Lecture 6, March 30, 2017 - Quantum Device Lab

||

Last week:

Brief revisit of the Transmon qubit Gate charge insensitivity Anharmonicity and driving of qubit Tuning by magnetic flux

Qubit-Qubit coupling in circuit QED 2-qubit gates by virtual photon interaction

This week:

Qubit-Qubit coupling in circuit QED The controlled NOT gate Creating entangled states The Toffoli gate

Single Photons generation and Qubit Photon Entanglement

30-Mar-17Andreas Wallraff, Quantum Device Lab 161

Lecture 6, March 30, 2017

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

||

Reading: BooksHaroche, S. & Raimond, J.-M.; Exploring the Quantum: Atoms, Cavities, and Photons, Oxford University Press, New York, USA, (2006)


Nielsen, M. A. & Chuang, I. L. Quantum Computation and Quantum Information, Cambridge University Press (2000)

Gerry, C. & Knight, P. L. Introductory Quantum Optics, Cambridge University Press (2005)

||

Reading: Papers, Reviews, Other Material


Read (some of) the research papers mentioned on the slides.

• First read abstract and discussion/summary• Try to understand essence of the paper reading

it once, not caring for the details• Don’t be put off by not understanding

everything immediately• Read a different paper to get another authors

view of the same subject• Research you will do in the lab (Semester

Thesis, Master Thesis) aims at going beyond (all of) the papers that you read in preparation.

E.g.:A. Blais, et al., PRA 69, 062320 (2004)

Quantum Machines: Measurement and Control of Engineered Quantum Systems: Lecture Notes of the Les Houches Summer School: Volume 96, July 2011chapters: (link on QIP II web site)• 3 Circuit QED: superconducting qubits coupled to

microwave photons S. M. Girvin Department of Physics, Yale University

• 4 Quantum logic gates in superconducting qubitsJ. M. Martinis Department of Physics, University of California, Santa Barbara, CA 93111, USA

• 6 Readout of superconducting qubitsD. Esteve Quantronics Group Service de Physique de l’Etat Condensé/IRAMIS/DSM (CNRS URA 2464) CEA Saclay

ETH Zurich, TU Delft, (Imperial College), RWTH Aachen IDEA league summer school series.Lectures slides, videos, homework sets: http://www.qei.ethz.ch/education/IDEA-School.htmlhttp://qischoolsidea.wikispaces.com/home

||

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 softwarehttp://www.economist.com/topics/quantum-computinghttp://www.economist.com/

||

Industry & StartupsIBM Qhttp://research.ibm.com/ibm-q/

Google/UCSBhttp://web.physics.ucsb.edu/~martinisgroup/

D-Wave Systemshttps://www.dwavesys.com/

Microsofthttps://stationq.microsoft.com/

Rigetti Computinghttp://rigetti.com/

Intelhttps://phys.org/news/2015-09-intel-mn-quantum.html


||

Virtual Photon Exchange Controlled by Detuning

Salathé et al., PRX 5, 021027 (2015)

qubit 1 qubit 2

Freq

uenc

y

evolution of states during interaction:

J

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

Initial state intermediate state final state


||

4 Qubit Device with Nearest Neighbor Resonator-Mediated Coupling

Salathé et al., PRX 5, 021027 (2015)

four qubitsfour resonators→ mediate couplingtwo readout linesfour microwave drive linesfour flux bias lines

→ tune qubit transition

1 mm30-Mar-17Andreas Wallraff, Quantum Device Lab 167

||

Virtual Photon Coupling (01-10): Calculation

Salathé et al., PRX 5, 021027 (2015)

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).


||

Virtual Photon Coupling (01-10): Experimental Data

Salathé et al., PRX 5, 021027 (2015)

maximally entangled stateIndicated by

state fidelity: 99.7 %

Maximal entanglement at (2n+1) π/2 for n = 0, 1,2,3,…

• Maximally mixed single qubit states• Maximal two qubit correlators• Maximal negativity• High fidelity with expected state

• Experimental data extracted from 2-qubit quantum state tomography


||

Virtual Photon Coupling (01-10): Calculation

Salathé et al., PRX 5, 021027 (2015)

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


||

Virtual Photon Coupling (01-10): Experimental Data

Salathé et al., PRX 5, 021027 (2015)

state fidelity: F = 99.4 %

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


||

Universal Two-Qubit Non-Adiabatic Controlled Phase Gate (11-20)

proposal: F. W. Strauch et al., Phys. Rev. Lett. 91, 167005 (2003).first implementation: L. DiCarlo et al., Nature 460, 240 (2010).

Interaction mediated by virtual photon exchange

through resonator

Tune levels into resonance using magnetic field

qubit A qubit B

Full 2π rotation induces phase factor -1


Make use of qubit states beyond 0, 1

||

Universal Two-Qubit Controlled Phase Gate

proposal: F. W. Strauch et al., Phys. Rev. Lett. 91, 167005 (2003).first implementation: L. DiCarlo et al., Nature 460, 240 (2010).

C-Phase gate:

Universal two-qubit gate. Used together with single-qubitgates to create any quantum operation.

qubit A qubit B

Qubits in states 01, 10 and 00 do not interact

and thus acquire no phase shift


Make use of qubit states beyond 0, 1

||

2011

Two-excitation manifold

Two-Excitation Manifold of System

Strauch et al., PRL (2003): proposed using interactions with higher levels for computation in phase qubitsslide adapted from L. DiCarlo (TUD)

• Spectroscopy of higher excited states

• Avoided crossing (160 MHz)

11 02↔

Flux bias on right transmon (a.u.)


||

Flux bias right transmon (a.u.)

01

10

11

2-excitationmanifold

1-excitationmanifold

ζ

2001 10f f+

11 1e1 11 iϕ→

01 e10 10iϕ→

10 1e0 01 iϕ→

0

2 ( )ft

a at

f t dtϕ π δ= − ∫

Adiabatic Controlled Phase Gate

slide credit: L. DiCarlo (TUD)

0

11 10 01 2 ( )ft

t

t dtϕ ϕ ϕ π ζ= + − ∫


||

1 0 0 00 1 0 00 0 1 00 0 0 1

U

−

Adjust timing of flux pulse so that only quantum amplitude of acquires a minus sign:

11

01

10

11

1 0 0 00 0 00 0 00 0

ˆ

0

i

i

i

eU

ee

ϕ

ϕ

ϕ

00 1001 11

00

10

01

11

Implementing the C-Phase Gate with One Flux Pulse

slide credit: L. DiCarlo (TUD)

How to verify the operation of this gate?


||

Process Tomography: C-Phase Gate

arbitrary quantum process

decomposed into operator basis positive semi definite Hermitian matrix characteristic for the process

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

χ


||

Re[χ] (|Im[χ]|<0.08)

Process Tomography of a C-NOT Gate

Measured χ-matrix:Controlled-NOT gate

=


||

GHZ State with 3 Qubits

This data: J. Heinsoo et al., ETHZF = 88%: DiCarlo et al. Nature 467, (2010)F = 62%: Neeley et al. Nature 467, (2010)F = 96%: Barends et al. Nature 508, (2014)

Fid(𝜎𝜎,𝜌𝜌) = Tr 𝜌𝜌𝜎𝜎 𝜌𝜌†2

= 88.9% (MLE)

Real Imaginary

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

GHZ class states, e.g. |000>+|111> created using:

• single qubit gates• C-PHASE gates


||

GHZ-like State with 4 Qubits

This data: J. Heinsoo et al., ETHZF = 86.3%: Barends et al. Nature, 2014, 508

Real Imaginary

Fid(𝜎𝜎,𝜌𝜌) = Tr 𝜌𝜌𝜎𝜎 𝜌𝜌†2

= 74.8% (MLE)

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


||

A Three Qubit Gate: The Toffoli Gateproposed 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)


||

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

Implementation of a Toffoli Gate


||

Circuit Diagram

A. Fedorov et al., Nature (London) 481, 170 (2012)

Initial state: Final state

B

A

C

π 3π

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

same amount of resources, more efficient


||

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


||

Process Tomography of Toffoli Gate


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

• 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)]

• Fidelity

68.5 +- 0.5 %

69 +- 3 %


||

Truth Table of Toffoli Gate


• characterizes the action of the Toffoli gate on the basis input states

• Fidelity


This implementation:• Realization and full characterization of 3 qubit

Toffoli gate, also with efficient process certificationA. 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 protocolM. 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)

||

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.


The DiVincenzo Criteria

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

||

Protocols: TeleportationL. Steffen et al., Nature 500, 319 (2013) M.. Baur et al., PRL 108, 040502 (2012)

Quantum Computing with Superconducting Circuits

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)

Deutsch & Grover Algorithms, Toffoli GateL. DiCarlo et al., Nature 460, 240 (2009)L. DiCarlo et al., Nature 467, 574 (2010)A. Fedorov et al., Nature 481, 170 (2012)

Error CorrectionM. Reed et al., Nature 481, 382 (2012)

Corcoles et al., Nat. Com. 6, 6979 (2015) Ristè et al., Nat. Com. 6, 6983 (2015) Kelly et al., Nature 519, 66-69 (2015)

Adiabatic Quantum ComputationR. Barends et al., Nature, 534, 222-226 (2016)


||

Quantum Simulation Applications with Superconducting Circuits

Salathe et al., PRX 5, 021027 (2015)

Solid State and Atomic Physics: Digital simulation of exchange, Heisenberg, Ising spin models

Solid State and Atomic Physics:

two-mode fermionic Hubbard models

Barends et al., Nat. Com. 6, 7654 (2015)

Photonics:Analog simulations with cavity

and/or qubit arraysHouck et al., Nat. Phys. 8, 292 (2012)

Raftery et al., PRX 4, 031043 (2014)

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

Quantum Chemistry: simulation of correlated systems using variational approach


||

Quantum Optics with Superconducting CircuitsStrong Coherent CouplingChiorescu et al., Nature 431, 159 (2004)Wallraff et al., Nature 431, 162 (2004)Schuster et al., Nature 445, 515 (2007)

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

Microwave Fock and Cat StatesHofheinz et al., Nature 454, 310 (2008)

Hofheinz et al., Nature 459, 546 (2009)Kirchmair et al., Nature 495, 205 (2013)Vlastakis et al., Science 342, 607 (2013)

Wang et al., Science 352, 1087 (2016)

Waveguide QED –Qubit Interactions in Free Space

Astafiev et al., Science 327, 840 (2010)I.-C. Hoi et al. PRL 107, 073601 (2011)

van Loo et al., Science 342, 1494 (2013)

Root n NonlinearitiesFink et al., Nature 454, 315 (2008)

Deppe et al., Nat. Phys. 4, 686 (2008)Bishop et al., Nat. Phys. 5, 105 (2009)


||

Hybrid Systems with Superconducting CircuitsQuantum Dots: CNT, Gate Defined 2DEG, nanowires Delbecq et al., PRL 107, 256804 (2011)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: Mi et al., Science 355, 156 (2017)Stockklauser et al., PRX 7, 011030 (2017)Bruhat et al., arXiv:1612.05214 (2016)

Spin Ensembles: e.g. NV centersSchuster et al., PRL 105, 140501 (2010)Kubo et al., PRL 105, 140502 (2010)

Nano-MechanicsTeufel et al., Nature 475, 359 (2011)Zhou et al., Nat. Phys. 9, 179(2013)

Polar Molecules, Rydberg, BECRabl et al, PRL 97, 033003 (2006)

Andre et al, Nat. Phys. 2, 636 (2006)Petrosyan et al, PRL 100, 170501 (2008)

Verdu et al, PRL 103, 043603 (2009)

Rydberg AtomsHoganet al., PRL 108, 063004 (2012)

zx

vz


||

105 Improvement in Coherence Time in 13 Years

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

||

Towards Quantum Error Correction

DM

DT

DB

encode

P

P

Discretize, signal errors using quantum parity checks

0

0

X

X

X

X

X

X

• IBM: Corcoles et al., Nat. Com. 6, 6979 (2015), ArXiv:1410.6419

• QuTech: Ristè, Poletto, Huang et al., Nat. Com. 6, 6983 (2015), ArXiv:1411.5542

• UCSB/Google: Kelly et al., Nature 519, 66-69 (2015), ArXiv:1411.7403

Slide courtesy of L. DiCarlo

AM

AT

000 111α β+

||

Design


||

Fabrication


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

Control


Automation


Cryogenics

||

Quantum Science and Engineering


Documents

Lecture 6, March 30, 2017 - Quantum Device Lab