Skip to main content

Performance of two different quantum annealing correction codes

Abstract

Quantum annealing is a promising approach for solving optimization problems, but like all other quantum information processing methods, it requires error correction to ensure scalability. In this work, we experimentally compare two quantum annealing correction (QAC) codes in the setting of antiferromagnetic chains, using two different quantum annealing processors. The lower-temperature processor gives rise to higher success probabilities. The two codes differ in a number of interesting and important ways, but both require four physical qubits per encoded qubit. We find significant performance differences, which we explain in terms of the effective energy boost provided by the respective redundantly encoded logical operators of the two codes. The code with the higher energy boost results in improved performance, at the expense of a lower-degree encoded graph. Therefore, we find that there exists an important trade-off between encoded connectivity and performance for quantum annealing correction codes.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23

Notes

  1. It was shown in Ref. [42] that in general this is related to the per-site percolation threshold of the encoded graph, though this is not relevant in the case of chains.

  2. Both processors have meanwhile been dismantled.

  3. See Ref. [41] for an analytically solvable model that exhibits an increased gap via this mechanism.

References

  1. Kelly, J., Barends, R., Fowler, A.G., Megrant, A., Jeffrey, E., White, T.C., Sank, D., Mutus, J.Y., Campbell, B., Chen, Y., Chen, Z., Chiaro, B., Dunsworth, A., Hoi, I.C., Neill, C., O’Malley, P.J.J., Quintana, C., Roushan, P., Vainsencher, A., Wenner, J., Cleland, A.N., Martinis, J.M.: State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519(7541), 66–69 (2015)

    Article  ADS  Google Scholar 

  2. Corcoles, A.D., Magesan, E., Srinivasan, S.J., Cross, A.W., Steffen, M., Gambetta, J.M., Chow, J.M.: Demonstration of a quantum error detection code using a square lattice of four superconducting qubits. Nat. Commun. 6 (2015). doi:10.1038/ncomms7979

  3. Lloyd, S.: Universal quantum simulators. Science 273(5278), 1073–1078 (1996)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Buluta, I., Nori, F.: Quantum simulators. Science 326(5949), 108–111 (2009)

    Article  ADS  Google Scholar 

  5. Barreiro, J.T., Muller, M., Schindler, P., Nigg, D., Monz, T., Chwalla, M., Hennrich, M., Roos, C.F., Zoller, P., Blatt, R.: An open-system quantum simulator with trapped ions. Nature 470(7335), 486–491 (2011)

    Article  ADS  Google Scholar 

  6. Finnila, A.B., Gomez, M.A., Sebenik, C., Stenson, C., Doll, J.D.: Quantum annealing: a new method for minimizing multidimensional functions. Chem. Phys. Lett. 219(5–6), 343–348 (1994). doi:10.1016/0009-2614(94)00117-0

    Article  ADS  Google Scholar 

  7. Kadowaki, T., Nishimori, H.: Quantum annealing in the transverse Ising model. Phys. Rev. E 58(5), 5355–5363 (1998). doi:10.1103/PhysRevE.58.5355

    Article  ADS  Google Scholar 

  8. Brooke, J., Bitko, D., Aeppli, G.: Quantum annealing of a disordered magnet. Science 284(5415), 779–781 (1999). doi:10.1126/science.284.5415.779

    Article  ADS  Google Scholar 

  9. Brooke, J., Rosenbaum, T.F., Aeppli, G.: Tunable quantum tunnelling of magnetic domain walls. Nature 413(6856), 610–613 (2001). doi:10.1038/35098037

    Article  ADS  Google Scholar 

  10. Kaminsky, W.M., Lloyd, S.: Scalable architecture for adiabatic quantum computing of NP-hard problems. In: Leggett, A., Ruggiero, B., Silvestrini, P. (eds.) Quantum Computing and Quantum Bits in Mesoscopic Systems. Kluwer Academic Publishers (2004). arXiv:quant-ph/0211152

  11. Johnson, M.W., Amin, M.H.S., Gildert, S., Lanting, T., Hamze, F., Dickson, N., Harris, R., Berkley, A.J., Johansson, J., Bunyk, P., Chapple, E.M., Enderud, C., Hilton, J.P., Karimi, K., Ladizinsky, E., Ladizinsky, N., Oh, T., Perminov, I., Rich, C., Thom, M.C., Tolkacheva, E., Truncik, C.J.S., Uchaikin, S., Wang, J., Wilson, B., Rose, G.: Quantum annealing with manufactured spins. Nature 473(7346), 194–198 (2011). doi:10.1038/nature10012

    Article  ADS  Google Scholar 

  12. Farhi, E., Goldstone, J., Gutmann, S., Sipser, M.: Quantum Computation by Adiabatic Evolution. arXiv:quant-ph/0001106

  13. Farhi, E., Goldstone, J., Gutmann, S., Lapan, J., Lundgren, A., Preda, D.: A quantum adiabatic evolution algorithm applied to random instances of an NP-complete problem. Science 292(5516), 472–475 (2001). doi:10.1126/science.1057726

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Barahona, F.: On the computational complexity of Ising spin glass models. J. Phys. A Math. Gen. 15(10), 3241–3253 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  15. Kato, T.: On the adiabatic theorem of quantum mechanics. J. Phys. Soc. Jpn. 5, 435 (1950). doi:10.1143/JPSJ.5.435

    Article  ADS  Google Scholar 

  16. Jansen, S., Ruskai, M.-B., Seiler, R.: Bounds for the adiabatic approximation with applications to quantum computation. J. Math. Phys. 48(10), 10211 (2007). doi:10.1063/1.2798382

    Article  MathSciNet  Google Scholar 

  17. Lidar, D.A., Rezakhani, A.T., Hamma, A.: Adiabatic approximation with exponential accuracy for many-body systems and quantum computation. J. Math. Phys. 50(10), 102106 (2009). doi:10.1063/1.3236685

    Article  ADS  MathSciNet  Google Scholar 

  18. Childs, A.M., Farhi, E., Preskill, J.: Robustness of adiabatic quantum computation. Phys. Rev. A 65(1), 012322 (2001). doi:10.1103/PhysRevA.65.012322

    Article  ADS  Google Scholar 

  19. Sarandy, M.S., Lidar, D.A.: Adiabatic quantum computation in open systems. Phys. Rev. Lett. 95(25), 250503 (2005). doi:10.1103/PhysRevLett.95.250503

    Article  ADS  MathSciNet  Google Scholar 

  20. Aberg, J., Kult, D., Sjöqvist, E.: Quantum adiabatic search with decoherence in the instantaneous energy eigenbasis. Phys. Rev. A 72(4), 042317 (2005). doi:10.1103/PhysRevA.72.042317

    Article  ADS  Google Scholar 

  21. Roland, J., Cerf, N.J.: Noise resistance of adiabatic quantum computation using random matrix theory. Phys. Rev. A 71, 032330 (2005). doi:10.1103/PhysRevA.71.032330

    Article  ADS  MathSciNet  Google Scholar 

  22. Amin, M.H.S., Averin, D.V., Nesteroff, J.A.: Decoherence in adiabatic quantum computation. Phys. Rev. A 79(2), 022107 (2009). doi:10.1103/PhysRevA.79.022107

    Article  ADS  Google Scholar 

  23. Albash, T., Lidar, D.A.: Decoherence in adiabatic quantum computation. Phys. Rev. A 91(6), 062320 (2015). doi:10.1103/PhysRevA.91.062320

    Article  ADS  Google Scholar 

  24. Young, K.C., Blume-Kohout, R., Lidar, D.A.: Adiabatic quantum optimization with the wrong Hamiltonian. Phys. Rev. A 88(6), 062314 (2013). doi:10.1103/PhysRevA.88.062314

    Article  ADS  Google Scholar 

  25. Lidar, D., Brun, T. (eds.): Quantum Error Correction. Cambridge University Press, Cambridge (2013)

    Google Scholar 

  26. Jordan, S.P., Farhi, E., Shor, P.W.: Error-correcting codes for adiabatic quantum computation. Phys. Rev. A 74(5), 052322 (2006). doi:10.1103/PhysRevA.74.052322

    Article  ADS  MathSciNet  Google Scholar 

  27. Lidar, D.A.: Towards fault tolerant adiabatic quantum computation. Phys. Rev. Lett. 100(16), 160506 (2008). doi:10.1103/PhysRevLett.100.160506

    Article  ADS  Google Scholar 

  28. Quiroz, G., Lidar, D.A.: High-fidelity adiabatic quantum computation via dynamical decoupling. Phys. Rev. A 86, 042333 (2012). doi:10.1103/PhysRevA.86.042333

    Article  ADS  Google Scholar 

  29. Ganti, A., Onunkwo, U., Young, K.: Family of [[6k,2k,2]] codes for practical, scalable adiabatic quantum computation. Phys. Rev. A 89(4), 042313 (2014). doi:10.1103/PhysRevA.89.042313

    Article  ADS  Google Scholar 

  30. Bookatz, A.D., Farhi, E., Zhou, L.: Error suppression in Hamiltonian-based quantum computation using energy penalties. Phys. Rev. A 92(2), 022317 (2015). doi:10.1103/PhysRevA.92.022317

    Article  ADS  Google Scholar 

  31. Young, K.C., Sarovar, M., Blume-Kohout, R.: Error suppression and error correction in adiabatic quantum computation: techniques and challenges. Phys. Rev. X 3(4), 041013 (2013). doi:10.1103/PhysRevX.3.041013

    Google Scholar 

  32. Sarovar, M., Young, K.C.: Error suppression and error correction in adiabatic quantum computation: non-equilibrium dynamics. New J. Phys. 15(12), 125032 (2013). doi:10.1088/1367-2630/15/12/125032

    Article  ADS  Google Scholar 

  33. Marvian, I., Lidar, D.A.: Quantum error suppression with commuting Hamiltonians: two local is too local. Phys. Rev. Lett. 113(26), 260504 (2013). doi:10.1103/PhysRevLett.113.260504

    Article  Google Scholar 

  34. Aliferis, P., Gottesman, D., Preskill, J.: Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput. 6, 097–165 (2006)

    MathSciNet  Google Scholar 

  35. Knill, E.: Quantum computing with realistically noisy devices. Nature 434(7029), 39–44 (2005)

    Article  ADS  Google Scholar 

  36. Mizel, A.: Fault-Tolerant, Universal Adiabatic Quantum Computation. arXiv:1403.7694

  37. Johnson, M.W., Bunyk, P., Maibaum, F., Tolkacheva, E., Berkley, A.J., Chapple, E.M., Harris, R., Johansson, J., Lanting, T., Perminov, I., Ladizinsky, E., Oh, T., Rose, G.: A scalable control system for a superconducting adiabatic quantum optimization processor. Supercond. Sci. Technol. 23(6), 065004 (2010). doi:10.1088/0953-2048/23/6/065004

    Article  ADS  Google Scholar 

  38. Berkley, A.J., Johnson, M.W., Bunyk, P., Harris, R., Johansson, J., Lanting, T., Ladizinsky, E., Tolkacheva, E., Amin, M.H.S., Rose, G.: A scalable readout system for a superconducting adiabatic quantum optimization system. Supercond. Sci. Technol. 23(10), 105014 (2010). doi:10.1088/0953-2048/23/10/105014

    Article  ADS  Google Scholar 

  39. Harris, R., Johnson, M.W., Lanting, T., Berkley, A.J., Johansson, J., Bunyk, P., Tolkacheva, E., Ladizinsky, E., Ladizinsky, N., Oh, T., Cioata, F., Perminov, I., Spear, P., Enderud, C., Rich, C., Uchaikin, S., Thom, M.C., Chapple, E.M., Wang, J., Wilson, B., Amin, M.H.S., Dickson, N., Karimi, K., Macready, B., Truncik, C.J.S., Rose, G.: Experimental investigation of an eight-qubit unit cell in a superconducting optimization processor. Phys. Rev. B 82, 024511 (2010). doi:10.1103/PhysRevB.82.024511

    Article  ADS  Google Scholar 

  40. Pudenz, K.L., Albash, T., Lidar, D.A.: Error-corrected quantum annealing with hundreds of qubits. Nat. Commun. 5, 3243 (2014). doi:10.1038/ncomms4243

    Article  ADS  Google Scholar 

  41. Pudenz, K.L., Albash, T., Lidar, D.A.: Quantum annealing correction for random Ising problems. Phys. Rev. A 91(4), 042302 (2015). doi:10.1103/PhysRevA.91.042302

    Article  ADS  Google Scholar 

  42. Vinci, W., Albash, T., Paz-Silva, G., Hen, I., Lidar, D.A.: Quantum annealing correction with minor embedding. Phys. Rev. A 92(4), 042310 (2015). doi:10.1103/PhysRevA.92.042310

    Article  ADS  Google Scholar 

  43. Lucas, A.: Ising formulations of many NP problems. Front. Phys. 2, 5 (2014). doi:10.3389/fphy.2014.00005

    Article  Google Scholar 

  44. Rønnow, T.F., Wang, Z., Job, J., Boixo, S., Isakov, S.V., Wecker, D., Martinis, J.M., Lidar, D.A., Troyer, M.: Defining and detecting quantum speedup. Science 345(6195), 420–424 (2014). doi:10.1126/science.1252319

    Article  ADS  Google Scholar 

  45. Matsuura, S., Nishimori, H., Albash, T., Lidar, D. A.: Mean Field Analysis of Quantum Annealing Correction. arXiv:1510.07709

  46. Albash, T., Boixo, S., Lidar, D.A., Zanardi, P.: Quantum adiabatic Markovian master equations. New J. Phys. 14(12), 123016 (2012). doi:10.1088/1367-2630/14/12/123016

    Article  ADS  MathSciNet  Google Scholar 

  47. Reed, M.D., Dicarlo, L., Nigg, S.E., Sun, L., Frunzio, L., Girvin, S.M., Schoelkopf, R.J.: Realization of three-qubit quantum error correction with superconducting circuits. Nature 482, 382–385 (2012). doi:10.1038/nature10786

    Article  ADS  Google Scholar 

  48. Amin, M.H.S., Love, P.J., Truncik, C.J.S.: Thermally assisted adiabatic quantum computation. Phys. Rev. Lett. 100, 060503 (2008). doi:10.1103/PhysRevLett.100.060503

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anurag Mishra.

Additional information

Access to the D-Wave Two quantum annealers was made available by the USC-Lockheed Martin Quantum Computing Center and D-Wave Systems Inc. This work was supported under ARO Grant Number W911NF-12-1-0523, ARO MURI Grant No. W911NF-11-1-0268, NSF Grant No. CCF-1551064, and Fermilab Grant No. 622302. A.M. was also supported by the USC Provost Ph.D. fellowship.

Appendices

Appendix 1: Optimizing \({\pmb \gamma }\)

For each chain instance, we identified the optimal penalty coupling strength \(\gamma \) by varying it in increments of 0.1 in the range [0, 1]. This is shown in Figs. 17, 18, 19, and 20 where we plot the success probability as a function of \(\gamma \) and \(\overline{N}\). We note that for the \({[4,1,4]}_{0}\) code the optimal penalty scales with \(\alpha \), i.e., \(\gamma _{\text {opt}}\propto \alpha \). Lower values of \(\gamma _{\text {opt}}\) are observed on the S6 device. For the \({[3,1,3]}_{1}\) code, the optimal \(\gamma \) is around \(\gamma \approx 0.2\)–0.3 for all \(\alpha \) values studied, and the optimal values are unchanged across the two devices.

Appendix 2: Comparing decoding strategies

In the main text, we compared four strategies: U, C, the \({[4,1,4]}_{0}\) code, and \({[3,1,3]}_{1}\) code. We also used different decoding strategies: EM, EP, and CT. Figures 21 and 22 show all these strategies for a few chosen values of the scaling parameter \(\alpha \) for the DW2-ISI and S6 devices, respectively. The U strategy is always worst. The \({[3,1,3]}_{1}\) code can be seen to outperform all other strategies at each \(\alpha \) value for sufficiently long chains. The \({[4,1,4]}_{0}\) code outperforms the C strategy below a device-dependent \(\alpha \) value and for sufficiently long chains. The fact that the success probabilities of the CT and EM strategies are nearly equal suggests that there are very few tied qubits in the \({[4,1,4]}_{0}\)-encoded chains, an observation that holds for both devices.

In the main text, we also presented indirect evidence for the small number of ties in the \({[4,1,4]}_{0}\) code. Figure 23 shows this directly.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mishra, A., Albash, T. & Lidar, D.A. Performance of two different quantum annealing correction codes. Quantum Inf Process 15, 609–636 (2016). https://doi.org/10.1007/s11128-015-1201-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11128-015-1201-z

Keywords

  • Quantum computation
  • Quantum error correction
  • Quantum annealing
  • Error correction
  • Quantum optimization
  • Superconducting flux qubits
  • Transverse field Ising model
  • Spin chains