1. Production and detection of few-qubit entanglement in circuit-QED processors Leo DiCarlo (in place of Rob Schoelkopf) Department of Applied Physics, Yale University Portland Convention Center Sunday, March 14 8:30 a.m. - 12:30 p.m.

2. Outline What is quantum entanglement? What is a quantum processor, and why must it produce (and why must we detect) highly-entangled qubit states? Going beyond two qubits Outlook How to detect it? the complete way: quantum state tomography the scalable way: entanglement witnesses One specific example: production and detection of two-qubit entanglement in a circuit-QED processor two-qubit conditional phase gate joint qubit readout

3. Defining a quantum processor Related MM 2010 talks: W6.00003 & T29.00012

4. What is a quantum processor? Quantum processor: programmable computing device using quantum superposition and entanglement in a qubit register. The program it executes is compiled into a sequence of one- and two- (perhaps more-) qubit gates, following an algorithm. quantum processors based on circuit QED 2009 model 2 qubits 2010 model 4 qubits

5. Anatomy of quantum algorithm Register qubits Ancilla qubits M create superposition encode function in a unitary analyze the function initialize measure involves entanglement between qubits Maintain quantum coherence 1) Start in superposition: all values at once! 2) Build complex transformation out of one- and two-qubit gates 3) Make the answer we seek result in an eigenstate of measurement involves disentangling the qubits UaUa UfUf UsUs

6. Example: The Grover quantum search Quantum circuit diagram of the algorithm quantum oracle Problem: Find the unknown root of Given: a quantum oracle that implements the unitary

7. Quantum debugging oracle bcdf e g Half-way along the algorithm, the two-qubit state ideally maximally entangled! The quality of a quantum processor relies on producing near- perfect multi-qubit entanglement at intermediate steps of the computation. A quantum computer engineer needs to detect this entanglement as a way to benchmark or debug the processor.

8. But what is quantum entanglement? `Entanglement is simply Schrodinger’s name for superposition in a multi-particle system.` GHZ, Physics Today 1993

9. for N=2 qubits: A pure 2-qubit pure state is fully described by 6 real #s Wavefunction description of pure two-qubit states normalization irrelevant global phase complex numbers -

10. When are two qubits entangled? Two qubits are entangled when their joint wavefunction cannot be separated into a product of individual qubit wavefunctions Some common terms: Entangled = non-separable = non-product state Unentangled = separable = product state vs

11. When are two qubits entangled? State Entangled? no yes no yes The singlet

12. Quantifying entanglement Two qubits in a pure state are entangled if they have nonzero concurrence

13. Quantifying entanglement – pure states The concurrence is an entanglement monotone: If, we say state a is more entangled than state b If, we say state a is maximally entangled Example: The Bell state is maximally entangled. The state, with, is entangled, but less entangled than a Bell state.

14. Reality check #1 : quantum states never pure!

15. Density-matrix description of mixed states Hermitian Unity trace for a pure state for a mixed state Fully describing a 2-qubit mixed state requires 15 real #s complex #s -

16. The decomposition is generally not unique! Warning: Example: 1 0 0 0101 1010 1 0 0101 1010 1 and Give the same City-scape (Manhattan plot)

17. Quantifying entanglement – mixed states The concurrence of a mixed state is given by where the minimization is over all possible decompositions of The are the eigenvalues of the matrix in decreasing order, and Can show: Yes, it’s completely non-intuitive! A very non-linear function of Hill and Wootters, PRL (2007) Wootters, PRL (2008) Horodecki 4, RMP (2009)

18. Getting : Two-qubit state tomography Review: the N=1 qubit case Can we generalize the Bloch vector to N>1? Answer: the Pauli set Extracting usual metrics from the Pauli set A quantum debugging tool Knowing all there is to know about the 2-qubit state Necessary to extract C Drawback: it is not scalable diagnostic tool!

19. Geometric visualization for N=1: The Bloch sphere Bloch vector (NMR) x y z pure state mixed state Is there a similarly intuitive description for N=2 qubits?

20. State tomography of qubit decay Steffen et al., PRL (2006) Related MM talks: Tomography of qutrits: Z26.00013 (phase qubits) Z26.00012 (transmon qubits)

21. One of them, always Generalizing the Bloch vector: The Pauli set Polarization of Qubit 1 Polarization of Qubit 2 Two-qubit correlations The two-qubit Pauli set can be divided into three sections: The Pauli set = the set of expectation values of the 16 2-qubit Pauli operators.  gives a full description of the 2-qubit state  is the extension of the Bloch vector to 2 qubits  generalizes to higher N

22. Visualizing N=2 states: product states 1 0 1 0 0 0101 1010 1 0 0101 1010 1 0 0101 1010 1 0 0101 1010 1

23. Visualizing N=2 states: Bell states 1 0 1 0 0 0101 1010 1 0 0101 1010 1 0 0101 1010 1 0 0101 1010 1

24. Extracting usual metrics from the Pauli Set State purity: Fidelity to a target state : Concurrence: Warning: for pure states only

25. Measuring the Pauli set with a joint qubit readout Related MM 2010 talks: W6.00003 & T29.00012 Filipp et al., PRL (2009) DiCarlo et al., Nature (2009) Chow et al., arXiv 0908.1955

26. 1 ns resolution transmon qubits DC - 2 GHz A two-qubit quantum processor T = 13 mK flux bias lines cavity: “entanglement bus,” driver, & detector

27. Two-qubit joint readout via cavity Cavity transmission Frequency “Strong dispersive cQED” Schuster, Houck et al., Nature (2007)

28. Prepare and measure Qubit-state dependent cavity resonance Chow et al., arXiv 0908.1955

29. Two qubit joint readout via cavity Cavity transmission Frequency Measure cavity transmission: “Are qubits both in their ground state?” Direct access to qubit-qubit correlations with a single measurement channel!

30. Direct access to qubit-qubit correlations How to reconstruct the two-qubit state from an ensemble measurement of the form ? Answer: Combine joint readout with one-qubit pre-rotations It is possible to acquire correlation info. with one measurement channel! + Apply &, then measure: Joint Dispersive Readout Example: How to extract Apply, then measure: All Pauli set components are obtained by linear operations on raw data.

31. Producing entanglement with the circuit QED processor Related MM 2010 talks: W6.00003 & T29.00012 First demonstration of 2Q entanglement in SC qubits: Steffen et al., Science (2006)

32. Spectroscopy of qubit 2 -cavity system Qubit-qubit swap interaction Majer et al., Nature (2007) cavity left qubit right qubit Cavity-qubit interaction Vacuum Rabi splitting Wallraff et al., Nature (2004)

33. Preparation 1-qubit rotations Measurement cavity Q One-qubit gates: X and Y rotations x y z

34. Preparation 1-qubit rotations Measurement cavity I One-qubit gates: X and Y rotations x y z

35. Preparation 1-qubit rotations Measurement cavity Q One-qubit gates: X and Y rotations x y z

36. Preparation 1-qubit rotations Measurement cavity I One-qubit gates: X and Y rotations J. Chow et al., PRL (2009) Fidelity = 99% x y z

37. cavity Two-qubit gate: turn on interactions Conditional phase gate Use control lines to push qubits near a resonance: A controlled interaction, a la NMR

38. Two-excitation manifold Two-excitation manifold of system Transmon “qubits” have multiple levels… Strauch et al. PRL (2003): proposed using interactions with higher levels for computation in phase qubits Avoided crossing (160 MHz) Two-excitation manifold

39. Adiabatic phase gate 2-excitation manifold 1-excitation manifold 01+10

40. Adjust timing of flux pulse so that only quantum amplitude of acquires a minus sign: Implementing C-Phase with 1 fancy pulse 30 ns 11

41. Apply C-Phase entangler: rotation on each qubit yields a maximal superposition: No longer a product state! How to create a Bell state using C-Phase

42. rotation on LEFT qubit yields: How to create a Bell state using C-Phase

43. Two-qubit entanglement experiment Ideally: wavefunction density matrix Expt’l state tomography

44. Bell state Fidelity Concurrence 91% 88% 94% 90% 86% 87% 81% Entanglement on demand

45. Switching to the Pauli set Related MM 2010 talks: T29.00012 & W6.00003

46. Experimental N=2 Pauli sets

47. Pauli set movies Look at evolutions of separable and entangled states ~98% visibility for separable states, ~92% visibility for entangled states a test for systematic errors in tomography, such as offsets and amplitude errors in Pauli bars

48. Working around Concurrence Related MM 2010 talks: W6.00003 &Y26.00013

49. Drawbacks of Concurrence  Requires full state tomography (knowing )  Is a very non-linear function of  It is difficult to propagate experimental errors in tomography to error (bias and noise) in Can we characterize entanglement without reliance on ? Can we place lower bounds on without performing full state tomography?

50. Witnessing entanglement with a subset of the Pauli set An entanglement witness is a unity-trace observable with a positive expectation value for all product states. state is entangled, guaranteed. witness simply doesn’t know Witness also gives a lower bound on : Witnesses require only a subset of the Pauli set! An optimal witness of an entangled state gives the strictest lower bound on Example: Is the optimal witness for the singlet

51. Witnessing entanglement Entanglement is witnessed (by some witness) at all angles in entangled-state movie!

52. Witnessing entanglement Control: entanglement is not witnessed by any witness in the separable movie!

53. Bell inequalities: another form of entanglement witness  x x’ z’ z state is clearly highly entangled! Clauser, Horne, Shimony & Holt (1969) Separable bound: Bell state Also UCSB group, closing detection loophole, Ansmann et al., Nature (2009) CHSH operator = entanglement witness not test of hidden variables… (loopholes abound)

54. Beyond two qubits Related MM 2010 talks: W6.00003, Y26.00013 (with transmon qubits) Z26.00003 (with phase qubits) DiCarlo et al., (2010) Reed et al., (2010)

55. Knowing the three-qubit state = expectation values of 63 Pauli operators Polarization of qubit 1 Polarization of qubit 2 2-qubit correlations 3-qubit correlations Polarization of qubit 3 The density matrix for N=3

56. Example Pauli set for N=3: product state 1 0 000000 111111 111111 000000

57. The GHZ state Example Pauli set for N=3: 3-qubit entangled state Greenberger Horne Zeilinger 1 0 000000 111111 000000 111111

58. Upgrading the processor: more qubits Q3Q3 Q2Q2 Q1Q1 Q4Q4 DiCarlo et al., (2010) Reed et al., (2010)  We have recently extended C-Phase gates and joint qubit readout to produce and detect 3-qubit entanglement. Invitation to MM talks: W6.00003 (w/ transmon qubits, and joint readout) Also, Z26.00003 (w/ phase qubits, and individual qubit readouts)

59. Measure cavity transmission: “Are all 3 qubits in the ground state?” Cavity transmission Frequency Joint 3-Qubit Readout Same concept: Use cavity transmission to learn a collective property, now of the three qubits

60. The trick still works! Combining joint readout with one-qubit “analysis” gives access to all 3-qubit Pauli operators, only more rotations are necessary. Example: extract no pre-rotation:  on Q1 and Q2: on Q1 and Q3: on Q2 and Q3: Joint Readout 3-Qubit state tomography

61. Outlook

62. 1 2 3 4 5 6 7 8 9 10 3 15 63 255 1,023 4,095 16,383 65,535 262,143 1,048,575 Is full state tomography scalable? In ion traps: Haffner, Nature (2005) Steffen et al., Science (2006) Filipp et al., PRL (2009) DiCarlo et al., Nature (2009) Chow et al., arXiv 0908.1955 Steffen et al., PRL (2007) M. Metcalfe, Ph.D. Thesis (2008) Neeley et al., ???? DiCarlo et al., (2010) Number of qubits Parameters in density matrix

63. How far will we push full state tomography? Up to N=8 qubits 10 hours of ensemble averaging!

64. Summary in pictures Few-qubit processors bbased on circuit QED C-phase gate Joint Readout Generation and detection of entanglement vs

65. Related MM 2010 talks by Yale cQED team L. DiCarlo et al. Realization of Simple Quantum Algorithms with Circuit QED W6.00003: Thursday 03/18, 12:27 PM, Room 253 J. M. Chow et al. Quantifying entanglement with a joint readout of superconducting qubits T29.00012: Wednesday 03/17, 4:42 PM, Room C123 M. D. Reed et al. Eliminating the Purcell effect in cQED NEW TITLE: ???? Y26.00013: Friday 03/19, 10:24 AM, Room D136 Lev S. Bishop et al. Latching behavior of the driven damped Jaynes-Cummings model in cQED T29.00012: Wednesday 03/17, 5:06 PM, Room C123

66. Acknowledgements Expt: Jerry Chow Matthew Reed Blake Johnson Luyan Sun Joseph Schreier Andrew Houck David Schuster Johannes Majer Luigi Frunzio Theory: Jay Gambetta Lev Bishop Andreas Nunnenkamp Jens Koch Alexandre Blais PI’s: Robert Schoelkopf, Michel Devoret, Steven Girvin Funding sources:

67. Circuit QED team members ‘09 Eran Ginossar Luigi Frunzio David Schuster Blake Johnson Jens Koch Hanhee Paik Andreas Nunnenkamp Matt Reed Steve Girvin Jerry Chow Leo DiCarlo Lev Bishop Adam Sears Andreas Fragner Luyan Sun Rob Schoelkopf Jay Gambetta