Production and detection of few-qubit entanglement in circuit-QED processors

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 148:30 a.m. - 12:30 p.m.

2. Outline • What is a quantum processor, and why must it produce • (and why must we detect) highly-entangled qubit states? • What is quantum entanglement? • 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 • Going beyond two qubits • Outlook

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. 2009 model 2 qubits quantum processors based on circuit QED 2010 model 4 qubits

5. Anatomy of quantum algorithm Register qubits Us Uf Ua M Ancilla qubits measure encode function in a unitary analyze the function create superposition initialize involves entanglement between qubits involves disentangling the 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

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

7. Quantum debugging 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. oracle b c d f g e

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

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

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

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

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

13. The concurrence is an entanglement monotone: Quantifying entanglement – pure states 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 for a pure state for a mixed state Hermitian complex #s - Unity trace Fully describing a 2-qubit mixed state requires 15 real #s

16. Warning: The decomposition is generally not unique! Example: City-scape (Manhattan plot) 1 0 -1 00 and 01 10 11 10 11 01 00 Give the same

17. The concurrence of a mixed state is given by Quantifying entanglement – mixed states where the minimization is over all possible decompositions of Can show: The are the eigenvaluesof the matrix in decreasing order, and Yes, it’s completely non-intuitive! A very non-linear function of Hill and Wootters, PRL (2007) Wootters, PRL (2008) Horodecki4, RMP (2009)

18. Getting : Two-qubit state tomography • 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! • Review: the N=1 qubitcase • Can we generalize the Bloch vector to N>1? • Answer: the Pauli set • Extracting usual metrics from the Pauli set

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

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

21. Generalizing the Bloch vector: The Pauli set • 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 One of them, always The two-qubit Pauli set can be divided into three sections: • Polarization of Qubit 1 • Polarization of Qubit 2 • Two-qubit correlations

22. Visualizing N=2 states: product states 1 0 -1 00 01 10 11 11 10 01 00 1 0 -1 00 01 10 11 11 10 01 00

23. Visualizing N=2 states: Bell states 1 0 -1 00 01 10 11 11 10 01 00 1 0 -1 00 01 10 11 11 10 01 00

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 Filipp et al., PRL (2009) Related MM 2010 talks: W6.00003 & T29.00012 DiCarlo et al., Nature (2009) Chow et al., arXiv 0908.1955

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

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

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

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

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

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

32. Spectroscopy of qubit2-cavity system right qubit Qubit-qubit swap interaction Majer et al., Nature(2007) left qubit Cavity-qubit interaction Vacuum Rabi splitting Wallraff et al., Nature(2004) cavity

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

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

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

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

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

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

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

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

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

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

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

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

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 • a test for systematic errors in tomography, such • as offsets and amplitude errors in Pauli bars ~98% visibility for separable states, ~92% visibility for entangled states

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 withoutperforming 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 : An optimal witness of an entangled state gives the strictest lower bound on Witnesses require only a subset of the Pauli set! Example: Is the optimal witness for the singlet