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Lecture 6, March 30, 2017

Last week: This week:

. Brief revisit of the . Qubit-Qubit coupling in circuit QED . Gate charge insensitivity . The controlled NOT gate . Anharmonicity and driving of qubit . Creating entangled states . Tuning by magnetic flux . The Toffoli gate . Qubit-Qubit coupling in circuit QED . Single Photons generation and Qubit Photon . 2-qubit gates by virtual photon interaction Entanglement

J. Koch et al., Phys. Rev. A 76, 042319 (2007) A. Blais, et al., Phys. Rev. A 69, 062320 (2004)

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 161 Reading: Books

Nielsen, M. A. & Chuang, I. L. Haroche, S. & Raimond, J.-M.; Gerry, C. & Knight, P. L. Introductory Quantum Computation and Exploring the Quantum: Atoms, , Cambridge Quantum Information, Cambridge Cavities, and Photons, Oxford University Press (2005) University Press (2000) University Press, New York, USA, (2006)

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 162 Reading: Papers, Reviews, Other Material Read (some of) the research papers mentioned on Quantum Machines: Measurement and Control of the slides. Engineered Quantum Systems: Lecture Notes of the • First read abstract and discussion/summary Les Houches Summer School: Volume 96, July 2011 chapters: (link on QIP II web site) • Try to understand essence of the paper reading 3 Circuit QED: superconducting coupled to it once, not caring for the details • microwave photons • Don’t be put off by not understanding S. M. Girvin Department of Physics, everything immediately • 4 Quantum logic gates in superconducting qubits • Read a different paper to get another authors J. M. Martinis Department of Physics, University of view of the same subject California, Santa Barbara, CA 93111, USA Research you will do in the lab (Semester • 6 Readout of superconducting qubits • D. Esteve Quantronics Group Service de Physique de Thesis, Master Thesis) aims at going beyond l’Etat Condensé/IRAMIS/DSM (CNRS URA 2464) CEA (all of) the papers that you read in preparation. Saclay E.g.: ETH Zurich, TU Delft, (Imperial College), RWTH A. Blais, et al., PRA 69, 062320 (2004) Aachen IDEA league summer school series. Lectures slides, videos, homework sets: http://www.qei.ethz.ch/education/IDEA-School.html http://qischoolsidea.wikispaces.com/home Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 163 The Economist Quantum leaps • An entangled web: The promise of quantum encryption • Cue bits: Why all eyes are on quantum computers • Here, there and everywhere: Quantum technology is beginning to come into its own • Commercial breaks: The uses of quantum technology • Program management: Quantum computers will require a whole new set of software http://www.economist.com/topics/quantum-computing http://www.economist.com/

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 164 Industry & Startups Google/UCSB http://web.physics.ucsb.edu/~martinisgroup/ IBM Q http://research.ibm.com/ibm-q/

Microsoft D-Wave Systems https://stationq.microsoft.com/ https://www.dwavesys.com/ Rigetti Computing http://rigetti.com/ Intel https://phys.org/news/2015- 09-intel-mn-quantum.html

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 165 Virtual Photon Exchange Controlled by Detuning

qubit 1 qubit 2 Frequency tuning by magnetic flux: • tunable interaction time τ • compensation of dynamic phase

Frequency J

evolution of states during interaction: Initial state intermediate state final state

Salathé et al., PRX 5, 021027 (2015) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 166 4 Qubit Device with Nearest Neighbor Resonator-Mediated Coupling

four qubits four resonators → mediate coupling two readout lines four microwave drive lines four flux bias lines → tune qubit transition

1 mm Salathé et al., PRX 5, 021027 (2015) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 167 Virtual Photon Coupling (01-10): Calculation

Initial condition: • Qubit 1: 0 Qubit 2: 1

Single qubit Bloch spheres • Pure state on surface • Fully mixed state in center Pauli operator expectation values • Single qubit IX, IY, IZ and XI, YI, ZI • Two qubit correlators XX, XY, XZ, YX, … Entanglement measure: negativity N G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).

Salathé et al., PRX 5, 021027 (2015) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 168 Virtual Photon Coupling (01-10):

Experimental Data maximally entangled state Maximal entanglement at (2n+1) π/2 Indicated by for n = 0, 1,2,3,… • Maximally mixed single qubit states • Maximal two qubit correlators • Maximal negativity • High fidelity with expected state

state fidelity: 99.7 %

• Experimental data extracted from 2-qubit tomography

Salathé et al., PRX 5, 021027 (2015) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 169 Virtual Photon Coupling (01-10): Calculation

Initial conditions: • Qubit 1 : 0 Qubit 2: (0+1)

Salathé et al., PRX 5, 021027 (2015) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 170 Virtual Photon Coupling (01-10): Experimental Data

Maximal entanglement at (2n+1) π/2 for n = 0, 1, 2, 3, … • Partially mixed single qubit states • Non-zero two qubit correlators • Non-zero negativity • High fidelity with expected state

state fidelity: F = 99.4 %

Salathé et al., PRX 5, 021027 (2015) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 171 Universal Two-Qubit Non-Adiabatic Controlled Phase Gate (11-20) Make use of qubit states beyond 0, 1

qubit A qubit B

Interaction mediated Full 2π rotation induces by virtual photon exchange phase factor -1 through resonator

Tune levels into resonance using magnetic field

proposal: F. W. Strauch et al., Phys. Rev. Lett. 91, 167005 (2003). first implementation: L. DiCarlo et al., 460, 240 (2010). Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 172 Universal Two-Qubit Controlled Phase Gate Make use of qubit states beyond 0, 1

qubit A qubit B

Qubits in states 01, 10 and 00 do not interact and thus acquire no phase shift

C-Phase gate:

Universal two-qubit gate. Used together with single-qubit gates to create any quantum operation. proposal: F. W. Strauch et al., Phys. Rev. Lett. 91, 167005 (2003). first implementation: L. DiCarlo et al., Nature 460, 240 (2010). Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 173 Two-Excitation Manifold of System

• Spectroscopy of higher 20 excited states Two-excitation 11 manifold

• Avoided crossing (160 MHz) 110↔ 2

Flux bias on right transmon (a.u.)

Strauch et al., PRL (2003): proposed using interactions with higher levels for computation in phase qubits slide adapted from L. DiCarlo (TUD) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 174 Adiabatic Controlled Phase Gate

20 t f + ff01 10 ϕ= − πδ 11 aa2∫ f () t dt t0

2-excitation ϕ manifold ζ 111 → ei 11 1 t f ϕ=+− ϕ ϕ πζ 11 10 01 2∫ ()t dt 10 t0 iϕ 1-excitation 10 → e 01 10 manifold 01 ϕ 001 → ei 10 1

Flux bias right transmon (a.u.)

Andreas Wallraff, Quantum Device Lab slide credit: L. DiCarlo (TUD) | 30-Mar-17 | 175 Implementing the C-Phase Gate with One Flux Pulse

00 01 10 11 10 0 000 100 0 iϕ  0e 01 0001 010 0 ˆ ˆ   U  ϕ U 00ei 10 010 001 0  iϕ11 − 00 0 e 11 000 1

Adjust timing of flux pulse so that only quantum amplitude of 1 1 acquires a minus sign: How to verify the operation of this gate?

slide credit: L. DiCarlo (TUD) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 176 Process Tomography: C-Phase Gate arbitrary decomposed into operator basis quantum χ positive semi definite process Hermitian matrix characteristic for the process

Controlled phase gate Measured χ-matrix: Re[χ] (|Im[χ]|<0.04)

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 177 Process Tomography of a C-NOT Gate

Controlled-NOT gate Measured χ-matrix: Re[χ] (|Im[χ]|<0.08)

=

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 178 GHZ State with 3 Qubits

Protocol Measured (color) and ideal (wireframe) density matrix:

Real Imaginary

GHZ class states, e.g. |000>+|111> created using: • single qubit gates • C-PHASE gates

This data: J. Heinsoo et al., ETHZ

F = 88%: DiCarlo et al. Nature 467, (2010) Fid( , ) = Tr 2 = 88.9% (MLE) F = 62%: Neeley et al. Nature 467, (2010) † 𝜎𝜎 𝜌𝜌 𝜌𝜌𝜎𝜎 𝜌𝜌 F = 96%: Barends et al. Nature 508, (2014) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 180 GHZ-like State with 4 Qubits

Protocol Measured (color) and ideal (wireframe) density matrix:

Real Imaginary

Fid( , ) = Tr 2 = 74.8% (MLE) † This data: J. Heinsoo et al., ETHZ 𝜎𝜎 𝜌𝜌 𝜌𝜌𝜎𝜎 𝜌𝜌 F = 86.3%: Barends et al. Nature, 2014, 508 Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 181 A Three Qubit Gate: The Toffoli Gate

proposed by Tommaso Toffoli in 1980 • any reversible computation can be performed with only the Toffoli gate

function: • inverts qubit C only if qubits A and B are in selected basis states

applications: • for universal reversible classical computation • for simplification of complex quantum circuits • used in quantum error-correction schemes (essential for any practical quantum processor)

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 182 Implementation of a Toffoli Gate

with only single and two-qubit gates requires: • 6 CNOT gates • 10 single qubit gates

• Inefficient decomposition • Not ideal at limited coherence

Alternative Approach suggested by T. C. Ralph et. al., PRA 75, 022313 (2007): • use higher levels (qutrits) for efficient decomposition

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 183 Circuit Diagram Alternative approach: use qubit-qutrit gates for the more efficient decomposition! • CC-PHASE – inverts the sign for only one basis state • Equivalent to Toffoli up to single qubit rotations

Initial state: Final state

A π 3π B

C

same amount of resources, more efficient

A. Fedorov et al., Nature (London) 481, 170 (2012) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 184 Implementation sequence of: • five resonant single qubit microwave pulses • three single qubit flux pulses realizing … • … qubit-qubit and qubit-qutrit gates making use of avoided crossing between 11 and 20 states

A. Fedorov et al., Nature (London) 481, 170 (2012) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 185 Process Tomography of Toffoli Gate

Fully characterizes the process by evaluating χ-matrix (ML)

• Fidelity 69 +- 3 %

• Monte Carlo process certification does not rely on maximum-likelihood procedures [da Silva et al., PRL 107, 210404 (2011), Steffen et al., Phys. Rev. Lett. 108, 260506 (2012)] 68.5 +- 0.5 % A. Fedorov et al., Nature (London) 481, 170 (2012) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 187 Truth Table of Toffoli Gate

• characterizes the action of the Toffoli gate on This implementation: the basis input states • Realization and full characterization of 3 qubit • Fidelity Toffoli gate, also with efficient process certification A. Fedorov et al., Nature (London) 481, 170 (2012) L. Steffen et al., Phys. Rev. Lett. 108, 260506 (2012)

Related work: • Toffoli gate used for correcting an artificial error in an error correction protocol M. D. Reed et al., Nature (London) 482, 382 (2012) • Realization of Toffoli-class gate with only two qubits (used resonator as 3rd qubit) and limited characterization (phase fidelity) M. Mariantoni et al., Science 334, 61 (2011)

A. Fedorov et al., Nature (London) 481, 170 (2012) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 188 The DiVincenzo Criteria for Implementing a quantum computer in the standard (circuit approach) to quantum information processing (QIP):

#1. A scalable physical system with well-characterized qubits. #2. The ability to initialize the state of the qubits. #3. Long (relative) decoherence times, much longer than the gate-operation time. #4. A universal set of quantum gates. #5. A qubit-specific measurement capability. plus two criteria requiring the possibility to transmit information:

#6. The ability to interconvert stationary and mobile (or flying) qubits. #7. The ability to faithfully transmit mobile qubits between specified locations.

David P. DiVincenzo, The Physical Implementation of Quantum Computation, arXiv:quant-ph/0002077 (2000)

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 189 Quantum Computing with Superconducting Circuits Protocols: Teleportation L. Steffen et al., Nature 500, 319 (2013) M.. Baur et al., PRL 108, 040502 (2012)

Architectures: Circuit QED A. Blais et al., PRA 69, 062320 (2004) A. Wallraff et al., Nature 431, 162 (2004) M. Sillanpaa et al., Nature 449, 438 (2007) H. Majer et al., Nature 449, 443 (2007) M. Mariantoni et al., Science 334, 61 (2011) R. Barends et al., Nature 508, 500 (2014)

Error Correction M. Reed et al., Nature 481, 382 (2012) Deutsch & Grover Algorithms, Toffoli Gate Corcoles et al., Nat. Com. 6, 6979 (2015) Adiabatic Quantum Computation Ristè et al., Nat. Com. 6, 6983 (2015) L. DiCarlo et al., Nature 460, 240 (2009) R. Barends et al., Nature, 534, 222-226 (2016) Kelly et al., Nature 519, 66-69 (2015) L. DiCarlo et al., Nature 467, 574 (2010) A. Fedorov et al., Nature 481, 170 (2012)

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 190 Quantum Simulation Applications with Superconducting Circuits

Quantum Chemistry: simulation of correlated systems using variational approach Solid State and Atomic Physics: two-mode fermionic Hubbard models Barends et al., Nat. Com. 6, 7654 (2015)

Eichleret al., PRX 5, 041044 (2015) O’Malley et al., PRX 6, 031007 (2016)

Solid State and Atomic Physics: Digital simulation of exchange, Photonics: Heisenberg, Ising spin models Analog simulations with cavity and/or qubit arrays Houck et al., Nat. Phys. 8, 292 (2012) Raftery et al., PRX 4, 031043 (2014)

Salathe et al., PRX 5, 021027 (2015) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 192 Quantum Optics with Superconducting Circuits Strong Coherent Coupling Chiorescu et al., Nature 431, 159 (2004) Wallraff et al., Nature 431, 162 (2004) Schuster et al., Nature 445, 515 (2007)

Root n Nonlinearities Microwave Fock and Cat States Fink et al., Nature 454, 315 (2008) Hofheinz et al., Nature 454, 310 (2008) Deppe et al., Nat. Phys. 4, 686 (2008) Hofheinz et al., Nature 459, 546 (2009) Bishop et al., Nat. Phys. 5, 105 (2009) Kirchmair et al., Nature 495, 205 (2013) Vlastakis et al., Science 342, 607 (2013) Wang et al., Science 352, 1087 (2016)

Parametric Amplification & Squeezing Castellanos-Beltran et al., Nat. Phys. 4, 928 (2008) Abdo et al., PRX 3, 031001 (2013)

Waveguide QED – Qubit Interactions in Free Space Astafiev et al., Science 327, 840 (2010)

I.-C. Hoi et al. PRL 107, 073601 (2011) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 193 van Loo et al., Science 342, 1494 (2013) Hybrid Systems with Superconducting Circuits Polar Molecules, Rydberg, BEC Rabl et al, PRL 97, 033003 (2006) Quantum Dots: CNT, Gate Defined 2DEG, Andre et al, Nat. Phys. 2, 636 (2006) nanowires Petrosyan et al, PRL 100, 170501 (2008) Delbecq et al., PRL 107, 256804 (2011) Verdu et al, PRL 103, 043603 (2009) Frey et al., PRL 108, 046807 (2012) Petersson et al., Nature 490, 380 (2012) Radiation Emission: Liu et al., Science 347, 285 (2015) Stockklauser et al., PRL 115, 046802 (2015) Strong Coupling Cavity QED: Spin Ensembles: e.g. NV centers Mi et al., Science 355, 156 (2017) Schuster et al., PRL 105, 140501 (2010) Stockklauser et al., PRX 7, 011030 (2017) Kubo et al., PRL 105, 140502 (2010) Bruhat et al., arXiv:1612.05214 (2016)

Nano-Mechanics Teufel et al., Nature 475, 359 (2011) Zhou et al., Nat. Phys. 9, 179(2013) Rydberg Atoms z Hoganet al., PRL 108, v 063004 (2012) x z

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 194 105 Improvement in Coherence Time in 13 Years

M. Devoret, R. Schoelkopf Science 339, 1169 (2013) Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 195 Towards

X AT DT X DM DB X • IBM: Corcoles et al., Nat. AM Com. 6, 6979 (2015), αβ000+ 111 Discretize, signal errors ArXiv:1410.6419 encode using quantum parity checks • QuTech: Ristè, Poletto, Huang et al., Nat. Com. 6, 0 X Pˆ 6983 (2015), ArXiv:1411.5542 X Pˆ • UCSB/Google: Kelly et al., Nature 519, 66-69 (2015), 0 X ArXiv:1411.7403

Slide courtesy of L. | DiCarlo| Design

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 197 Fabrication

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 198 Control

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 199 Automation

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 200 Cryogenics

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 201 Quantum Science and Engineering

Andreas Wallraff, Quantum Device Lab | 30-Mar-17 | 202