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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)
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Reading: BooksHaroche, S. & Raimond, J.-M.; Exploring the Quantum: Atoms, Cavities, and Photons, Oxford University Press, New York, USA, (2006)
30-Mar-17Andreas Wallraff, Quantum Device Lab 162
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)
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Reading: Papers, Reviews, Other Material
30-Mar-17Andreas Wallraff, Quantum Device Lab 163
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
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The Economist
30-Mar-17Andreas Wallraff, Quantum Device Lab 164
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/
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 165
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 166
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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
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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).
30-Mar-17Andreas Wallraff, Quantum Device Lab 168
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 169
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Virtual Photon Coupling (01-10): Calculation
Salathé et al., PRX 5, 021027 (2015)
Initial conditions:• Qubit 1 : 0 Qubit 2: (0+1)
30-Mar-17Andreas Wallraff, Quantum Device Lab 170
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 171
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 172
Make use of qubit states beyond 0, 1
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 173
Make use of qubit states beyond 0, 1
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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.)
30-Mar-17Andreas Wallraff, Quantum Device Lab 174
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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= + − ∫
30-Mar-17Andreas Wallraff, Quantum Device Lab 175
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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?
30-Mar-17Andreas Wallraff, Quantum Device Lab 176
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 177
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Re[χ] (|Im[χ]|<0.08)
Process Tomography of a C-NOT Gate
Measured χ-matrix:Controlled-NOT gate
=
30-Mar-17Andreas Wallraff, Quantum Device Lab 178
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 180
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 181
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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)
30-Mar-17Andreas Wallraff, Quantum Device Lab 182
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 183
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 184
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Implementation
A. Fedorov et al., Nature (London) 481, 170 (2012)
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
30-Mar-17Andreas Wallraff, Quantum Device Lab 185
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Process Tomography of Toffoli Gate
A. Fedorov et al., Nature (London) 481, 170 (2012)
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 %
30-Mar-17Andreas Wallraff, Quantum Device Lab 187
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Truth Table of Toffoli Gate
A. Fedorov et al., Nature (London) 481, 170 (2012)
• characterizes the action of the Toffoli gate on the basis input states
• Fidelity
30-Mar-17Andreas Wallraff, Quantum Device Lab 188
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)
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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.
30-Mar-17Andreas Wallraff, Quantum Device Lab 189
The DiVincenzo Criteria
David P. DiVincenzo, The Physical Implementation of Quantum Computation, arXiv:quant-ph/0002077 (2000)
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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)
30-Mar-17Andreas Wallraff, Quantum Device Lab 190
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 192
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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)
30-Mar-17Andreas Wallraff, Quantum Device Lab 193
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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
30-Mar-17Andreas Wallraff, Quantum Device Lab 194
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105 Improvement in Coherence Time in 13 Years
M. Devoret, R. Schoelkopf Science 339, 1169 (2013) 30-Mar-17Andreas Wallraff, Quantum Device Lab 195
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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+
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Design
30-Mar-17Andreas Wallraff, Quantum Device Lab 197
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Fabrication
30-Mar-17Andreas Wallraff, Quantum Device Lab 198
|| 30-Mar-17Andreas Wallraff, Quantum Device Lab 199
Control
|| 30-Mar-17Andreas Wallraff, Quantum Device Lab 200
Automation
|| 30-Mar-17Andreas Wallraff, Quantum Device Lab 201
Cryogenics
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Quantum Science and Engineering
30-Mar-17Andreas Wallraff, Quantum Device Lab 202