Publication: Iterative Phase Optimization of Elementary Quantum Error Correcting Codes
Full text at PDC
Advisors (or tutors)
American Physical Society (APS)
Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits, as errors can be fully characterized. For multiqubit operations, though, this is no longer the case, as in the most general case, analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments, additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum-information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g., a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript, we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum-information processing.
© Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. M. M. thanks M. Guta for valuable discussions. We gratefully acknowledge support by the Austrian Science Fund (FWF) through the SFB FoQuS (FWF Project No. F4002-N16), as well as the Institut für Quantenoptik und Quanteninformation GmbH. E. A. M. is a recipient of a DOC grant from the Austrian Academy of Sciences. P. S. was supported by the Austrian Science Foundation (FWF) Erwin Schrödinger Stipendium 3600-N27. The research is based upon work supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office Grant No. W911NF-16-1-0070. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the view of the U.S. Army Research Office. We also acknowledge support by U.S. A.R.O. through Grant No. W911NF-14-1-010, the Spanish MINECO Grant No. FIS2012-33152, and the CAM Research Consortium QUITEMAD+ S2013/ICE-2801.
1. T. D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe, and J. L. O’Brien, Quantum Computers, Nature (London) 464, 45 (2010). 2. H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, Cambridge, England, 2009). 3. L. Viola, E. Knill, and S. Lloyd, Dynamical Decoupling of Open Quantum Systems, Phys. Rev. Lett. 82, 2417 (1999). 4. L. Viola, E. M. Fortunato, M. A. Pravia, E. Knill, R. Laflamme, and D. G. Cory, Experimental Realization of Noiseless Subsystems for Quantum Information Processing, Science 293, 2059 (2001). 5. P. Facchi, S. Tasaki, S. Pascazio, H. Nakazato, A. Tokuse, and D. A. Lidar, Control of Decoherence: Analysis and Comparison of Three Different Strategies, Phys. Rev. A 71, 022302 (2005). 6. J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Laforest, J. Baugh, D. G. Cory, and R. Laflamme, Symmetrized Characterization of Noisy Quantum Processes, Science 317, 1893 (2007). 7. J. Wallman, C. Granade, R. Harper, and S. T. Flammia, Estimating the Coherence of Noise,New J. Phys. 17, 113020 (2015). 8. C. Altafini and F. Ticozzi, Modeling and Control of Quantum Systems: An Introduction, IEEE Trans. Autom. Control 57, 1898 (2012). 9. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, 2002). 10. A. Rivas and S. F. Huelga, Open Quantum Systems. An Introduction (Springer, Heidelberg, 2011). 11. H. Häffner, C. F. Roos, and R. Blatt, Quantum Computing with Trapped Ions, Phys. Rep. 469, 155 (2008). 12. N. F. Ramsey, A Molecular Beam Resonance Method with Separated Oscillating Fields, Phys. Rev. 78, 695 (1950). 13. C. J. Foot, Atomic Physics (Oxford University Press, Oxford, 2005). 14. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2000). 15. D. A. Lidar and T. A. Brun, Quantum Error Correction (Cambridge University Press, Cambridge, England, 2013). 16. D. Nigg, M. Müller, E. A. Martinez, P. Schindler, M. Hennrich, T. Monz, M. A. Martin-Delgado, and R. Blatt, Quantum Computations on a Topologically Encoded Qubit, Science 345, 302 (2014). 17. M. Cramer, M. B. Plenio, S. T. Flammia, R. Somma, D. Gross, S. D. Bartlett, O. Landon-Cardinal, D. Poulin, and Y.-K. Liu, Efficient Quantum State Tomography, Nat. Commun. 1, 149 (2010). 18. P. Schindler, D. Nigg, T. Monz, J. T. Barreiro, E. Martinez, S. X. Wang, S. Quint, M. F. Brandl, V. Nebendahl, C. F. Roos, M. Chwalla, M. Hennrich, and R. Blatt, A Quantum Information Processor with Trapped Ions, New J. Phys. 15, 123012 (2013). 19. E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, Topological Quantum Memory, J. Math. Phys. (N.Y.) 43, 4452 (2002). 20. B. M. Terhal, Quantum Error Correction for Quantum Memories, Rev. Mod. Phys. 87, 307 (2015). 21. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, Fast Quantum Gates for Neutral Atoms, Phys. Rev. Lett. 85, 2208 (2000). 22. M. Saffman, T. G. Walker, and K. Mølmer, Quantum Information with Rydberg Atoms, Rev. Mod. Phys. 82, 2313 (2010). 23. D. Crow, R. Joynt, and M. Saffman, Numerical Simulation of Coherent Error Correction,arXiv:1510.08359. 24. S. E. Anderson, K. C. Younge, and G. Raithel, Trapping Rydberg Atoms in an Optical Lattice,Phys. Rev. Lett. 107, 263001 (2011). 25. M. Viteau, M. G. Bason, J. Radogostowicz, N. Malossi, D. Ciampini, O. Morsch, and E. Arimondo, Rydberg Excitations in Bose-Einstein Condensates in Quasi-One-Dimensional Potentials and Optical Lattices, Phys. Rev. Lett. 107, 060402 (2011). 26. P. Schauss, M. Cheneau, M. Endres, T. Fukuhara, S. Hild, A. Omran, T. Pohl, C. Gross, S. Kuhr, and I. Bloch, Observation of Spatially Ordered Structures in a Two-Dimensional Rydberg Gas, Nature (London) 491, 87 (2012). 27. F. Nogrette, H. Labuhn, S. Ravets, D. Barredo, L. Béguin, A. Vernier, T. Lahaye, and A. Browaeys, Single-Atom Trapping in Holographic 2D Arrays of Microtraps with Arbitrary Geometries, Phys. Rev. X 4, 021034 (2014). 28. T. Xia, M. Lichtman, K. Maller, A. W. Carr, M. J. Piotrowicz, L. Isenhower, and M. Saffman, Randomized Benchmarking of Single-Qubit Gates in a 2D Array of Neutral-Atom Qubits, Phys. Rev. Lett. 114, 100503 (2015). 29. A. M. Steane, Error Correcting Codes in Quantum Theory, Phys. Rev. Lett. 77, 793 (1996). 30. H. Bombin and M. A. Martin-Delgado, Topological Quantum Distillation, Phys. Rev. Lett. 97, 180501 (2006). 31. R. Hanson and D. Awschalom, Coherent Manipulation of Single Spins in Semiconductors,Nature (London) 453, 1043 (2008). 32. A. D. Corcoles, E. Magesan, S. J. Srinivasan, A. W. Cross, M. Steffen, J. M. Gambetta, and J. M. Chow, Demonstration of a Quantum Error Detection Code Using a Square Lattice of Four Superconducting Qubits, Nat. Commun. 6, 6979 (2015). 33. J. M. Gambetta, J. M. Chow, and M. Steffen, Building Logical Qubits in a Superconducting Quantum Computing System, arXiv:1510.04375. 34. J. Kelly, R. Barends, A. G. Fowler, A. Megrant, E. Jeffrey, T. C. White, D. Sank, J. Y. Mutus, B. Campbell, Y. Chen et al., State Preservation by Repetitive Error Detection in a Superconducting Quantum Circuit, Nature (London) 519, 66 (2015). 35. G. Waldherr, Y. Wang, S. Zaiser, M. Jamali, T. Schulte-Herbruggen, H. Abe, T. Ohshima, J. Isoya, J. F. Du, P. Neumann, and J. Wrachtrup, Quantum Error Correction in a Solid-State Hybrid Spin Register, Nature (London) 506, 204 (2014). 36. A. Fedorov, L. Steffen, M. Baur, M. P. da Silva, and A. Wallraff, Implementation of a Toffoli Gate with Superconducting Circuits, Nature (London) 481, 170 (2012). 37. A. R. Calderbank and P. W. Shor, Good Quantum Error-Correcting Codes Exist, Phys. Rev. A54, 1098 (1996). 38. K. Mølmer and A. Sørensen, Multiparticle Entanglement of Hot Trapped Ions, Phys. Rev. Lett.82, 1835 (1999). 39. C. F. Roos, Ion Trap Quantum Gates with Amplitude-Modulated Laser Beams, New J. Phys.10, 013002 (2008). 40. L. M. K. Vandersypen and I. L. Chuang, NMR Techniques for Quantum Control and Computation, Rev. Mod. Phys. 76, 1037 (2005). 41. V. Nebendahl, H. Häffner, and C. F. Roos, Optimal Control of Entangling Operations for Trapped-Ion Quantum Computing, Phys. Rev. A 79, 012312 (2009). 42. E. Peik, T. Schneider, and C. Tamm, Laser Frequency Stabilization to a Single Ion, J. Phys. B39, 145 (2006). 43. S. J. Wright, Coordinate Descent Algorithms, Math. Program. 151, 3 (2015). 44. Y. Nesterov, Efficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems,SIAM J. Optim. 22, 341 (2012). 45. P. W. Shor, Fault-Tolerant Quantum Computation, in Proceedings of 37th Annual Symposium on the Foundations of Computer Science, Burlington, VT, 1996 (IEEE Computer Society, New York, 1996), p. 56. 46. J. Preskill, Fault-Tolerant Quantum Computation in Introduction to Quantum Computation and Information (World Scientific, Singapore, 1997).