Superconduting Qubits David DiVincenzo, IBM Ann Arbor Summer School, 6/2008
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Superconduting Qubits
David DiVincenzo, IBM
Ann Arbor Summer School, 6/2008
Description: L-R: A.J. Rutgers; Hendrik Casimir in an automobile they bought for $50.00 to drive from Ann Arbor, Michigan to New York City where they abandoned it. (photo: S. Goudsmit.)
Coherent control of macroscopic quantum states in a single-Cooper-pair boxY. Nakamura, Yu. A. Pashkin and J. S. TsaiNature 398, 786-788(29 April 1999)
Saclay Josephson junction qubit
Science 296, 886 (2002)
Oscillations show rotation of qubit at constant rate, with noise.
UCSB Josephson junction qubit (“phase”)
Simple electric circuit…small
L C
harmonic oscillator with resonantfrequency
LC/10 =ω
Quantum mechanically, like a kind of atom (with harmonic potential):
x is any circuit variable(capacitor charge/current/voltage,Inductor flux/current/voltage)
That is to say, it is a “macroscopic” variable that isbeing quantized.
Textbook (classical) SQUID characteristic: the “washboard”
small
Energy
Josephson phase ω
ω
1. Loop: inductance L, energy ω2/L
2. Josephson junction: critical current Ic,energy Ic cos ω
3. External bias energy(flux quantization effect): ωΦ/L
Φ
Textbook (classical) SQUID characteristic: the “washboard”
small
Energy
Josephson phase ω
ω
Junction capacitance C, plays role of particle mass
Energy ω Φ
1. Loop: inductance L, energy ω2/L
2. Josephson junction: critical current Ic,energy Ic cos ω
3. External bias energy(flux quantization effect): ωΦ/L
Quantum SQUID characteristic: the “washboard”
small
Energy
Josephson phase ω
ω
Junction capacitance C, plays role of particle mass
Quantum energy levels
But we will need to learn to deal with…
small
G. Burkard, R. H. Koch, and D. P. DiVincenzo, “Multi-level quantum description of decoherence in superconducting flux qubits,” Phys. Rev. B 69, 064503 (2004); cond-mat/0308025.
--Josephson junctions--current sources--resistances and impedances--mutual inductances--non-linear circuit elements?
Josephson junction circuits
small
Practical Josephson junction is a combination of three electrical elements:
Ideal Josephson junction (x in circuit):current controlled by difference in superconducting phase phi across the tunnel junction:
Completely new electrical circuitelement, right?
not really…small
What’s an inductor (linear or nonlinear)?
( )Φ=
Φ=
=Φ
−
−
1
1
,
LILILI
Φ is the magnetic fluxproduced by the inductor
V=Φ•
(Faraday)
ϕ
Ideal Josephson junction:
is the superconducting phase difference across the barrier
V=•
Φ ϕπ20
(Josephson’s second law)
eh /0 =Φ flux quantum
(instantaneous)
not really…small
What’s an inductor (linear or nonlinear)?
( )Φ=
Φ=
=Φ
−
−
1
1
,
LILILI
Φ is the magnetic fluxproduced by the inductor
V=Φ•
(Faraday)
ϕ
Ideal Josephson junction:
is the superconducting phase difference across the barrier
V=•
Φ ϕπ20
(Josephson’s second law)Phenomenologically, Josephson junctions are non-linear inductors.
Superconduting Qubits
David DiVincenzo, IBM
Ann Arbor Summer School, 6/2008
So, we now do the systematic quantum theory
small
Graph formalismsmall
1. Identify a “tree” of the graph – maximal subgraph containing all nodes and no loops
graphtree
Branches not in tree are called “chords”; each chord completes a loop
graph formalism, continuedsmall
e.g.,
NB: this introducessubmatix of F labeledby branch type
Circuit equations in the graph formalism:small
Kirchhoff’s current laws:
Kirchhoff’s voltage laws:
V: branch voltagesI: branch currentsΦ: external fluxes threading
loops
Equation of motion of a complex circuit:
small
The lossless parts of this equation arise from a simple Hamiltonian:
H; U=exp(iHt)
Burkard, Koch, DiVincenzo,PRB (2004).
the equation of motion (continued):small
All are complicated but straightforward functions of the topology (F matrices) and the inductance matrix
the equation of motion (continued):
small
Straightforward (but complicated!) functions of the topology (F matrices) and the inductance matrix
…
The physics of the coupling matrices
Linear electriccircuit
The physics of the coupling matrices
Linear electriccircuit
Cut out the Josephson junctions…
The physics of the coupling matrices
Linear electriccircuit
~~
~
~
~
Vi
MultiportElectric circuitGreen function:
( ) ( ) ( ) ( ) ( ) K
t
K +=+== −
ωωωωωω
iML
iYVYI ijijiijj
011,
the equation of motion (continued):small
The lossy parts of this equation arise from a bath Hamiltonian,Via a Caldeira-Leggett treatment:
Connecting Cadeira Leggett to circuit theory:small
Overview of what we’ve accomplished:small
We have a systematic derivation of a general system-bath Hamiltonian. From this we can proceed to obtain:
• system master equation• spin-boson approximation (two level)• Born-Markov approximation -> Bloch Redfield theory• golden rule (decay rates)• leakage rates
For example:
IBM Josephson junction qubit1
“qubit” = circulationof electric currentin one direction oranother (????)
DiVincenzo, Brito, and Koch, Phys. Rev. B (2006).
small-loop noise: gradiometrically protected
Large-loop noise: bad, but heavily filtered
“Charge”, “Phase”, and “Flux” Qubits?
CDF detector, Fermilab
Google datacenter“No power is required
to perform computation.”CH Bennett
“Quantum computers canoperate autonomously.”N Margolus
2% US electricity consumption
(inventor of “computronium”)
© 2007 IBM Corporation
Efficient one- and two-qubit pulsed gates for an oscillator stabilized Josephson qubit
Frederico Brito, David P. DiVincenzo, Matthias Steffen
and Roger H. Koch
IBM T. J. Watson Research Center
New J. Phys. 2008
Presenter
Presentation Notes
To replace the title / subtitle with your own: Click on the title block -> select all the text by pressing Ctrl+A -> press Delete key -> type your own text
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
PlanIn tribute to Hans...
R. Koch1950-2007
Presenter
Presentation Notes
Some very hard things have to happen in the laboratory to make even rudimentary quantum information processing a reality. I will give a report "from the trenches" to give some idea of how you start from scratch -- in a state of the art solid state physics lab -- and try to make a working qubit. I will also give a point of view on progress on other fronts where things seem to be going better, in particular in the atomic physics lab.
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
Ic=1.3 µA
L1=32pH
L3=680pH
M1cf=0.8pH
M3flux=0.5pH
ωT=2π 3.1 GHz
Z0= 110 Ω
LT=5.6 nH
MqT=200 pH
DiVincenzo, Brito, and Koch, Phys. Rev. B (2006).
small-loop noise: gradiometrically protected
Large-loop noise: bad, but heavily filtered
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
- IBM Qubit Koch et al. PRL 96, 127001 ’06; PRB 72, 092512 ’05.
Oscillator Stabilized Flux Qubit
- Three Josephson Junctions, Three loops
- High-Quality Superconducting Transmission Line
- T2 = 2.7µs (memory point); T1= ~10ns (measurement point)- Low-bandwidth operations
Presenter
Presentation Notes
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Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
- Level Dynamics
Portal
“Par
king
” Reg
ime“S-Line”
time [ns]
Follow-up Experiment, March 2007 (unpublished)
Should observe “Larmor precessions” which measure quality of harmonic oscillator
Sle
ft[a
.u.]
time [ns]
left right Q ~ 50,000
T2 ~ 2.3 μs
|L⟩ |L⟩+ |R⟩
|0⟩+ |1⟩
|R⟩ |L⟩ - |R⟩
|0⟩ - |1⟩(1) (2)(3)
(1) (2)(3)
time [ns]
Φc
T2 increased by 300x !!
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
- One-qubit Gates summary
0-1 measurement: Fidelity > 99.99%
+ - measurement: Fidelity ~ 99.8%
Z – Gate: Fidelity > 99.99%
Hadamard Gate: Fidelity ~ 98.7%
Memory Noise: 1% of error in ~10-3 s
~ 99.5%
Gaussian Probability
Distribution
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
-- Low frequency noise:-1/f flux noise (and other 1/f sources)- pulse jitter
Two qubit gate setup
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
A
A
B
B
- Two-qubit Gate
Flux-Flux Coupling: ZZ interaction
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
“|+>, |->” Measurement Gate
Non- Adiabatic Process
Leakage < 0.06%99.8%
Phase noise plays
the main role
Two qubit gate: adiabatic controlled phase noise
almost completely Z type
200,000xC
Logical CNOT
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
- Qubit Potential
L R LR
L R
Burkard, Koch, DiVincenzo PRB 69, 064503 ‘04
- Coupling qubit-Transmission Line
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
- “Martini” Plot
Portal
Smallest avoided crossing Gap0;, AS
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
1.38 1.4 1.42 1.44 1.46 1.48 1.5
1. ´ 10 - 8
0.00001
0.01
10
10000
1. ´ 107
1.38 1.40 1.42 1.46 1.48 1.5
107
104
10
10-2
10-5
10-8
T1
Tφ
Relaxation and Dephasing times (s)
Temperature = 30mK
Efficient one- and two-qubit pulse gates for an oscillator stabilized Josephson flux qubit
© 2007 IBM Corporation
L0
L1
L2
R0
R1
L0
R0
L1
R1
L2
- Band diagram
“S-Line”
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