Digital Feedback Control

  • Diego Ristè
  • Leonardo DiCarloEmail author
Part of the Quantum Science and Technology book series (QST)


This chapter covers the development of feedback control of superconducting qubits using projective measurement and a discrete set of conditional actions, here referred to as digital feedback. We begin with an overview of the applications of digital feedback in quantum computing . We then introduce an implementation of high-fidelity projective measurement of superconducting qubits. This development lays the ground for closed-loop control based on the binary measurement result.


Entangle State Feedback Controller Bell State Parity Measurement Microwave Pulse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank all the collaborators who have contributed to the experiments here presented: J.G. van Leeuwen, C.C. Bultink, M. Dukalski, C.A. Watson, G. de Lange, H.-S. Ku, M.J. Tiggelman, K.W. Lehnert, Ya. M. Blanter, and R.N. Schouten. We acknowledge L. Tornberg and G. Johansson for useful discussions. Funding for this research was provided by the Dutch Organization for Fundamental Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO, VIDI scheme), and the EU FP7 projects SOLID and SCALEQIT.


  1. 1.
    D. Ristè, J.G. van Leeuwen, H.-S. Ku, K.W. Lehnert, L. DiCarlo, Initialization by measurement of a superconducting quantum bit circuit. Phys. Rev. Lett. 109, 050507 (2012)CrossRefADSGoogle Scholar
  2. 2.
    D. Ristè, C.C. Bultink, K.W. Lehnert, L. DiCarlo, Feedback control of a solid-state qubit using high-fidelity projective measurement. Phys. Rev. Lett. 109, 240502 (2012)CrossRefADSGoogle Scholar
  3. 3.
    D. Ristè, M. Dukalski, C.A. Watson, G. de Lange, M.J. Tiggelman, Y.M. Blanter, K.W. Lehnert, R.N. Schouten, L. DiCarlo, Deterministic entanglement of superconducting qubits by parity measurement and feedback. Nature 502, 350 (2013)Google Scholar
  4. 4.
    G.G. Gillett et al., Experimental feedback control of quantum systems using weak measurements. Phys. Rev. Lett. 104, 080503 (2010)CrossRefADSGoogle Scholar
  5. 5.
    C. Sayrin et al., Real-time quantum feedback prepares and stabilizes photon number states. Nature 477, 73 (2011)CrossRefADSGoogle Scholar
  6. 6.
    P. Bushev et al., Feedback cooling of a single trapped ion. Phys. Rev. Lett. 96, 043003 (2006)CrossRefADSGoogle Scholar
  7. 7.
    M. Koch, C. Sames, A. Kubanek, M. Apel, M. Balbach, A. Ourjoumtsev, P.W.H. Pinkse, G. Rempe, Feedback cooling of a single neutral atom. Phys. Rev. Lett. 105, 173003 (2010)CrossRefADSGoogle Scholar
  8. 8.
    S. Brakhane, W. Alt, T. Kampschulte, M. Martinez-Dorantes, R. Reimann, S. Yoon, A. Widera, D. Meschede, Bayesian feedback control of a two-atom spin-state in an atom-cavity system. Phys. Rev. Lett. 109, 173601 (2012)CrossRefADSGoogle Scholar
  9. 9.
    G. de Lange, D. Ristè, M.J. Tiggelman, C. Eichler, L. Tornberg, G. Johansson, A. Wallraff, R.N. Schouten, L. DiCarlo, Reversing quantum trajectories with analog feedback. Phys. Rev. Lett. 112, 080501 (2014)CrossRefGoogle Scholar
  10. 10.
    M.S. Blok, C. Bonato, M.L. Markham, D.J. Twitchen, V.V. Dobrovitski, R. Hanson, Manipulating a qubit through the backaction of sequential partial measurements and real-time feedback. Nat. Phys. 10, 189 (2014)CrossRefGoogle Scholar
  11. 11.
    J.P. Groen, D. Ristè, L. Tornberg, J. Cramer, P.C. de Groot, T. Picot, G. Johansson, L. DiCarlo, Partial-measurement backaction and nonclassical weak values in a superconducting circuit. Phys. Rev. Lett. 111, 090506 (2013)CrossRefADSGoogle Scholar
  12. 12.
    D.P. DiVincenzo, The physical implementation of quantum computation. Fortschr. Phys. 48, 771 (2000)CrossRefzbMATHGoogle Scholar
  13. 13.
    C. Monroe, D. Meekhof, B. King, S. Jefferts, W. Itano, D. Wineland, P. Gould, Resolved-sideband Raman cooling of a bound atom to the 3D zero-point energy. Phys. Rev. Lett. 75, 4011 (1995)CrossRefADSGoogle Scholar
  14. 14.
    M. Atatüre, J. Dreiser, A. Badolato, A. Högele, K. Karrai, A. Imamoglu, Quantum-dot spin-state preparation with near-unity fidelity. Science 312, 551 (2006)CrossRefADSGoogle Scholar
  15. 15.
    S.O. Valenzuela, W.D. Oliver, D.M. Berns, K.K. Berggren, L.S. Levitov, T.P. Orlando, Microwave-induced cooling of a superconducting qubit. Science 314, 1589 (2006)CrossRefADSGoogle Scholar
  16. 16.
    V.E. Manucharyan, J. Koch, L.I. Glazman, M.H. Devoret, Fluxonium: single cooper-pair circuit free of charge offsets. Science 326, 113 (2009)CrossRefADSGoogle Scholar
  17. 17.
    M.D. Reed, B.R. Johnson, A.A. Houck, L. DiCarlo, J.M. Chow, D.I. Schuster, L. Frunzio, R.J. Schoelkopf, Fast reset and suppressing spontaneous emission of a superconducting qubit. Appl. Phys. Lett. 96, 203110 (2010)CrossRefADSGoogle Scholar
  18. 18.
    M. Mariantoni et al., Implementing the quantum von Neumann architecture with superconducting circuits. Science 334, 61 (2011)CrossRefADSGoogle Scholar
  19. 19.
    L. Robledo, L. Childress, H. Bernien, B. Hensen, P.F.A. Alkemade, R. Hanson, High-fidelity projective read-out of a solid-state spin quantum register. Nature 477, 574 (2011)CrossRefADSGoogle Scholar
  20. 20.
    P. Schindler, J.T. Barreiro, T. Monz, V. Nebendahl, D. Nigg, M. Chwalla, M. Hennrich, R. Blatt, Experimental repetitive quantum error correction. Science 332, 1059 (2011)CrossRefADSGoogle Scholar
  21. 21.
    D. Ristè, C.C. Bultink, M.J. Tiggelman, R.N. Schouten, K.W. Lehnert, L. DiCarlo, Millisecond charge-parity fluctuations and induced decoherence in a superconducting transmon qubit. Nat. Commun. 4, 1913 (2013)Google Scholar
  22. 22.
    L. Sun et al., Tracking photon jumps with repeated quantum non-demolition parity measurements. Nature 511, 444 (2014)CrossRefADSGoogle Scholar
  23. 23.
    R. Ruskov, A.N. Korotkov, Entanglement of solid-state qubits by measurement. Phys. Rev. B 67, 241305 (2003)CrossRefADSGoogle Scholar
  24. 24.
    M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000)zbMATHGoogle Scholar
  25. 25.
    H.-J. Briegel, W. Dür, J.I. Cirac, P. Zoller, Quantum repeaters: the role of imperfect local operations in quantum communication. Phys. Rev. Lett. 81, 5932 (1998)CrossRefADSGoogle Scholar
  26. 26.
    C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Mixed-state entanglement and quantum error correction. Phys. Rev. A 54, 3824 (1996)CrossRefADSMathSciNetGoogle Scholar
  27. 27.
    D. Mermin, Quantum Computer Science: An Introduction, 1st edn. (Cambridge University Press, Cambridge, 2007)CrossRefGoogle Scholar
  28. 28.
    A.G. Fowler, M. Mariantoni, J.M. Martinis, A.N. Cleland, Surface codes: towards practical large-scale quantum computation. Phys. Rev. A 86, 032324 (2012)CrossRefADSGoogle Scholar
  29. 29.
    D.S. Wang, A.G. Fowler, L.C.L. Hollenberg, Surface code quantum computing with error rates over 1%. Phys. Rev. A 83, 020302 (2011)CrossRefADSGoogle Scholar
  30. 30.
    J. Kelly et al., State preservation by repetitive error detection in a superconducting quantum circuit. Nature 519, 66 (2015)CrossRefADSGoogle Scholar
  31. 31.
    D. Ristè, S. Poletto, M.-Z. Huang, A. Bruno, V. Vesterinen, O.-P. Saira, L. DiCarlo, Detecting bit-flip errors in a logical qubit using stabilizer measurements. Nat. Commun. 6, 6983 (2015)Google Scholar
  32. 32.
    A.G. Fowler, Time-optimal quantum computation (2012). arXiv:1210.4626
  33. 33.
    H.J. Briegel, D.E. Browne, W. Dür, R. Raussendorf, M. Van den Nest, Measurement-based quantum computation. Nat. Phys. 5, 19 (2009)CrossRefGoogle Scholar
  34. 34.
    M. Riebe, T. Monz, K. Kim, A. Villar, P. Schindler, M. Chwalla, M. Hennrich, R. Blatt, Deterministic entanglement swapping with an ion-trap quantum computer. Nat. Phys. 4, 839 (2008)CrossRefGoogle Scholar
  35. 35.
    A. Furusawa, J.L. Sørensen, S.L. Braunstein, C.A. Fuchs, H.J. Kimble, E.S. Polzik, Unconditional quantum teleportation. Science 282, 706 (1998)CrossRefADSGoogle Scholar
  36. 36.
    M.D. Barrett et al., Deterministic quantum teleportation of atomic qubits. Nature 429, 737 (2004)CrossRefADSGoogle Scholar
  37. 37.
    M. Riebe et al., Deterministic quantum teleportation with atoms. Nature 429, 734 (2004)CrossRefADSGoogle Scholar
  38. 38.
    J.F. Sherson, H. Krauter, R.K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, E.S. Polzik, Quantum teleportation between light and matter. Nature 443, 557 (2006)CrossRefADSGoogle Scholar
  39. 39.
    H. Krauter, D. Salart, C.A. Muschik, J.M. Petersen, H. Shen, T. Fernholz, E.S. Polzik, Deterministic quantum teleportation between distant atomic objects. Nat. Phys. 9, 400 (2013)CrossRefGoogle Scholar
  40. 40.
    M.S. Tame, R. Prevedel, M. Paternostro, P. Böhi, M.S. Kim, A. Zeilinger, Experimental realization of Deutsch’s algorithm in a one-way quantum computer. Phys. Rev. Lett. 98, 140501 (2007)CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    R. Prevedel, P. Walther, F. Tiefenbacher, P. Böhi, R. Kaltenbaek, T. Jennewein, A. Zeilinger, High-speed linear optics quantum computing using active feed-forward. Nature 445, 65 (2007)CrossRefADSGoogle Scholar
  42. 42.
    K. Chen, C.-M. Li, Q. Zhang, Y.-A. Chen, A. Goebel, S. Chen, A. Mair, J.-W. Pan, Experimental realization of one-way quantum computing with two-photon four-qubit cluster states. Phys. Rev. Lett. 99, 120503 (2007)CrossRefADSGoogle Scholar
  43. 43.
    G. Vallone, E. Pomarico, F. De Martini, P. Mataloni, Active one-way quantum computation with two-photon four-qubit cluster states. Phys. Rev. Lett. 100, 160502 (2008)CrossRefADSGoogle Scholar
  44. 44.
    R. Ukai, N. Iwata, Y. Shimokawa, S.C. Armstrong, A. Politi, J.-I. Yoshikawa, P. van Loock, A. Furusawa, Demonstration of unconditional one-way quantum computations for continuous variables. Phys. Rev. Lett. 106, 240504 (2011)CrossRefADSGoogle Scholar
  45. 45.
    B.A. Bell, D.A. Herrera-Martí, M.S. Tame, D. Markham, W.J. Wadsworth, J.G. Rarity, Experimental demonstration of a graph state quantum error-correction code. Nat. Commun. 5, 3658 (2014)ADSGoogle Scholar
  46. 46.
    C. Vitelli, N. Spagnolo, L. Aparo, F. Sciarrino, E. Santamato, L. Marrucci, Joining the quantum state of two photons into one. Nat. Photonics 7, 521 (2013)CrossRefADSGoogle Scholar
  47. 47.
    R. Vijay, C. Macklin, D.H. Slichter, K.W. Murch, R. Naik, A.N. Korotkov, I. Siddiqi, Stabilizing Rabi oscillations in a superconducting qubit using quantum feedback. Nature 490, 77 (2012)CrossRefADSGoogle Scholar
  48. 48.
    P. Campagne-Ibarcq, E. Flurin, N. Roch, D. Darson, P. Morfin, M. Mirrahimi, M.H. Devoret, F. Mallet, B. Huard, Persistent control of a superconducting qubit by stroboscopic measurement feedback. Phys. Rev. X 3, 021008 (2013)Google Scholar
  49. 49.
    L. Steffen et al., Deterministic quantum teleportation with feed-forward in a solid state system. Nature 500, 319 (2013)CrossRefADSGoogle Scholar
  50. 50.
    W. Pfaff et al., Unconditional quantum teleportation between distant solid-state quantum bits. Science 345, 532 (2014)CrossRefADSMathSciNetGoogle Scholar
  51. 51.
    A. Blais, R.-S. Huang, A. Wallraff, S.M. Girvin, R.J. Schoelkopf, Cavity quantum electrodynamics for superconducting electrical circuits: an architecture for quantum computation. Phys. Rev. A 69, 062320 (2004)CrossRefADSGoogle Scholar
  52. 52.
    A. Wallraff, D.I. Schuster, A. Blais, L. Frunzio, R.-S. Huang, J. Majer, S. Kumar, S.M. Girvin, R.J. Schoelkopf, Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics. Nature 431, 162 (2004)CrossRefADSGoogle Scholar
  53. 53.
    H. Paik et al., Observation of high coherence in Josephson junction qubits measured in a three-dimensional circuit QED architecture. Phys. Rev. Lett. 107, 240501 (2011)CrossRefADSGoogle Scholar
  54. 54.
    M.A. Castellanos-Beltran, K.D. Irwin, G.C. Hilton, L.R. Vale, K.W. Lehnert, Amplification and squeezing of quantum noise with a tunable Josephson metamaterial. Nat. Phys. 4, 929 (2008)CrossRefGoogle Scholar
  55. 55.
    R. Vijay, M.H. Devoret, I. Siddiqi, Invited review article: the Josephson bifurcation amplifier. Rev. Sci. Instrum. 80, 111101 (2009)CrossRefADSGoogle Scholar
  56. 56.
    D.I. Schuster et al., Resolving photon number states in a superconducting circuit. Nature 445, 515 (2007)CrossRefADSGoogle Scholar
  57. 57.
    J. Majer et al., Coupling superconducting qubits via a cavity bus. Nature 449, 443 (2007)CrossRefADSGoogle Scholar
  58. 58.
    A.A. Houck et al., Controlling the spontaneous emission of a superconducting transmon qubit. Phys. Rev. Lett. 101, 080502 (2008)CrossRefADSGoogle Scholar
  59. 59.
    A. Lupaşcu, S. Saito, T. Picot, P.C. de Groot, C.J.P.M. Harmans, J.E. Mooij, Quantum non-demolition measurement of a superconducting two-level system. Nat. Phys. 3, 119 (2007)CrossRefGoogle Scholar
  60. 60.
    N. Boulant et al., Quantum nondemolition readout using a Josephson bifurcation amplifier. Phys. Rev. B 76, 014525 (2007)CrossRefADSGoogle Scholar
  61. 61.
    J.E. Johnson, C. Macklin, D.H. Slichter, R. Vijay, E.B. Weingarten, J. Clarke, I. Siddiqi, Heralded state preparation in a superconducting qubit. Phys. Rev. Lett. 109, 050506 (2012)CrossRefADSGoogle Scholar
  62. 62.
    J.M. Chow et al., Implementing a strand of a scalable fault-tolerant quantum computing fabric. Nat. Commun. 5, 4015 (2014)CrossRefADSGoogle Scholar
  63. 63.
    E. Jeffrey et al., Fast accurate state measurement with superconducting qubits. Phys. Rev. Lett. 112, 190504 (2014)CrossRefADSGoogle Scholar
  64. 64.
    Y. Lin, J.P. Gaebler, F. Reiter, T.R. Tan, R. Bowler, A.S. Sørensen, D. Leibfried, D.J. Wineland, Dissipative production of a maximally entangled steady state of two quantum bits. Nature 504, 415 (2013)CrossRefADSGoogle Scholar
  65. 65.
    K. O’Brien, C. Macklin, I. Siddiqi, X. Zhang, Resonant phase matching of Josephson junction traveling wave parametric amplifiers. Phys. Rev. Lett. 113, 157001 (2014)CrossRefADSGoogle Scholar
  66. 66.
    J.Y. Mutus et al., Strong environmental coupling in a Josephson parametric amplifier. Appl. Phys. Lett. 104, 263513 (2014)CrossRefADSGoogle Scholar
  67. 67.
    C. Eichler, Y. Salathe, J. Mlynek, S. Schmidt, A. Wallraff, Quantum-limited amplification and entanglement in coupled nonlinear resonators. Phys. Rev. Lett. 113, 110502 (2014)CrossRefADSGoogle Scholar
  68. 68.
    D. Hover, S. Zhu, T. Thorbeck, G.J. Ribeill, D. Sank, J. Kelly, R. Barends, J.M. Martinis, R. McDermott, High fidelity qubit readout with the superconducting low-inductance undulatory galvanometer microwave amplifier. Appl. Phys. Lett. 104, 152601 (2014)CrossRefADSGoogle Scholar
  69. 69.
    V. Schmitt, X. Zhou, K. Juliusson, B. Royer, A. Blais, P. Bertet, D. Vion, D. Esteve, Multiplexed readout of transmon qubits with Josephson bifurcation amplifiers. Phys. Rev. A 90, 062333 (2014)Google Scholar
  70. 70.
    E. Garrido, L. Riesebos, J. Somers, S. Visser, Feedback system for three-qubit bit-flip code. MSc project, Delft University of Technology (2014)Google Scholar
  71. 71.
    D.T. McClure, H. Paik, L.S. Bishop, M. Steffen, J.M. Chow, J.M. Gambetta, Rapid driven reset of a qubit readout resonator. Phys. Rev. Applied 5, 011001 (2016)Google Scholar
  72. 72.
    N. Ofek et al., Demonstrating real-time feedback that enhances the performance of measurement sequence with cat states in a cavity. Bull. Am. Phys. Soc. (2015).
  73. 73.
    A.D. Córcoles, J.M. Chow, J.M. Gambetta, C. Rigetti, J.R. Rozen, G.A. Keefe, M. Beth Rothwell, M.B. Ketchen, M. Steffen, Protecting superconducting qubits from radiation. Appl. Phys. Lett. 99, 181906 (2011)CrossRefADSGoogle Scholar
  74. 74.
    R.T. Thew, K. Nemoto, A.G. White, W.J. Munro, Qudit quantum-state tomography. Phys. Rev. A 66, 012303 (2002)CrossRefADSMathSciNetGoogle Scholar
  75. 75.
    R. Bianchetti, S. Filipp, M. Baur, J.M. Fink, M. Göppl, P.J. Leek, L. Steffen, A. Blais, A. Wallraff, Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics. Phys. Rev. A 80, 043840 (2009)CrossRefADSGoogle Scholar
  76. 76.
    C. Rigetti et al., Superconducting qubit in a waveguide cavity with a coherence time approaching 0.1 ms. Phys. Rev. B 86, 100506 (2012)CrossRefADSGoogle Scholar
  77. 77.
    B. Trauzettel, A.N. Jordan, C.W.J. Beenakker, M. Büttiker, Parity meter for charge qubits: an efficient quantum entangler. Phys. Rev. B 73, 235331 (2006)CrossRefADSGoogle Scholar
  78. 78.
    R. Ionicioiu, Entangling spins by measuring charge: a parity-gate toolbox. Phys. Rev. A 75, 032339 (2007)CrossRefADSGoogle Scholar
  79. 79.
    N.S. Williams, A.N. Jordan, Entanglement genesis under continuous parity measurement. Phys. Rev. A 78, 062322 (2008)CrossRefADSGoogle Scholar
  80. 80.
    G. Haack, H. Förster, M. Büttiker, Parity detection and entanglement with a Mach-Zehnder interferometer. Phys. Rev. B 82, 155303 (2010)CrossRefADSGoogle Scholar
  81. 81.
    C.W.J. Beenakker, D.P. DiVincenzo, C. Emary, M. Kindermann, Charge detection enables free-electron quantum computation. Phys. Rev. Lett. 93, 020501 (2004)CrossRefADSGoogle Scholar
  82. 82.
    H.-A. Engel, D. Loss, Fermionic Bell-state analyzer for spin qubits. Science 309, 586 (2005)CrossRefADSGoogle Scholar
  83. 83.
    C. Ahn, A.C. Doherty, A.J. Landahl, Continuous quantum error correction via quantum feedback control. Phys. Rev. A 65, 042301 (2002)CrossRefADSGoogle Scholar
  84. 84.
    W. Pfaff, T.H. Taminiau, L. Robledo, H. Bernien, M. Markham, D.J. Twitchen, R. Hanson, Demonstration of entanglement-by-measurement of solid-state qubits. Nat. Phys. 9, 29 (2013)CrossRefGoogle Scholar
  85. 85.
    C.L. Hutchison, J.M. Gambetta, A. Blais, F.K. Wilhelm, Quantum trajectory equation for multiple qubits in circuit QED: generating entanglement by measurement. Can. J. Phys. 87, 225 (2009)CrossRefADSGoogle Scholar
  86. 86.
    K. Lalumière, J.M. Gambetta, A. Blais, Tunable joint measurements in the dispersive regime of cavity QED. Phys. Rev. A 81, 040301 (2010)CrossRefADSGoogle Scholar
  87. 87.
    S. Filipp et al., Two-qubit state tomography using a joint dispersive readout. Phys. Rev. Lett. 102, 200402 (2009)CrossRefADSGoogle Scholar
  88. 88.
    L. Tornberg, G. Johansson, High-fidelity feedback-assisted parity measurement in circuit QED. Phys. Rev. A 82, 012329 (2010)CrossRefADSGoogle Scholar
  89. 89.
    K.W. Murch, S.J. Weber, C. Macklin, I. Siddiqi, Observing single quantum trajectories of a superconducting quantum bit. Nature 502, 211 (2013)CrossRefADSGoogle Scholar
  90. 90.
    R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Quantum entanglement. Rev. Mod. Phys. 81, 865 (2009)CrossRefADSMathSciNetzbMATHGoogle Scholar
  91. 91.
    M. Sarovar, H.-S. Goan, T.P. Spiller, G.J. Milburn, High-fidelity measurement and quantum feedback control in circuit QED. Phys. Rev. A 72, 062327 (2005)CrossRefADSGoogle Scholar
  92. 92.
    N. Roch et al., Observation of measurement-induced entanglement and quantum trajectories of remote superconducting qubits. Phys. Rev. Lett. 112, 170501 (2014)CrossRefADSGoogle Scholar
  93. 93.
    Y. Liu, S. Shankar, N. Ofek, M. Hatridge, A. Narla, K. Sliwa, L. Frunzio, R.J. Schoelkopf, M.H. Devoret, Comparing and combining measurement-based and driven-dissipative entanglement stabilization (2015). arxiv:1509.00860
  94. 94.
    O.-P. Saira, J.P. Groen, J. Cramer, M. Meretska, G. de Lange, L. DiCarlo, Entanglement genesis by Ancilla-based parity measurement in 2D circuit QED. Phys. Rev. Lett. 112, 070502 (2014)CrossRefADSGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Raytheon BBN TechnologiesCambridgeUSA
  2. 2.QuTech Advanced Research Center and Kavli Institute of NanoscienceDelft University of TechnologyDelftThe Netherlands

Personalised recommendations