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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. 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 Defining a quantum processor Related MM 2010 talks: W6.00003 & T29.00012 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 oneand two- (perhaps more-) qubit gates, following an algorithm. 2009 model 2 qubits quantum processors based on circuit QED 2010 model 4 qubits Anatomy of quantum algorithm Register qubits Us Uf Ua M Ancilla qubits initialize create superposition encode function in a unitary involves entanglement between qubits analyze the function measure 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 Example: The Grover quantum search 1, x x0 f ( x) 0, x x0 Problem: Find the unknown root x0 of f Given: a quantum oracle that implements the unitary U f x (1) f ( x )0 x Quantum circuit diagram of the algorithm 0 Ry /2 Ry /2 ij 0 Ry /2 00 Ry /2 quantum oracle Ry /2 Ry /2 Quantum debugging Half-way along the algorithm, the two-qubit state ideally maximally entangled! The quality of a quantum processor relies on producing nearperfect 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 Ry /2 0 b 0 /2 Ry Ry /2 c 10 d e Ry /2 00 /2 Ry f /2 Ry g ideal 1 00 11 2 But what is quantum entanglement? `Entanglement is simply Schrodinger’s name for superposition in a multi-particle system.` GHZ, Physics Today 1993 Wavefunction description of pure two-qubit states for N=2 qubits: c00 00 c01 01 c10 10 c11 11 2 2 complex numbers - • normalization 1 • irrelevant global phase A pure 2-qubit pure state is fully described by 6 real #s When are two qubits entangled? Two qubits are entangled when their joint wavefunction cannot be separated into a product of individual qubit wavefunctions 1 2 vs 1a 2a 1b 2b Some common terms: Unentangled = separable = product state Entangled = non-separable = non-product state When are two qubits entangled? Entangled? State 1 1 11 1 01 10 2 no The singlet 1 00 01 10 11 2 1 1 0 1 0 1 2 2 1 00 01 10 11 2 yes no yes Quantifying entanglement Two qubits in a pure state c00 00 c01 01 c10 10 c11 11 are entangled if they have nonzero concurrence C C ( ) 2 c00c11 c01c10 11 1 01 10 2 1 00 01 10 11 2 1 00 01 10 11 2 C 0 C 1 C 0 C 1 Quantifying entanglement – pure states The concurrence is an entanglement monotone: 0 C ( ) 2 c00c11 c01c10 1 If C ( a ) C ( b ) , we say state a is more entangled than state b If C ( a ) 1, we say state a is maximally entangled Example: 1 01 10 is maximally entangled. The Bell state 2 1 1 1 The state 00 10 11 , with C 2 / 3 , is 3 3 3 entangled, but less entangled than a Bell state. Reality check #1 : quantum states never pure! Density-matrix description of mixed states for a pure state pi i i for a mixed state i pi [0,1], pi 1 i Dim[ ] 2 N 2 N 22 N complex #s - † Tr[ ] 1 Hermitian Unity trace Fully describing a 2-qubit mixed state requires 15 real #s Warning: The decomposition pi i i is generally not unique! i Example: 1 p1 , 2 1 p2 , 2 City-scape (Manhattan plot) 1 00 2 11 1 0 -1 00 01 10 11 and 1 p1 , 2 1 p2 , 2 Im( ) Re( ) 1 1 00 11 2 1 2 00 11 2 00 01 10 11 Give the same Quantifying entanglement – mixed states The concurrence of a mixed state is given by C min pi C i , i where the minimization is over all possible decompositions of Can show: C ( ) max 0, 1 2 3 4 The i are the eigenvalues of the matrix in decreasing order, and * YY YY Yes, it’s completely non-intuitive! A very non-linear function of Hill and Wootters, PRL (2007) Wootters, PRL (2008) Horodecki4, RMP (2009) 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 qubit case • Can we generalize the Bloch vector to N>1? • Answer: the Pauli set • Extracting usual metrics from the Pauli set Geometric visualization for N=1: The Bloch sphere 1 j j 2 j{i , x , y , z} z I 1 ( X , Y , Z ) ( X ,Y , Z ) 2 2 v Bloch v Bloch Bloch vector (NMR) y x v Bloch 1 pure state v Bloch 1 mixed state Is there a similarly intuitive description for N=2 qubits? State tomography of qubit decay Steffen et al., PRL (2006) Related MM talks: Tomography of qutrits: Z26.00013 (phase qubits) Z26.00012 (transmon qubits) Generalizing the Bloch vector: The Pauli set 1 1 2 1 2 j k j k 4 j ,k{i , x , y , z} The Pauli set P = 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, II 1 always The two-qubit Pauli set can be divided into three sections: • Polarization of Qubit 1 • Polarization of Qubit 2 P1 XI , YI , ZI P2 IX , IY , IZ • Two-qubit correlations P12 XX , XY , XZ , YX , YY , YZ , ZX , ZY , ZZ Visualizing N=2 states: product states 00 Re( ) 1 0 P1 P2 P12 -1 00 01 10 11 10 11 00 01 1 0 1 0 1 2 1 0 -1 00 01 10 11 10 11 00 01 Im( ) Visualizing N=2 states: Bell states 1 00 11 2 Re( ) 1 0 -1 00 01 10 11 10 11 00 01 1 00 01 10 11 2 1 0 -1 00 01 10 11 10 11 00 01 Im( ) Extracting usual metrics from the Pauli Set State purity: 1 Tr[ ] P P 4 2 Fidelity to a target state T F T T : 1 P PT 4 Concurrence: P12 P12 1 C ( ) 2 C0 Warning: for pure states only C 1 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 A two-qubit quantum processor flux bias lines fR 1 ns resolution VR DC - 2 GHz fC VH fL fC T = 13 mK VL cavity: “entanglement bus,” driver, & detector transmon qubits Two-qubit joint readout via cavity 1 1 fC VH Cavity transmission fC 11 10 01 1 0 00 2 R 2 L 2 L 2 R “Strong dispersive cQED” Schuster, Houck et al., Nature (2007) Frequency Qubit-state dependent cavity resonance Prepare 00 and measure 01 10 11 Chow et al., arXiv 0908.1955 Two qubit joint readout via cavity fC Cavity transmission fC 11 10 01 00 VH Measure cavity transmission: “Are qubits both in their ground state?” M 00 00 ZI IZ ZZ Frequency Direct access to qubit-qubit correlations with a single measurement channel! Direct access to qubit-qubit correlations How to reconstruct the two-qubit state from an ensemble measurement of the form VH 1 ZI 2 IZ 12 ZZ ? Answer: Combine joint readout with one-qubit pre-rotations Example: How to extract YZ Rx /2 Rx0, Apply Apply /2 , then measure: x /2 + & , then measure: x x R R R Joint Dispersive Readout 1 ZI 2 IZ 12 ZZ 1 YI 2 IZ 12 YZ 1 YI 2 IZ 12 YZ 212 YZ It is possible to acquire correlation info. with one measurement channel! All Pauli set components are obtained by linear operations on raw data. First demonstration of 2Q entanglement in SC qubits: Steffen et al., Science (2006) C 0.55 Producing entanglement with the circuit QED processor Related MM 2010 talks: W6.00003 & T29.00012 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 VR One-qubit gates: X and Y rotations fR z Preparation 1-qubit rotations Measurement y x cavity Q VR sin(2 f q t ) One-qubit gates: X and Y rotations fR z Preparation 1-qubit rotations Measurement y x cavity I VR cos(2 f q t ) One-qubit gates: X and Y rotations fR fL z Preparation 1-qubit rotations Measurement y x cavity Q VR sin(2 f q t ) One-qubit gates: X and Y rotations fL z Preparation 1-qubit rotations Measurement y x cavity I VR J. Chow et al., PRL (2009) Fidelity = 99% cos(2 f q t ) Two-qubit gate: turn on interactions VR Use control lines to push qubits near a resonance: Conditional phase gate A controlled z z interaction, a la NMR cavity VR Two-excitation manifold of system 02 Two-excitation manifold 11 02 11 Two-excitation manifold • Avoided crossing (160 MHz) 11 02 20 Transmon “qubits” have multiple levels… Strauch et al. PRL (2003): proposed using interactions with higher levels for computation in phase qubits Adiabatic phase gate tf 02 a 2 f a (t )dt 11 t0 2-excitation manifold 11 ei11 11 tf 11 10 01 2 (t )dt t0 01 1-excitation manifold 10 01 e 10 e i01 i10 01 10 Implementing C-Phase with 1 fancy pulse 00 01 1 0 0 ei01 U 0 0 0 0 10 0 0 ei10 0 11 30 ns 00 01 10 i11 11 e 0 0 0 00 01 10 11 Adjust timing of flux pulse so that only quantum amplitude of 11 acquires a minus sign: 1 0 U 0 0 0 1 0 0 0 0 00 0 0 01 1 0 10 0 1 11 11 How to create a Bell state using C-Phase 0 0 Ry /2 rotation on each qubit yields a maximal superposition: 1 1 0 1 0 1 00 01 10 11 2 2 Apply C-Phase entangler: 1 0 0 0 0 1 0 0 1 00 01 10 11 0 0 1 0 2 0 0 0 1 No longer a product state! How to create a Bell state using C-Phase 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 00 01 10 11 0 2 1 RY ( / 2) rotation on LEFT qubit yields: 1 Bell 10 01 2 Two-qubit entanglement experiment Ry /2 0 Ry /2 10 Ry /2 0 Re( ) Ideally: wavefunction 1 00 11 2 density matrix 1 00 00 00 11 11 00 11 11 2 Expt’l state tomography Entanglement on demand 0 Ry /2 Ry /2 ij 0 Ry /2 Re( ) Bell state Fidelity Concurrence 00 11 91% 88% 00 11 94% 94% 01 10 90% 86% 01 10 87% 81% Switching to the Pauli set Related MM 2010 talks: T29.00012 & W6.00003 Experimental N=2 Pauli sets 10 1 00 01 10 11 2 1 00 11 2 1 00 01 10 11 2 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 Ry 0 0 0 Ry /2 Ry 10 0 Ry /2 ~98% visibility for separable states, ~92% visibility for entangled states Working around Concurrence Related MM 2010 talks: W6.00003 &Y26.00013 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 C Can we characterize entanglement without reliance on C ? Can we place lower bounds on state tomography? C without performing full Witnessing entanglement with a subset of the Pauli set An entanglement witness is a unity-trace observable a positive expectation value for all product states. W 0 W 0 W with state is entangled, guaranteed. witness simply doesn’t know Witness also gives a lower bound on C: 2 W C An optimal witness of an entangled state gives the strictest lower bound on C Witnesses require only a subset of the Pauli set! Example: W opt 1 II XX YY ZZ 4 Is the optimal witness for the singlet Witnessing entanglement Entanglement is witnessed (by some witness) at all angles in entangled-state movie! 0 Ry /2 Ry 10 0 Ry /2 Witnessing entanglement Control: entanglement is not witnessed by any witness in the separable movie! 0 0 Ry Bell inequalities: another form of entanglement witness z’ z CHSH operator = entanglement witness x’ CHSH XX XZ ' ZX ZZ ' x Clauser, Horne, Shimony & Holt (1969) CHSH XX XZ ' ZX ZZ ' Bell state Separable bound: CHSH 2 state is clearly highly entangled! not test of hidden variables… (loopholes abound) Also UCSB group, closing detection loophole, Ansmann et al., Nature (2009) 2.61 0.04 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) The density matrix for N=3 1 1 2 3 1 2 3 j k l j k l 8 j ,k ,l{i , x , y , z} Knowing the three-qubit state = expectation values of 63 Pauli operators P1 XII , YII , ZII P2 IXI , IYI , IZI P3 IIX , IIY , IIZ Polarization of qubit 1 Polarization of qubit 2 Polarization of qubit 3 2-qubit correlations P12 XXI , XYI , XZI , YXI , YYI , YZI , ZXI , ZYI , ZZI P13 XIX , XIY , XIZ , YIX , YIY , YIZ , ZIX , ZIY , ZIZ P23 IXX , IXY , IXZ , IYX , IYY , IYZ , IZX , IZY , IZZ 3-qubit correlations P123 XXX , XXY , XXZ , ... , ZZX , ZZY , ZZZ Example Pauli set for N=3: product state 111 P1 P2 P3 P12 P13 P123 P23 Re( ) 1 0 -1 000 111 111 000 Example Pauli set for N=3: 3-qubit entangled state The GHZ state Greenberger Horne Zeilinger P1 P2 P3 P12 P13 1 000 111 2 P123 P23 1 Re( ) 0 -1 000 111 111 000 Upgrading the processor: more qubits Q2 Q1 Q3 Q4 We have recently extended C-Phase gates and joint qubit readout to produce and detect 3-qubit entanglement. DiCarlo et al., (2010) Reed et al., (2010) Invitation to MM talks: W6.00003 (w/ transmon qubits, and joint readout) Also, Z26.00003 (w/ phase qubits, and individual qubit readouts) Joint 3-Qubit Readout Cavity transmission Same concept: Use cavity transmission to learn a collective property, now of the three qubits 111 000 Frequency Measure cavity transmission: “Are all 3 qubits in the ground state?” M 000 000 ZII IZI IIZ ZZI ZIZ IZZ ZZZ 3-Qubit state tomography The trick still works! Combining joint readout with one-qubit “analysis” gives access to all 3-qubit Pauli operators, only more rotations are necessary. Rx0, Joint Readout Rx0, M Rx0, 000 000 Example: extract ZZZ Rx ( ) on Q1 and Q2: Rx ( ) on Q1 and Q3: Rx ( ) on Q2 and Q3: no pre-rotation: ZII ZII ZII ZII IZI IZI IZI IZI IIZ IIZ IIZ IIZ ZZI ZZI ZZI ZZI ZIZ ZIZ ZIZ ZIZ IZZ IZZ IZZ IZZ ZZZ ZZZ ZZZ ZZZ 4 ZZZ Outlook Is full state tomography scalable? Number of qubits N 1 2 3 4 5 6 7 8 9 10 Parameters in density matrix 22 N 1 Steffen et al., PRL (2007) 3 M. Metcalfe, Ph.D. Thesis (2008) 15 Steffen et al., Science (2006) 63 Filipp et al., PRL (2009) 255 DiCarlo et al., Nature (2009) 1,023 Chow et al., arXiv 0908.1955 4,095 Neeley et al., ???? 16,383 DiCarlo et al., (2010) 65,535 262,143 In ion traps: Haffner, Nature (2005) 1,048,575 How far will we push full state tomography? Up to N=8 qubits 10 hours of ensemble averaging! Summary in pictures C0 1 2 C 1 vs 1a 2a 1b 2b C-phase gate Joint Readout Few-qubit processors bbased on circuit QED Generation and detection of entanglement Related MM 2010 talks by Yale cQED team J. M. Chow et al. Quantifying entanglement with a joint readout of superconducting qubits T29.00012: Wednesday 03/17, 4:42 PM, Room C123 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 M. D. Reed et al. Eliminating the Purcell effect in cQED NEW TITLE: ???? Y26.00013: Friday 03/19, 10:24 AM, Room D136 L. DiCarlo et al. Realization of Simple Quantum Algorithms with Circuit QED W6.00003: Thursday 03/18, 12:27 PM, Room 253 Acknowledgements PI’s: Robert Schoelkopf, Michel Devoret, Steven Girvin 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 Funding sources: Circuit QED team members ‘09 Lev Bishop Matt Rob Leo Reed DiCarlo Hanhee Schoelkopf Paik Steve Girvin Jens Koch Jerry Chow Andreas Fragner Blake Johnson Jay Gambetta Eran Ginossar Adam Sears Luyan Sun David Schuster Luigi Frunzio Andreas Nunnenkamp