Homological quantum rotor codes: logical qubits from torsion. (English) Zbl 1541.81048
Summary: We formally define homological quantum rotor codes which use multiple quantum rotors to encode logical information. These codes generalize homological or CSS quantum codes for qubits or qudits, as well as linear oscillator codes which encode logical oscillators. Unlike for qubits or oscillators, homological quantum rotor codes allow one to encode both logical rotors and logical qudits in the same block of code, depending on the homology of the underlying chain complex. In particular, a code based on the chain complex obtained from tessellating the real projective plane or a Möbius strip encodes a qubit. We discuss the distance scalling for such codes which can be more subtle than in the qubit case due to the concept of logical operator spreading by continuous stabilizer phase-shifts. We give constructions of homological quantum rotor codes based on 2D and 3D manifolds as well as products of chain complexes. Superconducting devices being composed of islands with integer Cooper pair charges could form a natural hardware platform for realizing these codes: we show that the \(0 - \pi\) qubit as well as Kitaev’s current-mirror qubit – also known as the Möbius strip qubit – are indeed small examples of such codes and discuss possible extensions.
MSC:
| 81P70 | Quantum coding (general) |
| 03B35 | Mechanization of proofs and logical operations |
| 81P65 | Quantum gates |
| 81P45 | Quantum information, communication, networks (quantum-theoretic aspects) |
| 34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |
| 68Q12 | Quantum algorithms and complexity in the theory of computing |
| 13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |
| 18G35 | Chain complexes (category-theoretic aspects), dg categories |
| 51E15 | Finite affine and projective planes (geometric aspects) |
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
| 81Q60 | Supersymmetry and quantum mechanics |
| 12F05 | Algebraic field extensions |
References:
| [1] | Terhal, BM; Conrad, J.; Vuillot, C., Towards scalable bosonic quantum error correction, Quantum Sci. Technol., 5, 4, (2020) · doi:10.1088/2058-9565/ab98a5 |
| [2] | Albert, V.V.: Bosonic coding: introduction and use cases. arXiv e-prints (2022) arXiv:2211.05714 |
| [3] | Gottesman, D.; Kitaev, A.; Preskill, J., Encoding a qubit in an oscillator, Phys. Rev. A, 64, (2001) · doi:10.1103/PhysRevA.64.012310 |
| [4] | Albert, VV; Covey, JP; Preskill, J., Robust encoding of a qubit in a molecule, Phys. Rev. X, 10, 3, (2020) · doi:10.1103/PhysRevX.10.031050 |
| [5] | Raynal, P.; Kalev, A.; Suzuki, J.; Englert, B-G, Encoding many qubits in a rotor, Phys. Rev. A, (2010) · doi:10.1103/physreva.81.052327 |
| [6] | Vuillot, C.; Asasi, H.; Wang, Y.; Pryadko, LP; Terhal, BM, Quantum error correction with the toric Gottesman-Kitaev-Preskill code, Phys. Rev. A, 99, 3, (2019) · doi:10.1103/PhysRevA.99.032344 |
| [7] | Bermejo-Vega, J.; Lin, CY-Y; Van Den Nest, M., Normalizer circuits and a Gottesman-Knill theorem for infinite-dimensional systems, Quantum Inf. Comput., 16, 5-6, 361-422, (2016) |
| [8] | Haah, J.: Two generalizations of the cubic code model (2017). https://online.kitp.ucsb.edu/online/qinfo_c17/haah/ |
| [9] | Albert, VV; Pascazio, S.; Devoret, MH, General phase spaces: from discrete variables to rotor and continuum limits, J. Phys. A Math. Theor., 50, 50, (2017) · Zbl 1383.81119 · doi:10.1088/1751-8121/aa9314 |
| [10] | Hayden, P.; Nezami, S.; Popescu, S.; Salton, G., Error correction of quantum reference frame information, PRX Quantum, 2, 1, (2021) · doi:10.1103/PRXQuantum.2.010326 |
| [11] | Faist, P.; Nezami, S.; Albert, VV; Salton, G.; Pastawski, F.; Hayden, P.; Preskill, J., Continuous symmetries and approximate quantum error correction, Phys. Rev. X, 10, 4, (2020) · doi:10.1103/PhysRevX.10.041018 |
| [12] | Brooks, P.; Kitaev, A.; Preskill, J., Protected gates for superconducting qubits, Phys. Rev. A At. Mol. Opt. Phys., 87, 5, (2013) · doi:10.1103/PhysRevA.87.052306 |
| [13] | Dempster, JM; Fu, B.; Ferguson, DG; Schuster, DI; Koch, J., Understanding degenerate ground states of a protected quantum circuit in the presence of disorder, Phys. Rev. B, 90, 9, (2014) · doi:10.1103/PhysRevB.90.094518 |
| [14] | Gyenis, A.; Mundada, PS; Di Paolo, A.; Hazard, TM; You, X.; Schuster, DI; Koch, J.; Blais, A.; Houck, AA, Experimental realization of a protected superconducting circuit derived from the \(0- \pi\) qubit, PRX Quantum, 2, 1, (2021) · doi:10.1103/PRXQuantum.2.010339 |
| [15] | Kitaev, A.: Protected qubit based on a superconducting current mirror. arXiv e-prints (2006) arXiv:cond-mat/0609441 |
| [16] | Weiss, DK; Li, ACY; Ferguson, DG; Koch, J., Spectrum and coherence properties of the current-mirror qubit, Phys. Rev. B, 100, 22, (2019) · doi:10.1103/PhysRevB.100.224507 |
| [17] | Kitaev, A., Moore, G.W., Walker, K.: Noncommuting flux sectors in a tabletop experiment (2007). arXiv:0706.3410 [hep-th] |
| [18] | Hall, BC, Quantum Theory for Mathematicians. Graduate Texts in Mathematics, (2013), New York: Springer, New York · Zbl 1273.81001 · doi:10.1007/978-1-4614-7116-5 |
| [19] | Calderbank, AR; Shor, PW, Good quantum error-correcting codes exist, Phys. Rev. A, 54, 2, 1098-1105, (1996) · Zbl 07918813 · doi:10.1103/PhysRevA.54.1098 |
| [20] | Steane, A., Multiple-particle interference and quantum error correction, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 452, 1954, 2551-2577, (1997) · Zbl 0876.94002 · doi:10.1098/rspa.1996.0136 |
| [21] | Conrad, J.; Eisert, J.; Arzani, F., Gottesman-Kitaev-Preskill codes. A lattice perspective, Quantum, 6, 648, (2022) · doi:10.22331/q-2022-02-10-648 |
| [22] | Noh, K.; Girvin, SM; Jiang, L., Encoding an oscillator into many oscillators, Phys. Rev. Lett., 125, 8, (2020) · doi:10.1103/PhysRevLett.125.080503 |
| [23] | Hurewicz, W.; Wallman, H., Dimension Theory (PMS-4), (1941), Princeton: Princeton University Press, Princeton · JFM 67.1092.03 |
| [24] | Brown, EH; Comenetz, M., Pontrjagin duality for generalized homology and cohomology theories, Am. J. Math., 98, 1, 1, (1976) · Zbl 0325.55008 · doi:10.2307/2373610 |
| [25] | Hatcher, A., Algebraic Topology, (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1044.55001 |
| [26] | Munkres, JR, Elements of Algebraic Topology, (2019), Boca Raton: CRC Press, Boca Raton · doi:10.1201/9780429493911 |
| [27] | Kaczynski, T.; Mischaikow, KM; Mrozek, M., Computational Homology. Applied Mathematical Sciences, (2004), New York: Springer, New York · Zbl 1039.55001 |
| [28] | https://github.com/cianibegood/quantum-rotor-codes.git |
| [29] | Jochym-O’Connor, T.; Kubica, A.; Yoder, TJ, Disjointness of stabilizer codes and limitations on fault-tolerant logical gates, Phys. Rev. X, 8, 2, (2018) · doi:10.1103/PhysRevX.8.021047 |
| [30] | Rymarz, M.; Bosco, S.; Ciani, A.; DiVincenzo, DP, Hardware-encoding grid states in a nonreciprocal superconducting circuit, Phys. Rev. X, 11, (2021) · doi:10.1103/PhysRevX.11.011032 |
| [31] | Freedman, MH; Meyer, DA, Projective plane and planar quantum codes, Found. Comput. Math., 1, 3, 325-332, (2001) · Zbl 0995.94037 · doi:10.1007/s102080010013 |
| [32] | Leverrier, A.; Londe, V.; Zémor, G., Towards local testability for quantum coding, Quantum, 6, 661, (2022) · doi:10.22331/q-2022-02-24-661 |
| [33] | Conrad, J.; Chamberland, C.; Breuckmann, NP; Terhal, BM, The small stellated dodecahedron code and friends, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., 376, 2123, 20170323, (2018) · Zbl 1404.81080 · doi:10.1098/rsta.2017.0323 |
| [34] | Hodgson, CD; Weeks, JR, Symmetries, isometries and length spectra of closed hyperbolic three-manifolds, Exp. Math., 3, 4, 261-274, (1994) · Zbl 0841.57020 · doi:10.1080/10586458.1994.10504296 |
| [35] | Culler, M., Dunfield, N.M., Goerner, M., Weeks, J.R.: SnapPy, a computer program for studying the geometry and topology of \(3\)-manifolds. Available at http://snappy.computop.org (24/02/2023) |
| [36] | Tillich, J-P; Zémor, G., Quantum LDPC codes with positive rate and minimum distance proportional to the square root of the blocklength, IEEE Trans. Inf. Theory, 60, 2, 1193-1202, (2014) · Zbl 1364.94630 · doi:10.1109/TIT.2013.2292061 |
| [37] | Audoux, B.; Couvreur, A., On tensor products of CSS codes, Ann. Inst. Henri Poincaré D, 6, 2, 239-287, (2019) · Zbl 1460.81013 · doi:10.4171/aihpd/71 |
| [38] | Vool, U.; Devoret, M., Introduction to quantum electromagnetic circuits, Int. J. Circuit Theory Appl., 45, 7, 897-934, (2017) · doi:10.1002/cta.2359 |
| [39] | Burkard, G.; Koch, RH; DiVincenzo, DP, Multilevel quantum description of decoherence in superconducting qubits, Phys. Rev. B, 69, 6, (2004) · doi:10.1103/PhysRevB.69.064503 |
| [40] | Rasmussen, SE; Christensen, KS; Pedersen, SP; Kristensen, LB; Bækkegaard, T.; Loft, NJS; Zinner, NT, Superconducting circuit Companion—an introduction with worked examples, PRX Quantum, 2, 4, (2021) · doi:10.1103/PRXQuantum.2.040204 |
| [41] | Douçot, B.; Ioffe, LB, Physical implementation of protected qubits, Rep. Prog. Phys., 75, 7, (2012) · doi:10.1088/0034-4885/75/7/072001 |
| [42] | Le, DT; Grimsmo, A.; Müller, C.; Stace, TM, Doubly nonlinear superconducting qubit, Phys. Rev. A, 100, (2019) · doi:10.1103/PhysRevA.100.062321 |
| [43] | Conrad, J., Twirling and Hamiltonian engineering via dynamical decoupling for Gottesman-Kitaev-Preskill quantum computing, Phys. Rev. A, 103, (2021) · doi:10.1103/PhysRevA.103.022404 |
| [44] | Kolesnikow, X.C., Weda Bomantara, R., Doherty, A.C., Grimsmo, A.L.: Gottesman-Kitaev-Preskill state preparation using periodic driving. arXiv e-prints (2023) arXiv:2303.03541 |
| [45] | Sameti, M.; Potočnik, A.; Browne, DE; Wallraff, A.; Hartmann, MJ, Superconducting quantum simulator for topological order and the toric code, Phys. Rev. A, 95, 4, 042330, (2017) · doi:10.1103/PhysRevA.95.042330 |
| [46] | Suchara, M.; Bravyi, S.; Terhal, B., Constructions and noise threshold of topological subsystem codes, J. Phys. A Math. Theor., 44, 15, (2011) · Zbl 1213.81111 · doi:10.1088/1751-8113/44/15/155301 |
| [47] | Aliferis, P.; Cross, AW, Subsystem fault tolerance with the Bacon-Shor code, Phys. Rev. Lett., 98, (2007) · doi:10.1103/PhysRevLett.98.220502 |
| [48] | Cottet, A.: Implementation d’un bit quantique dans un circuit supraconducteur. PhD thesis (2002) |
| [49] | Koch, J.; Yu, TM; Gambetta, J.; Houck, AA; Schuster, DI; Majer, J.; Blais, A.; Devoret, MH; Girvin, SM; Schoelkopf, RJ, Charge-insensitive qubit design derived from the Cooper pair box, Phys. Rev. A At. Mol. Opt. Phys., 76, 4, (2007) · doi:10.1103/PhysRevA.76.042319 |
| [50] | Gyenis, A.; Di Paolo, A.; Koch, J.; Blais, A.; Houck, AA; Schuster, DI, Moving beyond the transmon: noise-protected superconducting quantum circuits, PRX Quantum, 2, 3, (2021) · doi:10.1103/PRXQuantum.2.030101 |
| [51] | Paolo, A.D., Grimsmo, A.L., Groszkowski, P., Koch, J., Blais, A.: Control and coherence time enhancement of the \(0-\pi\) qubit. New J. Phys. 21(4), 043002 (2019). doi:10.1088/1367-2630/ab09b0 |
| [52] | Groszkowski, P.; Paolo, AD; Grimsmo, AL; Blais, A.; Schuster, DI; Houck, AA; Koch, J., Coherence properties of the \(0- \pi\) qubit, New J. Phys., 20, 4, (2018) · doi:10.1088/1367-2630/aab7cd |
| [53] | Smith, WC; Kou, A.; Xiao, X.; Vool, U.; Devoret, MH, Superconducting circuit protected by two-Cooper-pair tunneling, NPJ Quantum Inf., 6, 1, 8, (2020) · doi:10.1038/s41534-019-0231-2 |
| [54] | Bravyi, S.; Hastings, MB; Michalakis, S., Topological quantum order: stability under local perturbations, J. Math. Phys., 51, 9, (2010) · Zbl 1309.81067 · doi:10.1063/1.3490195 |
| [55] | Dennis, E.; Kitaev, A.; Landahl, A.; Preskill, J., Topological quantum memory, J. Math. Phys., 43, 9, 4452-4505, (2002) · Zbl 1060.94045 · doi:10.1063/1.1499754 |
| [56] | Toner, J.; DiVincenzo, DP, Super-roughening: a new phase transition on the surfaces of crystals with quenched bulk disorder, Phys. Rev. B, 41, 632-650, (1990) · doi:10.1103/PhysRevB.41.632 |
| [57] | Atalaya, J.; Bahrami, M.; Pryadko, LP; Korotkov, AN, Bacon-Shor code with continuous measurement of noncommuting operators, Phys. Rev. A At. Mol. Opt. Phys., 95, 3, (2017) · doi:10.1103/PhysRevA.95.032317 |
| [58] | Hastings, MB; Haah, J., Dynamically generated logical qubits, Quantum, 5, 564, (2021) · doi:10.22331/q-2021-10-19-564 |
| [59] | Davydova, M.; Tantivasadakarn, N.; Balasubramanian, S., Floquet codes without parent subsystem codes, PRX Quantum, 4, 2, (2023) · doi:10.1103/PRXQuantum.4.020341 |
| [60] | Kesselring, M.S., de la Fuente, J.C.M., Thomsen, F., Eisert, J., Bartlett, S.D., Brown, B.J.: Anyon condensation and the color code. arXiv (2022). doi:10.48550/arXiv.2212.00042 |
| [61] | Willsch, D., Rieger, D., Winkel, P., Willsch, M., Dickel, C., Krause, J., Ando, Y., Lescanne, R., Leghtas, Z., Bronn, N.T., Deb, P., Lanes, O., Minev, Z.K., Dennig, B., Geisert, S., Günzler, S., Ihssen, S., Paluch, P., Reisinger, T., Hanna, R., Bae, J.H., Schüffelgen, P., Grützmacher, D., Buimaga-Iarinca, L., Morari, C., Wernsdorfer, W., DiVincenzo, D.P., Michielsen, K., Catelani, G., Pop, I.M.: Observation of Josephson harmonics in tunnel junctions. arXiv e-prints (2023) arXiv:2302.09192 |
| [62] | Golubov, AA; Kupriyanov, MY; Il’ichev, E., The current-phase relation in Josephson junctions, Rev. Mod. Phys., 76, 2, 411-469, (2004) · doi:10.1103/RevModPhys.76.411 |
| [63] | Gladchenko, S.; Olaya, D.; Dupont-Ferrier, E.; Douçot, B.; Ioffe, LB; Gershenson, ME, Superconducting nanocircuits for topologically protected qubits, Nat. Phys., 5, 1, 48-53, (2008) · doi:10.1038/nphys1151 |
| [64] | Kovalev, AA; Pryadko, LP, Quantum Kronecker sum-product low-density parity-check codes with finite rate, Phys. Rev. A, 88, 1, (2013) · doi:10.1103/PhysRevA.88.012311 |
| [65] | Panteleev, P.; Kalachev, G., Degenerate quantum LDPC codes with good finite length performance, Quantum, 5, 585, (2021) · doi:10.22331/q-2021-11-22-585 |
| [66] | Panteleev, P.; Kalachev, G., Quantum LDPC codes with almost linear minimum distance, IEEE Trans. Inf. Theory, 68, 1, 213-229, (2022) · Zbl 1489.94162 · doi:10.1109/TIT.2021.3119384 |
| [67] | Hastings, M.B., Haah, J., O’Donnell, R.: Fiber bundle codes: breaking the \(n^{1/2}\) polylog(n) barrier for Quantum LDPC codes. In: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing. STOC 2021, pp. 1276-1288. Association for Computing Machinery, New York, NY, USA (2021). doi:10.1145/3406325.3451005 · Zbl 07765248 |
| [68] | Breuckmann, NP; Eberhardt, JN, Balanced product quantum codes, IEEE Trans. Inf. Theory, 67, 10, 6653-6674, (2021) · Zbl 1487.81056 · doi:10.1109/TIT.2021.3097347 |
| [69] | Panteleev, P., Kalachev, G.: Asymptotically good Quantum and locally testable classical LDPC codes. In: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing. STOC 2022, pp. 375-388. Association for Computing Machinery, New York, NY, USA (2022). doi:10.1145/3519935.3520017 · Zbl 07774346 |
| [70] | Leverrier, A., Zemor, G.: Quantum Tanner codes. In: 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 872-883. IEEE, Denver, CO, USA (2022). doi:10.1109/FOCS54457.2022.00117 |
| [71] | Guth, L.; Lubotzky, A., Quantum error correcting codes and 4-dimensional arithmetic hyperbolic manifolds, J. Math. Phys., 55, 8, (2014) · Zbl 1298.81052 · doi:10.1063/1.4891487 |
| [72] | Freedman, M.H.: \(Z_2\)-systolic freedom. In: Proceedings of the Kirbyfest, pp. 113-123. Mathematical Sciences Publishers, MSRI, Berkeley, California, USA (1999). doi:10.2140/gtm.1999.2.113 · Zbl 1024.53029 |
| [73] | Freedman, M.H., Meyer, D.A., Luo, F.: \(Z_2\)-systolic freedom and quantum codes. In: Mathematics of Quantum Computation, pp. 287-320 (2002) · Zbl 1075.81508 |
| [74] | Freedman, M.; Hastings, M., Building manifolds from quantum codes, Geom. Funct. Anal., 31, 4, 855-894, (2021) · Zbl 1485.53115 · doi:10.1007/s00039-021-00567-3 |
| [75] | Arfken, GB; Weber, H-J; Harris, FE, Mathematical Methods for Physicists, (2012), Oxford: Academic Press, Oxford |
| [76] | Quintavalle, A.O., Webster, P., Vasmer, M.: Partitioning qubits in hypergraph product codes to implement logical gates. Quantum 7, 1153 (2023). doi:10.22331/q-2023-10-24-1153 |
| [77] | Bravyi, S.; DiVincenzo, DP; Loss, D., Schrieffer-Wolff transformation for quantum many-body systems, Ann. Phys., 326, 10, 2793-2826, (2011) · Zbl 1230.81030 · doi:10.1016/j.aop.2011.06.004 |
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