the toric Gottesman-Kitaev-Preskill code. Phys. Rev. A, 99: 032344, Mar 2019. 10.1103/​PhysRevA.99.032344. URL https:/​/​doi.org/​10.1103/​PhysRevA.99.032344.
https:/​/​doi.org/​10.1103/​PhysRevA.99.032344

[46] Yuchen Wang, Zixuan Hu, Barry C Sanders, and Sabre Kais. Qudits and high-dimensional quantum computing. Frontiers in Physics, 8: 589504, 2020. 10.3389/​fphy.2020.589504.
https:/​/​doi.org/​10.3389/​fphy.2020.589504

[47] Wojciech Wasilewski and Konrad Banaszek. Protecting an optical qubit against photon loss. Phys. Rev. A, 75: 042316, Apr 2007. 10.1103/​PhysRevA.75.042316. URL https:/​/​doi.org/​10.1103/​PhysRevA.75.042316.
https:/​/​doi.org/​10.1103/​PhysRevA.75.042316

[48] 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, May 2012. 10.1103/​RevModPhys.84.621. URL https:/​/​doi.org/​10.1103/​RevModPhys.84.621.
https:/​/​doi.org/​10.1103/​RevModPhys.84.621

Cited by

[1] Erik J. Gustafson, Henry Lamm, and Felicity Lovelace, "Primitive quantum gates for an SU(2) discrete subgroup: Binary octahedral", Physical Review D 109 5, 054503 (2024).

[2] Dong-Sheng Li, Yi-Hao Kang, Ye-Hong Chen, Yang Liu, Cheng Zhang, Yu Wang, Jie Song, and Yan Xia, "One-step parity measurement of N cat-state qubits via reverse engineering and optimal control", Physical Review A 109 2, 022437 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2024-03-29 03:34:34) and SAO/NASA ADS (last updated successfully 2024-03-29 03:34:35). The list may be incomplete as not all publishers provide suitable and complete citation data.