Playing Quantum Games with Superconducting Circuits Quantum Computing with... · Bad Honnef Physics...
Transcript of Playing Quantum Games with Superconducting Circuits Quantum Computing with... · Bad Honnef Physics...
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Quantum Computing
with
Superconducting Circuits
Rudolf Gross
Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften
andTechnische Universität München
Bad Honnef Physics School onQuantum Technologies5 – 10 August 2018
09.08.2018/RG - 2www.wmi.badw.de Bad Honnef Physics School on Quantum Technologies, 5 – 10 August 2018, ©WMI
F. Deppe, K. Fedorov, H. Huebl, A. Marx
Michael FischerPhilip SchmidtPeter EderBehdad Ghaffari
Stephan PogorzalekDaniel SchwienbacherEdwar XieMinxing Xu
• U. Las Heras, M. Sanz, E. Solano, R. Di Candia• (Bilbao)• T. Ramos, J. J. Garcia-Ripoll (Madrid)• M. Hartmann (Edinburgh)• I. Cirac (MPQ Garching)
theory support
• M. Möttönen (Aalto)
• K. Inomata, T. Yamamoto, Y. Nakamura(NEC, Tokyo University)
• M. Aspelmeyer (U. Vienna)
• E. Weig, J.P. Kotthaus (U. Konstanz/LMU Munich)
experimental partners
former group members:Jan Goetz (Aalto University, Finland)Elisabeth Hoffmann (attocube)Matteo Mariantoni (Waterloo, Canada)Edwin P. Menzel (Rohde & Schwarz)Tomasz Niemczyk (BMW Group)Manuel Schwarz (IAV GmbH)Thomas Weißl (Inst. Néel, Grenoble)Karl-Friedrich Wulschner (Univ. of Vienna)Ling Zhong (Yale University)Christoph Zollitsch (UC London)
WMI Team & Partners
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Intel Core i9 (14 nm)
Why Quantum Computing ?
What is next?
new architectures neuromorphic computing quantum computing
end of Moore‘s law
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Quantum Advantage
• many problems in science and industrial applications/business are too complex for classical computing systems
• Quantum Computing may help to solve “hard” problems
algebraic algorithms e.g. factorization, cryptography, systems of equations, …
optimization problems e.g. logistics (traveling salesman), business processes, risk analysis, …
simulation of quantum systems e.g. quantum chemistry, material science, drug design, …
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Simulation of Quantum SystemsHow much memory is needed to store a quantum state?How much time does it take to calculate dynamics of a quantum system?
by courtesy of S. Filipp (IBM Research)
Richard Feynman (1981):
“...trying to find a computer simulation of physics, ……, you'd better make it quantum mechanical, …..”
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Information is inevitably tied to a physical representation and therefore to restrictions and possibilities related to the laws of physics and the parts available in the universe.
Quantum mechanical superpositions of information bearing states can be used, and the real utility of that needs to be understood. Quantum parallelism in computation is one possibility and will be assessed pessimistically.
Rolf Landauer, Physics Letters A 217, 188 - 193 (1996)
Information is Physical
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M. H. Devoret, R. J. Schoelkopf, Science 339, 1169 – 1174 (2013)J. I. Cirac, H. J. Kimble, Nature Photonics 11, 18 – 20 (2017)
Roadmap to Quantum Computing
we do not go beyond this point in this tutorial
status in 2013
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Jay M. Gambetta, Jerry M. Chow & Matthias Steffen npj Quantum Information 3, 2 (2017)
Roadmap to Quantum Computing
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• Superconducting quantum circuits
• Trapped ion systems
• Optical lattices
• Quantum dot computer, spin-based
• Quantum dot computer, spatial-based
• Nuclear magnetic resonance on molecules in solution (liquid-state NMR)
• Solid-state NMR, Kane quantum computers
• Electrons-on-helium quantum computers
• Cavity quantum electrodynamics (c-QED)
• Molecular magnets
• Fullerene-based ESR quantum computer
• Linear optical quantum computer
• Diamond-based quantum computer (NV centers)
• …..
Hardware Platforms
….. many proposals, only some will be successful !!
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Quantum Computing: Implementation Proposals
NMR Photons Atoms Solid State
linear optics cavity QED
trapped ions optical lattices
semiconductors superconductorsNV centerselectrons on LHe
flux charge phasenuclearspins
electronspins
orbitalstates
molecularmagnets
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outline
• hype & hope: recent press releases
• solid state circuits go quantum
superconducting quantum electronics
advantages & drawbacks
• quantum computing
qubits, gates, readout, …
• experimental techniques
• outlook
Press Releases
Nov. 2017
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An IBM cryostat wired for a 50 qubit system.
IBM scientists have successfully built and measured a processor prototype with 50 quantum bits. It is the first time any company has built a quantum computer at this scale
Nov. 2017
IBM reached milestone in quantum computing
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Intel’s 17-qubit superconducting test chip for quantum computing has unique features for improved connectivity and better electrical and thermo-mechanical performance.
chip with advanced packaging delivered to QuTech
Intel Delivers 17-Qubit Superconducting Chip
Oct. 2017
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Intel's new superconducting quantum chip called Tangle Lake has enough qubits to make things very interesting from a scientific standpoint
Intel Fabricates Quantum Chip “Tangle Lake”
Jan. 2018
Google has lifted the lid on its new quantum processor, Bristlecone. The project could play a key role in making quantum computers "functionally useful."
72 qubit processor
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D-Wave 2000 Q:2 000 superconducting qubits,operating temperature: 30 mK
Quantum “Annealing” @ mK temperature
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Superconducting Quantum Circuits
Quant
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100 nm
Superconducting Quantum Circuits
Solid State CircuitsGo Quantum
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multi
electron, spin, fluxon, photon
devices
single/few
electron, spin, fluxon, photon
devices
quantum
electron, spin, fluxon, photon
devices
today near future far future
quantifiable,but not quantum
classicaldescription
quantumdescription
65 nm process 2005 superconducting qubitsingle electron transistor
PTBIntel
• quantumconfinement
• tunneling• …
• superposition of states• entanglement• quantized em-fields
WMI 20062 µm
... solid state circuits go quantum
quantum1.0 quantum2.0
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2nd Quantum Revolution
Quantum Information Theory
Solid-State Physics
Mathematics
…realization and full control of quantum systems !!
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David J. WinelandSerge Haroche
The Nobel Prize in Physics 2012 was awarded jointly to Serge Haroche and David J. Wineland "for ground-breaking experimental methods that enable measuring and manipulation of individual quantum systems"
Nobel Prize in Physics 2002
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http://web.physics.ucsb.edu/~martinisgroup/photos/BBCReZQu1103.jpg
quantumcomputing
http://research.physics.illinois.edu/QI/Photonics/research/
quantumcommunication
quantumsensing
Application fields
……. and more to come
quantummatter
https://www.mpq.mpg.de/4572004/profil
quantumsimulation
Credit: Francis Pratt / ISIS / STFC
quantummetrology
http://www.npl.co.uk/news/
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Research Programs in QST @ Europe
• United Kingdom: £ 270 million for a five-year program
• Netherlands: € 146 million for a ten-year program
• ERANet program QUANTERA – Cofund Initiative in quantum science and technologies (launch: January 2018): € 30 million
• ….
Planned Research Programs in QST
• QUTE Flagship: Call for Flagship ramp-up phase early in 2018: > € 1 000 million
• BMBF program QUTEGA: about € 300 million(Quantum Technology – Foundations & Applications)
starting in 2018
Public Funding of QST
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Is China winning race with the US to develop quantum computers?Chinese funding to research the next generation in computing may be dwarfing American efforts, according to US experts
PUBLISHED : Monday, 09 April, 2018, 12:37pm
China is building the world’s largest quantum research facility to develop a quantum computer and other “revolutionary” forms of technology that can be used by the military for code-breaking
or on stealth submarines, according to scientists and authorities involved in the project.
World’s Biggest Quantum Research Facility in China
Hefei Evening News
350.000 m2
China Quantum Center in Hefei – 10 Billion Euro Funding
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TUM gets new Center for Quantum Engineering
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Industry starts to get interested in important future technology field of Quantum Science & Technology
already existing industry efforts
• Spin qubits in semiconductors: Intel, HRL Laboratories, NTT, …
• Superconducting quantum circuits: Google, IBM, Intel, Anyon Systems Inc., Quantum Circuits Inc., Raytheon BBN Technologies, Rigetti Computing, …
• Superconducting quantum annealer: D-Wave
• Topological qubits: Microsoft
• Trapped ions interfaced with photons: Lockheed Martin
• …..
Interest of Industry
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The Belief in New Technologies
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..... have a large probability to be wrong !!
“I think there is a world market for maybe five computers”
Thomas J. Watson, chairman of IBM, 1943
“Whereas a calculator on the Eniac is equipped with 18000 vacuumtubes and weighs 30 tons, computers in the future may have only 1000 tubes and weigh only 1½ tons”
Popular Mechanics, March 1949
“There is no reason anyone would want a computer in their home”
Ken Olson, president, chairman and founder of DEC, 1977
Long-term Predictions.....
SuperconductingQuantum Electronics
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©WMI
Superconductivity in a Nutshell
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Superconducting Quantum Electronics
key physical ingredients
𝑰𝒔 = 𝑰𝒄𝑱 𝐬𝐢𝐧𝝋 ,𝒅𝝋
𝒅𝒕=𝟐𝒆𝑽
ℏර
𝑪
.
𝚲𝐉𝐬 ⋅ 𝒅ℓ + න
𝑭
.
𝑩 ⋅ ෝ𝒏 𝑑𝐹 = 𝒏𝚽𝟎
𝚿 𝐫, 𝐭 = 𝚿𝟎𝒆𝒊𝜽 𝒓,𝒕
𝚿 𝐫, 𝐭 𝟐 = 𝒏𝒔 𝒓, 𝒕
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Superconductivity in a Nutshell
http://www.wmi.badw.de/teaching/Lecturenotes/index.html
Lecture Notes on • Superconductivity & Low Temperature Physics I & II• Applied Superconductivity
Books • Festkörperphysik
R. Gross, A. Marx
Tutorials • Superconducting Quantum Circuits
• DPG-Frühjahrstagung Sektion Kondensierte Materie, Berlin, 11.03. – 16.03.2018• Summer School NanoQI 2017, 24.07. – 28.07.2017, San Sebastian, Spain
http://www.wmi.badw.de/teaching/Talks/index.html
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conventional electronic circuits
• classical physics• no quantization of fields• no superposition of states• no entanglement
... from Conventional to Quantum Electronics
𝑯 =𝚽𝟐
𝟐𝑳+𝑸𝟐
𝟐𝑪
2
1
quantum electronic circuits
• quantum mechanics• quantization of fields• coherent superposition of states• entanglement
2
1
Y. Nakamura et al., Nature 398, 786 (1999)
𝑯 =𝚽𝟐
𝟐𝑳+𝑸𝟐
𝟐𝑪= ℏ𝝎 ෝ𝒂† ෝ𝒂 +
𝟏
𝟐
𝚽, 𝑸 = 𝒊ℏ
LC oscillator
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Superconducting Quantum Electronics
capacitors
inductors
tunable, lossless, nonlinear inductor ≡ Josephson Junction
𝑰𝒔 = 𝑰𝒄𝑱 𝐬𝐢𝐧𝝋
𝝏𝝋
𝝏𝒕=𝟐𝒆𝑽
ℏ
𝑺𝟏
𝑺𝟐𝑰
𝜕𝐼𝑠𝜕𝑡
= 𝐼𝑐𝐽 cos𝜑𝜕𝜑
𝜕𝑡= 𝐼𝑐𝐽 cos𝜑
2𝑒𝑉
ℏ
𝑳𝑱 =𝑽
𝝏𝑰𝒔/𝝏𝒕=
ℏ
𝟐𝒆𝑰𝒄𝑱𝐜𝐨𝐬𝝋
𝑰𝒔𝟐 =𝑰𝒄𝑱 𝐬𝐢𝐧𝝋𝟐
𝑰𝒔𝟏 =𝑰𝒄𝑱 𝐬𝐢𝐧𝝋𝟏
𝐼𝑠 = 𝐼𝑐 cos 𝜋Φ
Φ0sin
𝜑1 + 𝜑22
𝚽 𝝋⋆𝑺𝟏
𝑺𝟐
𝑺𝟏
𝑺𝟐
𝑳𝑱(𝚽) =𝑽
𝝏𝑰𝒔/𝝏𝒕=
ℏ
𝟐𝒆𝑰𝒄(𝚽)𝐜𝐨𝐬𝝋⋆
𝑰𝒄(𝚽)
nonlinearity:
tunability:
𝑰𝒄 = 𝟐𝑰𝒄𝑱
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𝑬𝐩𝐨𝐭 = 𝑬𝑱 𝟏 − 𝐜𝐨𝐬 𝝋𝑬𝐩𝐨𝐭 = 𝑬𝑱 𝟏 − 𝐜𝐨𝐬 𝝋 − 𝑰𝝋
0.0 0.5 1.0 1.5 2.0-8
-6
-4
-2
0
2
/ 2
Ep
ot /
EJ0
I = 0
I = 0.5 Ic
I = Ic
𝜑/2𝜋
Josephson Junction (JJ)
full quantum treatment (quantum2.0):
𝑯 = 𝑬𝑱 𝟏 − 𝐜𝐨𝐬 ෝ𝝋 + 𝑬𝑪 𝑵𝟐
𝝓, 𝑸 = 𝒊ℏ𝝓 =
ෝ𝝋
𝟐𝝅𝚽𝟎
≡ position ≡ momentum
quasi-classical treatment (quantum1.0)
external force
nonlinear quantum harmonic oscillator
𝑸 = 𝑵 𝟐𝒆(𝟐𝒆)𝟐
𝟐𝑪
𝚽𝟎𝑰𝒄 𝚽
𝟐𝝅
classical motion of “phase particle”in tilt washboard potential
𝚽𝟎𝑰𝒄 𝚽
𝟐𝝅
𝟐𝑬𝑱
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JJ: Classical vs. Quantum Treatment
𝑬𝑱
𝟐𝑬𝐉
classical treatment valid for 𝟐𝑬𝑱
ℏ𝝎𝒑≃
𝑬𝑱
𝑬𝑪
𝟏/𝟐≫ 𝟏 (level spacing ≪ 𝑘B𝑇, potential depth)
enter quantum regime by decreasing junction area 𝑨 and reducing 𝑻
harmonic oscillator potentialclose to minimum- level spacing: ℏ𝜔p- lowest energy: ℏ𝜔p/2
≃ ℏ𝝎𝐩/𝟐
≃ ℏ𝝎𝐩
𝑬𝑱 𝟏 − 𝐜𝐨𝐬𝝋
ℏ𝝎𝒑 =ℏ
𝑳𝑱𝑪= 𝟐𝑬𝑪𝑬𝑱
𝑬𝑱 =𝚽𝟎𝑰𝒄𝟐𝝅
∝ 𝑨
𝑬𝑪 =𝟐𝒆 𝟐
𝟐𝑪∝𝟏
𝑨
nanotechnology & low temperatures required
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1E-4 1E-3 0.01 0.1 1 1010
-2
10-1
100
101
102
10-2
10-1
100
101
102
Ai (m
2)
Ec / k
B
EJ0 / k
B
𝑬𝑱 ∝ 𝑨
𝑬𝑪 ∝ 𝟏/𝑨
𝑬𝑪 > 𝑬𝑱
𝑬𝑪 < 𝑬𝑱
JJ: Classical vs. Quantum Treatment
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harmonic LC oscillator
E
|0>|1>|2>|3>|4>|5>
|g>
|e>
„artificial solid-state atom“„artificial solid-state photon“
„quantum optics“ on a chip
quantum2-levelsystem
=qubit
Linear and Nonlinear Quantum Electronic Circuits
𝑯 = ℏ𝝎 ෝ𝒂†ෝ𝒂 +𝟏
𝟐𝑳𝑱 𝚽 =
𝚽𝟎
𝟐𝝅𝑰𝒄 𝐜𝐨𝐬 𝝅𝚽𝚽𝟎
tunable, anharmonic LC oscillator
E
tunable,lossless
Josephsoninductance
𝚽
𝑰tunable
Josephsonjunction
….
….
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photon box:
microwave resonator
artificial atom:
solid state quantum circuit
A. Wallraff et al., Nature 431, 162 (2004).S. Girvin, R. Schoelkopf, Nature 451, 664-669 (2008) .
anharmonic level structure(quantum two-level system: qubit)
quantum coherence(coherence time: < 500 µs)
persistent current flux qubit
coplanar waveguide (CPW) resonator
small mode volume(Vmod/l3 10-5 – 10-6)
high quality factor(Q 104 – 106)
Circuit QED
Tunable Artificial Atoms & Photon Boxes
75 µm
many more: quantronium, fluxonium, transmon, x-mon, …
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e.g. Kimble and Mabuchi groups at CaltechRempe group at MPQ Garching, ….
cavity QED natural atom in optical cavity
Rempe group
circuit QED solid state circuit in µ-wave cavity
e.g. Wallraff (ETH), Martinis (UCSB), Schoelkopf (Yale), Nakamura (Tokyo), ….
WMI
resonator QED
Cavity & Circuit QED
Gross group
MPQ
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geometrical representation on Bloch sphere
𝟏 or 𝒆
𝟎 or 𝒈
Quantum Bit
𝜳 = 𝐜𝐨𝐬𝜽
𝟐𝟏 + 𝒆𝒊𝝋 𝐬𝐢𝐧
𝜽
𝟐𝟎
𝝋(𝒕) phase coherence
𝜽 𝒕 amplitude energy, population
Bloch angles:
𝜑 𝑡 =𝐸𝑒 − 𝐸𝑔
ℏ𝑡 = 𝜔ge𝑡
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two qubit gate (C-NOT)Bloch sphere 𝟏
𝟎
𝟏
𝟎
U1
𝟎 𝟏
readout
𝟏
𝟎
U1
𝟏
𝟎
Qubit
single qubit gate
U1
M.A. Nielsen, I.L. Chuang, Quantum Computation andQuantum Information (Cambridge Univ. Press, 2000)
Quantum Processor
𝜳 = 𝐜𝐨𝐬𝚯
𝟐𝟏 + 𝒆𝒊𝝓 𝐬𝐢𝐧
𝚯
𝟐𝟎
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superconductingquantum circuits
resonators qubits
couplers interferometers
switches JPAs
hybrid systems
qubits
Superconducting Quantum Electronics
resonators
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• High-Q quantum harmonic oscillators
linear circuit, not a qubit, not directly useful for quantum computation!
quantumsimulation of
manybodyHamiltonians
L C
ancilla qubit/nonlinearity explore quantumphysics (Fock states,
squeezing etc.)
typically longcoherence times
quantum memory
mediate couplingbetween qubits quantum bus
qubit readout(„dispersivereadout“)
identifydecoherence sources in
superconductingquantum circuits
indirect use
Superconducting Resonators
SC Quantum Circuits
Advantages & Drawbacks
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2D / h ≈ 10 GHz – 1 THz
ℏ𝝎𝐠𝐞 ≈ 1 – 10 GHz
normal metal superconductorEF
E E E
D >> kBT
|g>|e>co
ntinuum
of
exc
itations
ℏ𝝎𝐠𝐞
Superconducting Quantum Circuits
1. Macroscopic quantum nature of superconducting ground state 2. Energy gap in excitation spectrum
e.g. Al:2Δ
ℎ= 50 GHz
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• exploit macroscopic quantum nature of sc ground state andgap in excitation spectrum long coherence time
M. H. Devoret and R. J. Schoelkopf, Science 339, 1169 (2013)
Moore‘s Law for QubitLifetime
Superconducting Quantum Circuits
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Coherence Time
N. Ofek et al., Nature 536, 441–445 (2016)
Extending the lifetime of a quantum bit with error correction in superconducting circuits
Yale group:3D circuit QEDarchitecture
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2000 2004 2008 2012 201610-9
10-8
10-7
10-6
10-5
10-4
10-3
coh
ere
nce
tim
e (s
)
year
best T2 times
reproducible T2 times
CPB
quantronium
cQED
transmon
3D transmon
fluxonium
Coherence Time of SC-Qubits
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2 µm
flux qubit(Al)
transmon qubit(Al)
3D coplanarwaveguideresonator (Al)
coplanar waveguideresonator (Nb)
𝑻𝟐 ≲ 𝟏𝟎𝟎𝛍𝐬Megrant et al., APL 100, 113510 (2012)
𝑻𝟐 ≲ 𝟏𝟎𝐦𝐬M. Reagor et al., APL 102, 192604 (2013)
N. Ofek et al., Nature 536, 441 (2016)
𝑻𝟐 ≲ 𝟓𝟎𝟎𝛍𝐬
fabricate tailor-made quantum circuits
3. Established fabrication technology: thin film & nanotechnology
4. Superb design flexibility, tunability and scalability
Superconducting Quantum Circuits
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Superconducting Quantum Circuits
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Superconducting Quantum Circuits
quantum circuit with 8 resonators and 24 qubits (multiplexing readout)
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State preservation by repetitive error detection in a superconducting quantum circuit,J. Kelly et al., Nature 519, 66-69 (2015)
UCSB&
chip with9 X-mon qubits
Superconducting Quantum Circuit
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interaction energy = dipole moment ⋅ respective field
ℏ𝒈 = 𝐩 ⋅ 𝐄𝐫𝐦𝐬
ℏ𝒈 = 𝛍 ⋅ 𝐁𝐫𝐦𝐬
make electric (𝒑) or magnetic dipolemoment (𝝁) as big as possible
„big atoms“µm-sized circuits
„small cavities“quasi 1D cavities
make mode volume of cavity as small aspossible
𝑬𝐫𝐦𝐬𝐯𝐚𝐜 =
ℏ𝝎
𝝐𝟎𝑽𝐦𝐨𝐝
𝑩𝐫𝐦𝐬𝐯𝐚𝐜 =
𝝁𝟎ℏ𝝎
𝑽𝐦𝐨𝐝
5. Strong and ultrastrong coupling due to large dipole moments6. Fast manipulation by control pulses
Superconducting Quantum Circuits
T. Niemczyk et al., Nature Phys. 6, 772 (2010)
strong coupling: 𝒈 ≫ 𝜿, 𝜸(loss rates)
ultrastrong coupling: 𝒈 ≃ 𝝎𝒒, 𝝎𝒓
(system frequencies)
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superconducting resonator
T. Niemczyk et al., Nature Phys. 6, 772 (2010)
flux qubit
X. Zhou, et al., Nature Physics 9 , 179 (2013)
Si3N4 nanomechanical beam
Superconducting Quantum Circuits7. Realization of hybrid quantum systems by combination with
other degrees of freedom (e.g. spin, photonic, phononic, plasmonic, ….)
examples from WMI
Ch. Zollitsch et al., Appl. Phys. Lett. 107, 142105 (2015)
paramagnetic spins
phosphorousdonors in Si
H. Huebl et al., PRL 111, 127003 (2013)
ferrimagneticspin ensemble
YIG
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interferometer
nano-electromechanical circuit
Si3N4 nanobeam coupled to CPW resonator
Superconducting Quantum Circuits
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Superconducting Quantum Circuits
resonator atom
𝝎𝒓 𝝎𝐠𝐞
𝝎𝒓
𝟐𝝅≃
𝝎𝐠𝐞
𝟐𝝅≃ few GHz
1 GHz ↔ 50 mK
ℏ𝝎𝒓 ≃ 10-24 J
ultra-low temperatures
ultra-sensitive µ-wave experiments
challenges
nano-fabrication
1. Low energy scales
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Superconducting Quantum Circuits
2. Strong coupling to environment
protection against thermal microwave fieldse.g. cold attenuators, circulators, „Purcell filtering“ by cavity, ….
reduction of two-level fluctuatorse.g. substrate cleaning, avoid oxide layers, ….
strategies
optimum choice of operation pointe.g. operation @ qubit „sweet spot“, …
Qubit Design
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Qubit Design
Josephson Junction (JJ)≡
tunable, lossless nonlinear inductor+
parallel capacitor
Qubit Hamiltonian
𝑯 = 𝑬𝑱 𝟏 − 𝐜𝐨𝐬 ෝ𝝋 + 𝑬𝑪 𝑵𝟐+ 𝑬𝐞𝐱
𝝓, 𝑸 = 𝒊ℏ
𝝓 =ෝ𝝋
𝟐𝝅𝚽𝟎 ≡ position
≡ momentum𝑸 = 𝑵 𝟐𝒆
(𝟐𝒆)𝟐
𝟐𝑪
𝚽𝟎𝑰𝒄 𝚽
𝟐𝝅
𝑳𝑱
𝑪
bias circuit
Josephson Junction
externalcircuit
Qubit design ≡ engineering of the qubit Hamiltonian
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phase qubit
(EJ >> EC)current biased JJ
flux qubit
(EJ > EC)fluxon boxes
charge qubit
(EJ < EC)Cooper pair boxes
I I
I
V
J. Martinis (NIST) H. Mooij (Delft) V. Bouchiat (Quantronics)
nowadays superconducting qubit zoo is larger
transmon, camel-back, capacitively shunted 3JJ-FQB, quantronium, fluxonium…
“traditional” classification via 𝑬𝑱/𝑬𝑪 is increasingly difficult
Flexibility in Qubit Design (@2003)
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phase qubit
(EJ >> EC)current biased JJ
flux qubit
(EJ > EC)fluxon boxes
charge qubit
(EJ < EC)Cooper pair boxes
Qubit Design by Potential Engineering
engineered qubit potential
I
V
I
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𝑳𝑱
𝑪
add inductance
𝑯 = 𝑬𝑱 𝟏 − 𝐜𝐨𝐬 ෝ𝝋 + 𝑬𝑪 𝑵𝟐+ 𝑬𝐞𝐱
unsuitable for TLS!
add junctions
• flux/phase engineering
add bias current
add gate capacitor
• charge engineering
𝐸J naturally induces
anticrossings
add shunt capacitor change curvature ofcharge parabola
(3JJ flux qubit)
(rf SQUID &phase qubit)
(phase qubit)
(charge qubit)
(transmon qubit)
𝑳𝑱
𝑪
𝑳𝑱 𝜶𝑳𝑱
𝑳𝑱
𝑪
𝑳
𝑳𝑱
𝑪𝑱
𝑰
𝑵
𝑬𝑱𝑪𝐠
𝑽𝐠
𝑪𝑱
Qubit Design by Potential Engineering
𝑵
𝑬𝑱𝑪𝐠
𝑽𝐠
𝑪𝑱
𝑪𝒔
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𝑯𝐂𝐏𝐁 = 𝑬𝑪 𝑵 − 𝑵𝐠𝟐+ 𝑬𝐉 𝟏 − 𝐜𝐨𝐬 ෝ𝝋
• gate charge 𝑵𝐠 ≡𝑪𝐠𝑽𝐠
𝟐𝒆
induced by gate voltage 𝑽𝐠
adds/removes excess CP to/fromisland
classical quantity
may assume fractional values!
charge qubit – the Cooper pair box (CPB)
charge regime 𝑬𝑪 ≳ 𝑬𝐉 charge is good quantum number
Example: Cooper Pair Box (CPB)
𝑵−𝑵𝒈
𝑬𝑱𝑪𝐠
𝑽𝐠
𝑪𝑱
additional term due gate voltage small
𝑵 = −𝒊𝝏
𝝏𝝋
• charge energy: 𝑬𝑪 ≡(𝟐𝒆)𝟐
𝟐 𝑪𝒈+𝑪𝑱
superconductingisland
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tunable JJ (dc SQUID)
𝑪𝐉/𝟐
readout
• dc SQUID with 𝜷𝑳 ≪ 𝟏 tunable JJ with effective 𝑬𝑱(𝚽) and 𝑬𝑪
• gate voltage 𝑽𝐠 is control knob
• readout with additional JJ
detect number of excess Cooper pairs on island
• Josephson energy 𝑬𝑱 𝐜𝐨𝐬 ෝ𝝋
couples charge states/parabolas
avoided level crossings
island
𝑪𝐠
𝑽𝐠
typical prameters:𝐸𝐶/ℎ ≃ 5 GHz, 𝐸J/ℎ ≃ 5 GHz
𝑪𝐉/𝟐
charge qubit – the split Cooper pair box (CPB)
𝑯𝐂𝐏𝐁 = 𝑬𝑪 𝑵 − 𝑵𝐠𝟐+ 𝑬𝐉(𝚽) 𝟏 − 𝐜𝐨𝐬 ෝ𝝋
0.0 0.5 1.0 1.5 2.00.0
0.5
1.0
E / E
C
CVe / 2e
E
E+
E
𝑬𝐉
𝑬𝑪= 𝟎. 𝟎𝟔
𝑬𝐉(𝚽)
𝑵𝐠
𝐸/𝐸
𝐶
E+
𝑵=𝟎
𝑵=𝟏
𝑵=𝟐
Example: Cooper Pair Box (CPB)
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from the Cooper pair box to the transmon qubit
advantages of the CPB:
• simple design (2JJ, 𝛽𝐿 ≪ 1)
• level splitting Δ = 𝐸J ∝ 𝐼c(flux qubit: 𝛥 ∝ exp − Τ𝐸𝐽 𝐸𝐶 )
• voltages convenient for coupling to other qubits coupling to readout circuitry coupling to control signals
• large anharmonicity (few GHz)
• in first order insensitive to charge fluctuations at „sweet spot“ 𝑁g = 𝑛 +1
2
disadvantages:
• coherence times short due to susceptibility to 1/𝑓 charge noise
• in practice: coherence times of only a few 10 ns even at the sweet spot!
• idea flatten energy dispersion
Example: Cooper Pair Box (CPB)
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J. Koch et al., PRA 76, 042319 (2007).
take a CPB geometry andincrease Τ𝑬𝐉𝟎 𝑬𝑪 by shunt capacitor
charge dispersion decreases exponentially with Τ𝐸J 𝐸𝐶 less sensitive to charge noise
anharmonicity decreases only polynomially with Τ𝐸J 𝐸𝐶 optimum trade-off for Τ𝐸J 𝐸𝐶 ≈ 50
few hundreds of MHz anharmonicity left charge no longer good quantum number not tunable via gate voltage anymore tune via flux (dc SQUID)
transmission line shuntedplasma oscillation qubit
𝑵𝒈 = 𝑪𝒈𝑽𝒈/𝟐𝒆 𝑵𝒈 = 𝑪𝒈𝑽𝒈/𝟐𝒆
𝑵−𝑵𝒈
𝑬𝑱𝑪𝐠
𝑽𝐠
𝑪𝑱
𝑪𝒔superconductingisland
Example: Transmon Qubit
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• embed into a resonator for readout filtering control
2D geometries: 𝟏𝟎 − 𝟒𝟎 𝛍𝐬3D geometries: up to 𝟓𝟎𝟎 𝛍𝐬
• the transmon is currentlymost successful qubit withrespect to coherence times
• coherence of transmonsmostly limited by spuriousTLS (defects) in substrateand metal-substrate interface
J. Koch et al., Phys. Rev. A 76, 042319 (2007).
Example: Transmon Qubit
Qubit Coherence
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• interaction with environment for control purposes and readout
uncontrolled interactions (noise) also exist quantum effects (population oscillations, quantum interference,
superpositions, entanglement) unobservable after characteristic time after decoherence time quantum effects have decayed to Τ1 𝑒 of their original
level term “decoherence” originally only referred to phase nowadays sloppily comprises both phase and amplitude effects
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
𝝋(𝒕) phase coherence
𝜽 𝒕 amplitude energy, population
• ideal quantum system
completely isolated in reality, however, …
Quantum Coherence
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• population energy relaxation time 𝑻𝟏 decay from |e⟩ to |g⟩ nonadiabatic (irreversible) processes induced by high-frequency fluctuations (𝜔 ≈ 𝜔ge)
• phase pure dephasing time 𝑻𝝋 adiabatic (reversible) processes induced by low-frequency fluctuations (𝜔 → 0) often encountered: 1/f-noise real measurements always contain 𝑇1-effects
𝑻𝟐−𝟏 = 𝟐𝑻𝟏
−𝟏 + 𝑻𝝋−𝟏
nomenclature not very consistent in literature!
energy relaxation and dephasing
Qubit Lifetime
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
𝝋(𝒕) phase coherence
𝜽 𝒕 amplitude energy, population
𝝋 𝒕 =𝑬𝒆 − 𝑬𝒈
ℏ𝒕 = 𝝎𝐠𝐞𝒕
𝜹𝝋 = 𝜹𝝎𝐠𝐞𝑻𝝋 ≃ 𝟐𝝅
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Qubits strongly couple to electromagnetic fields decoherence due to environmental fluctuations
• place qubit in cavity: „Purcell filtering“
𝝎𝒓 𝝎𝝎𝒒
large detuning𝛿 = 𝜔𝑟 − 𝜔𝑞 ≫ 𝑔
strongly reduced „photon DOS“ @ 𝝎𝒒
𝜔𝑞/2𝜋
(GH
z)
𝜆 = 𝛿Φ/Φ0
𝜹𝝎𝒒
𝜹𝝎𝒒
• operate qubit @ sweet spot: 1st order coupling to noise vanishes
𝜹𝝎𝒒 =𝝏𝝎𝒒
𝝏𝝀𝜹𝝀 +
𝟏
𝟐
𝝏𝟐𝝎𝒒
𝝏𝝀𝟐𝜹𝝀𝟐 +⋯
1st ordercoupling
2nd ordercoupling
Qubit Lifetime
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𝑵𝒈 = 𝑪𝒈𝑽𝒈/𝟐𝒆 𝑵𝒈 = 𝑪𝒈𝑽𝒈/𝟐𝒆
Example: Transmon Qubit
𝜹𝝎𝒒 =𝟏
𝟐
𝝏𝟐𝝎𝒒
𝝏𝑵𝒈𝟐 𝜹𝑵𝒈
𝟐 is large𝝎𝒒
𝜹𝝎𝒒 =𝟏
𝟐
𝝏𝟐𝝎𝒒
𝝏𝑵𝒈𝟐 𝜹𝑵𝒈
𝟐 is small𝝎𝒒
𝑻𝟐 < 𝟓𝟎𝟎 𝛍𝐬
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Photon Statistics from Dephasing
𝑪 𝝉 ∝ 𝐕𝐚𝐫(𝒏)
𝜸𝝋𝒏 ∝ 𝐕𝐚𝐫(𝒏𝒓)
𝜸𝝋𝒏𝐭𝐡 𝒏𝒓 ∝ 𝒏𝒓
𝟐 + 𝒏𝒓𝜸𝝋𝒏𝐜𝐨𝐡 𝒏𝒓 ∝ 𝟐𝒏𝒓
𝜸𝝋𝒏𝐬𝐡𝐨𝐭 𝒏𝒓 ∝ 𝒏𝒓
thermal field
classicallimit
Poissonian
𝐕𝐚𝐫(𝐧) 𝒏𝟐 + 𝐧 𝒏𝟐 𝐧
J. Goetz et al., Phys. Rev. Lett. 118, 103602 (2017)
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• superconducting resonators
can be fabricated in various geometries with high quality factors
thin film based resonators: 𝑻𝟐 ≤ 𝟐𝟎 𝛍𝐬 (Nb on Si)
𝑻𝟐 ≤ 𝟏𝟎𝟎 𝛍𝐬 (Al on sapphire)
3D (bulk based) resonators: 𝑻𝟐 ≤ 𝟏𝟎𝐦𝐬 (Al)
• superconducting qubits
large variety of different qubits due to flexible potential engineering
transmon qubits presently show best coherence times: 𝑻𝟐 ≤ 𝟓𝟎𝟎 𝛍𝐬
𝑻𝟐 Times of Resonators & Qubits
reduction of two-level fluctuators is important taske.g. substrate cleaning, avoid oxide layers, remove surface spins….
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𝑻𝟐 > 𝟐𝟎 µs
https://quantumexperience.ng.bluemix.net/qx/devices
𝑻𝟐 Times of Qubits
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𝑻𝟐 Times of Qubits
Qubit
Connectivity𝑻𝟏 (µsec) 𝑻𝟐/𝑻𝟐
∗ (µsec)
Computer Qubit # Min Max Ave Min Max Ave Min Max Ave
IBM QX2 5 2 4 2.4 44.9 63.1 53.2 27.7 61.4 44.5
IBM QX4 5 2 4 2.4 36.2 54.8 48.1 14.9 55.7 31.1
IBM QX5 16 2 3 2.75 28.3 69.9 42.8 14.5 127.3 59.0
IBM QS1_1 20 2 6 3.9 47.5 173.5 80.1 15.6 94.2 41.3
Rigetti 19Q 19 1 3 2.21 8.2 31.0 20.3 4.9 26.8 10.9
Google indicated that their 𝑻𝟏 times are roughly 2-4x worse than IBM’s, but that their single and two qubit gate fidelities are 2-10x better, and their measurement fidelities are roughly 10x better.
https://quantumcomputingreport.com/scorecards/qubit-quality/
Single Qubit Gates
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important states on the Bloch sphere
𝒙
𝒚
𝒛
|𝐠⟩
|𝐞⟩
𝐞 + 𝐠
𝟐
𝐞 − 𝒊 𝐠
𝟐
𝐞 − 𝐠
𝟐
𝐞 + 𝒊 𝐠
𝟐
𝒙
𝒚
𝒛
|𝟏⟩
|𝟎⟩
𝟎 + 𝟏
𝟐
𝟎 − 𝒊 𝟏
𝟐
𝟎 − 𝟏
𝟐
𝟎 + 𝒊|𝟏⟩
𝟐
𝚿 𝒕 = 𝐜𝐨𝐬𝜽
𝟐𝐞 + 𝒆𝒊𝝋 𝐬𝐢𝐧
𝜽
𝟐𝐠 𝚿 𝒕 = 𝐜𝐨𝐬
𝜽
𝟐𝟎 + 𝒆𝒊𝝋 𝐬𝐢𝐧
𝜽
𝟐𝟏
ITphysics
Single Qubit Gates
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• single qubit gate
unitary operation 𝑈 on state |𝛹⟩ described by rotations on Bloch sphere + global phase technical implementation by microwave pulses
• rotation matrices
about x-axis 𝑅𝑥 𝛼 ≡ e−𝑖𝛼ෝ𝜎𝑥2 =
cos𝛼
2−𝑖 sin
𝛼
2
−𝑖 sin𝛼
2cos
𝛼
2
about y-axis 𝑅𝑦 𝛼 ≡ e−𝑖𝛼ෝ𝜎𝑦
2 =cos
𝛼
2−sin
𝛼
2
sin𝛼
2cos
𝛼
2
about z-axis 𝑅𝑧 𝛼 ≡ e−𝑖𝛼ෝ𝜎𝑧2 = 𝑒−𝑖 Τ𝛼 2 0
0 𝑒𝑖 Τ𝛼 2
In general unitary expressed by rotations
𝑈 = 𝑒𝑖𝛼 𝑅𝑧 𝛽 𝑅𝑦 𝛾 𝑅𝑧 𝛿 with 𝛼, 𝛽, 𝛾, 𝛿 ∈ ℝ
Z-Y decomposition (others possible) 𝛼 is a global phase (unobservable)
Single Qubit Gates
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examples for 1-qubit gates
NOT
graphical representation example
matrix representation (taken from QI theroy books) typically follow IT convention!
Single Qubit Gates
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unitary operations
𝑼|𝜳⟩ expressed via the Hermitian Pauli spin matrices 𝟏, ෝ𝝈𝒙, ෝ𝝈𝒚, ෝ𝝈𝒛
ෝ𝝈𝒙 ≡𝟎 𝟏𝟏 𝟎
ෝ𝝈𝒚 ≡𝟎 −𝒊𝒊 𝟎
ෝ𝝈𝒛 ≡𝟏 𝟎𝟎 −𝟏
𝟏 ≡𝟏 𝟎𝟎 𝟏
|𝐠⟩ and |𝐞⟩ are the eigenvectors of ෝ𝝈𝒛
pseudo spin
|𝜳⟩ is equivalent to spin wave function in external magnetic field
pseudo spin and Pauli matrices
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
Single Qubit Gatessu
pp
lem
enta
rym
ater
ial
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ෝ𝝈𝒙 ≡𝟎 𝟏𝟏 𝟎
ෝ𝝈𝒚 ≡𝟎 −𝒊𝒊 𝟎
ෝ𝝈𝒛 ≡𝟏 𝟎𝟎 −𝟏
𝟏 ≡𝟏 𝟎𝟎 𝟏
conventions: Pauli matrices and Bloch sphere
these definitons contain several conventions, such as
the global scaling factor the positon of the minus sign in 𝜎𝑧 here, we show two examples with fixed 𝜎𝑧
physics convention 𝐠 ≡𝟎𝟏
, 𝐞 ≡𝟏𝟎
ground state energy negative (more „physical“)
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝐞 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝐠
𝜳 𝒕 = 𝐜𝐨𝐬𝜽(𝒕)
𝟐𝟎 + 𝒆𝒊𝝋(𝒕) 𝐬𝐢𝐧
𝜽(𝒕)
𝟐𝟏
information theory (IT) convention
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press
𝟎 ≡𝟏𝟎
, 𝟏 ≡𝟎𝟏
ground state energy positive („unphysical“) easily generalized (more „logical”)
unless otherwise mentioned physics convention! formal resolution equate g to 1 and e to 0 used in this lecture!
Single Qubit Gatessu
pp
lem
enta
rym
ater
ial
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ෝ𝝈𝒙 ≡𝟎 𝟏𝟏 𝟎
ෝ𝝈𝒚 ≡𝟎 −𝒊𝒊 𝟎
ෝ𝝈𝒛 ≡𝟏 𝟎𝟎 −𝟏
𝟏 ≡𝟏 𝟎𝟎 𝟏
interpretation of the Pauli matrices
1 = |g⟩⟨g| + e e
ො𝜎𝑥 = ො𝜎− + ො𝜎+
ො𝜎𝑧 = |e⟩⟨e| − g g
ො𝜎𝑦 = 𝑖 ො𝜎− − ො𝜎+
• Pauli matrices can expressed in terms of projection operators
ො𝜎− = g e
ො𝜎+ = e g
induce transitions between |g⟩ and |e⟩
puts an excitation into the qubit
removes an excitation from the qubit
⟨ ො𝜎𝑧⟩ gives the qubit population
reflects normalization
• combination of basis definition and operator description in terms of projection operators matrix form of operators
• in this lecture, we fix the matrix definitions of the Pauli matrices “physical” intuition in g , e -notation notation consistent with Nielsen & Chuang and most physics papers!
g
e
g
e
g
e
g
e? ?
Single Qubit Gatessu
pp
lem
enta
rym
ater
ial
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Hadamard gate 𝑯 is of particular importance
𝑯 𝐠 =𝟏
𝟐( 𝐞 − |𝐠⟩)
𝑯 𝐞 =𝟏
𝟐(|𝐞⟩ + |𝐠⟩)
𝑯 ≡𝟏
𝟐
𝟏 𝟏𝟏 −𝟏
=𝟏
𝟐ෝ𝝈𝒙 + ෝ𝝈𝒛
𝒙
𝒚
𝒛
|𝐠⟩
|𝐞⟩
𝐞 + 𝐠
𝟐
𝐞 − 𝐠
𝟐
• physics convention
• applied to one of the basis states |g⟩ or |e⟩, it results in a superposition state ofthe basis states
𝑯 =𝟏
𝟐𝒆 𝒆 − 𝒈 𝒈 + 𝒆 𝒈 + 𝒈 𝒆
Single Qubit Gates
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https://quantumexperience.ng.bluemix.net/qx/editor
Single Qubit Gate Errors
gate fidelity > 99.7%
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Single Qubit Gate Errors
gate fidelity > 99.6%
https://quantumexperience.ng.bluemix.net/qx/devices
Two-Qubit Gates
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Two-Qubit Gates
• for quantum processor we need gates that perform conditional logic between qubits
state of one qubit depends on the state of another
• most relevant: Controlled-NOT or CNOT gate control
target
flips the target qubit only if the control qubit is |1⟩, otherwise it does nothing
controltarget
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• mitigate frequency crowding• optimal pulse control• flux-noise mitigation
Two-Qubit Gates
Idea: Parametric frequency modulation of a coupling element (flux tunable qubit)
realization by tunable coupler:
Bertet et al., Phys. Rev. B 73, 064512 (2006); Tian et al., NJP 10, 115001 (2008); Kapit et al., Phys. Rev. A 87, 062336 (2013); Roushan et al., Nat. Phys 13 146 (2017); McKay et al., Phys. Rev. Applied 6, 064007 (2016); Roth et al, arXiv:1708.02090 (2017)
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Two-Qubit Gates
tunable coupler
Bertet et al., Phys. Rev. B 73, 064512 (2006); Tian et al., NJP 10, 115001 (2008); Kapit et al., Phys. Rev. A 87, 062336 (2013); Roushan et al., Nat. Phys 13 146 (2017); McKay et al., Phys. Rev. Applied 6, 064007 (2016); Roth et al, arXiv:1708.02090 (2017)
𝑯𝐂 =ℏ𝝎𝟏
𝟐ෝ𝝈𝒛,𝟏 +
ℏ𝝎𝟐
𝟐ෝ𝝈𝒛,𝟐 +
ℏ𝑱(𝒕)
𝟐ෝ𝝈𝒙,𝟏ෝ𝝈𝒙,𝟐
𝑱 𝒕 =𝒈𝟏𝒈𝟐𝟐
𝟏
𝚫𝟏 𝒕+
𝟏
𝚫𝟐 𝒕
𝚫𝟏,𝟐 𝒕 = 𝝎𝟏,𝟐 −𝝎𝑪 𝚽 𝒕
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Two-Qubit Gates
• iSWAP operation between states |𝟎𝟏⟩ and 𝟎𝟏𝜔𝑑 = 𝜔2 − 𝜔1 (qubits’ difference frequency)
∼ ෝ𝝈+𝟏 ෝ𝝈−
𝟐 + ෝ𝝈−𝟏 ෝ𝝈+
𝟐= 𝐗𝟏𝐗𝟐 + 𝐘𝟏𝐘𝟐
• 2-photon transition (bSWAP): 𝟎𝟎 ↔ 𝟏𝟏𝜔𝑑 = 𝜔2 +𝜔1 (qubits’ difference frequency)
∼ ෝ𝝈+𝟏 ෝ𝝈+
𝟐+ ෝ𝝈−
𝟏 ෝ𝝈−𝟐 = 𝐗𝟏𝐗𝟐 − 𝐘𝟏𝐘𝟐
• Phase gate: shift of 𝟏𝟏𝜔𝑑 = 𝜔2 − 𝜔1 + 𝛼 (difference + anharmonicity)
∼ ෝ𝝈𝒛𝟏 ෝ𝝈𝒛
𝟐= 𝐙𝟏𝐙𝟐
tunable coupler can create XX, YY & ZZ interactions
𝟎𝟎
𝟏𝟏
𝟐𝟎
𝟏𝟎
𝟎𝟐
𝟎𝟏
𝜔2−𝜔1
𝜔2+𝜔1
𝜔2−𝜔1+𝛼
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https://quantumexperience.ng.bluemix.net/qx/editor
Two-Qubit Gate Errors
gate fidelity > 96.0%
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Two-Qubit Gate Errors
gate fidelity > 95.1%
https://quantumexperience.ng.bluemix.net/qx/devices
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1-Qubit Gate Fidelity 2-Qubit Gate Fidelity Read Out Fidelity
Computer Min Max Ave Min Max Ave Min Max Ave
IBM QX2 99.71% 99.88% 99.79% 94.22% 97.12% 95.33% 92.20% 98.20% 96.24%
IBM QX4 99.83% 99.96% 99.88% 95.11% 98.39% 97.11% 94.80% 97.10% 95.60%
IBM QX5 99.59% 99.87% 99.77% 91.98% 97.29% 95.70% 88.53% 96.66% 93.32%
IBM QS1_1 96.93% 99.92% 99.48% 82.28% 98.87% 95.68% 69.05% 93.55% 83.95%
Rigetti 19Q 94.96% 99.42% 98.63% 79.00% 93.60% 87.50% 84.00% 97.00% 93.30%
Single and Two-Qubit Gate Fidelity
Google indicated that their 𝑇1 times are roughly 2-4x worse than IBM’s, but that their single and two qubit gate fidelities are 2-10x better, and their measurement fidelities are roughly 10x better.
https://quantumcomputingreport.com/scorecards/qubit-quality/
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Technical Roadmap for Fault-Tolerant Quantum ComputingAmir Fruchtman, Iris Choi, University of Oxford (2016)
Two-Qubit Gate Fidelity
Qubit Readout
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Qubit Readout
𝝎 ≠ 𝝎𝐠𝐞
rf signal in
qubit𝒈 or 𝒆
dispersive readout strategy
qubit state is encoded into phase of outgoing rf-signal no energy is dissipated on chip repeat with enough photons to beat noise, use low-noise amplifiers
already demonstrated
multiplexed readout of several qubits, 80 ns readout pulse, fidelity >97%
resonator
or
tran
smis
sio
n
frequency (GHz)
Blais et al. PRA 2004, Walraff et al., Nature 2004
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Qubit Readout
courtesy: Olivier Buisson
𝑻𝐑𝐎 = 𝟓𝟐𝟎 ns
single shot readout
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1-Qubit Gate Fidelity 2-Qubit Gate Fidelity Read Out Fidelity
Computer Min Max Ave Min Max Ave Min Max Ave
IBM QX2 99.71% 99.88% 99.79% 94.22% 97.12% 95.33% 92.20% 98.20% 96.24%
IBM QX4 99.83% 99.96% 99.88% 95.11% 98.39% 97.11% 94.80% 97.10% 95.60%
IBM QX5 99.59% 99.87% 99.77% 91.98% 97.29% 95.70% 88.53% 96.66% 93.32%
IBM QS1_1 96.93% 99.92% 99.48% 82.28% 98.87% 95.68% 69.05% 93.55% 83.95%
Rigetti 19Q 94.96% 99.42% 98.63% 79.00% 93.60% 87.50% 84.00% 97.00% 93.30%
Qubit Readout Fidelity
https://quantumcomputingreport.com/scorecards/qubit-quality/
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Qubit Readoutsystem/technique fidelity 𝑻𝐑𝐎 (ns)
JC nonlinearity of a superconducting cavityReed et al., Phys. Rev. Lett. 105, 173601 (2010)
87% 500
Internal Josephson Bifurcation AmplifierSchmitt et al., Phys. Rev. A 90, 062333 (2014)
96% 50-500
Josephson phase-locked parametric oscillatorLin et al., Nat. Commun. 5, 4480 (2014)
89% 100
External Josephson Parametric AmplifierJeffrey et al., Phys. Rev. Lett. 112, 190504 (2014)
98.7% 140
Internal Josephson Parametric OscillatorKrantz et al., Nat. Commun. 7, 11417(2016)
81.5% 600
External Josephson Parametric Dimer AmplifierWalter et al., Phys. Rev. Appl. 7, 054020 (2017)
99.2% 88
Multiplexed readout with TWPAHeinsoo et al., arXiv:1801.07904
97% 250
Longitudinal coupling + bifurcation amplifierBuisson group, unpublished (2018)
97% 50
External Josephson TWPABultink et al., APL 112, 092601 (2018)
99.8% 1100
ExperimentalTechniques
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resonator atom
𝝎𝒓 𝝎𝐠𝐞
𝝎𝒓
𝟐𝝅≃
𝝎𝐠𝐞
𝟐𝝅≃ few GHz
1 GHz 50 mK
ℏ𝝎𝒓 ≃ 10-24 J
ultra-low temperatures
ultra-sensitive µ-wave experiments
energy scales
Drawbacks of sc quantum circuits
experimentalchallenges
nano-fabrication
sup
ple
men
tary
mat
eria
l
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Experimental Techniques
ultra-low Ttechniques
microwavetechnology
nano-technology
key physical ingredients and technological challenges
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Optical “table” @ mK temperature
1 GHz ≃ 50 mK
ħωr ≃ 10-24 J
mK
tech
no
logy
fo
r sc
qu
antu
m c
ircu
its
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mK technology for circuit QED
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IBM cryostat wired for a 50 qubit system
µ-wave Technology + mK temperature
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• materials for superconducting circuits
• Typical superconductors Nb
type-II superconductor, 𝑇c ≈ 9K fast measurements at 4K possible shadow evaporation for nanoscale junction not possible (without hard mask)
Al type-I superconductor, 𝑇c ≈ 1.2 K measurements require millikelvin temperatures shadow evaporation possible (stable oxide)
• Normal metals mainly Au (no natural oxide layer) for on-chip resistors and passivation layers
• Dielectric substrates silicon, sapphire contribute to dielectric losses (𝑇1)
Experimental Techniques
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• micro- and nanopatterning of superconducting circuits
• Lithography define pattern optical lithography (UV) electron beam lithography (EBL)
• Thin-film deposition deposit materials DC sputtering (metals, e.g. Nb) RF sputtering (insulators) electron beam evaporation (metals, e.g. Al) epitaxial growth (molecular beam epitaxy, higher substrate temperatures)
• Processing positive pattern Lift-off
deposit material only where you want it negative pattern Etching
deposit material everywhere remove what you don‘t want
Experimental Techniques
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WMI EBL system
• Nanobeam nb5• up to 100 kV acceleration voltage
strongly reduced „natural“ undercut frombackscattered electrons
undercut now deliberately designedduring the process
• large beam current fast• few nm resolution (in practice mostly resist
limited)• heavily automated (operated „from the office“)
advantage: fewer user-dependentparameers in the process
better reproducibility
Experimental Techniques
electron beam lithography (EBL)
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resist mask first layer second layer tunnel junction
ghost
structures
small
junctions
large
junction
J. Schuler, PhD ThesisTU Munich (2005)
key fabrication technique for Al/AlOx/Al Josephson junctions with submicron lateral dimensions
Experimental Techniques
qubit fabrication by shadow evaporation technique
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Experimental Techniques
qubit fabrication by shadow evaporation technique
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Experimental Techniques
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500 μm
ground
center
20 μm
1 μm
Nb Si3N4 Si
Fredrik Hocke et al., New J. Phys. 14 , 123037 (2012)
Xiaoqing Zhou, et al., Nature Physics 9 , 179 (2013)
Matthias Perpeintner, et al., APL 105, 123106 (2014)
Fredrik Hocke, et al., APL 105, 133102 (2014)
M. Abdi et al., PRL 114, 173602 (2015)
Experimental Techniques
CPW resonator coupled to nanomechanical beam
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CPW resonator with inductively coupled beam
Experimental Techniques
Challenges&
Problems
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100
101
102
103
104
105
106
107
108
102
101
100
10-1
102
101
100
10-1
erro
r co
rre
ctio
n g
ain
wo
rst
qu
bit
err
or
number of qubits
10−1
10−2
10−3
10−4
Quantum Computing: Quantity & Quality
logical qubit 𝟏𝟎−𝟏𝟐 quantum computer
increase quantity ??
error correction threshold
Google 9
GoogleSupremacy Device
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Major Challenges
• increase number of qubits while at the same time improving gate and readout fidelities as well as qubit coherence
• reduce crosstalk, frequency collision
• improve materials:density of two-level fluctuators and surface spins, dielectric losses, …
• develop system architecture e.g. 3D integration and packaging of multi-qubit chips, circuits with small footprint, …
• develop control circuitrye.g. scalable classical control electronics, cryogenic (microwave) components (switches, circulators, etc), optimal control pulses, ….
• develop alternative types of qubits with better performance
• develop new types of gatese.g. geometric gates, single-step multi-qubit entanglement generation, multi-qubit interactions based on n-body terms, …
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The WMI team
Thank you !