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Superconducting Qubits and Quantum computing.
M. GrajcarComenius University, SlovakiaIPHT Jena, Germany
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Maxwell deamon and Lanauer principle
2lnkS =∆
Landauer principle : any logically irreversible manipulation of informa tion, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy incr ease in non-informationbearing degrees of freedom
2lnkTTSW =⋅∆=
0 1Maxwell’s deamon
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Switching energy of bistable system
Pot
entia
l ene
rgy
1
Es
0
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Switching energy of bistable system
Pot
entia
l ene
rgy
1 0
Es
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Classical logic gate
NANDa
bc
a b c
0 0 1
0 1 1
1 0 1
1 1 0
Logic operations used in our classical computers ar e irreversible since we lose some information during a logic operation.
Consequence – maximal speed of irreversible computer is limited b y the powerwhich can be transferred to the environment
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Dissipated power of irreversible computer
22
4
1
4
1s
tTcTSP ∆∆ == αα
ssd t
kT
t
WP
2ln==
THz 10~2ln4
3
2
kT
Tc∆α
Heat transfer
Dissipated power
Maximal computer frequency
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Switching Energy in CMOS Logic
Saibal Mukhopadhyay et al., Switching Energy in CMOS Logic:How far are we from physical limit?
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Switching energy and interconnect (wiring) capacitance
GHz 100~4
3
2
sE
Tc∆α
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Superconducting computer
We can use superconductor instead of semiconductor
Frequency of superconducting, still irreversible co mputer based on RSFQ ~100 GHz
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Single-junction interferometer (RF-SQUID)as nonlinear quartic oscillator
eΦ eΦ
Potential energy Kinetic energy
mC
pQ
↔↔
2 , 200 Φ
Φ=ΦΦ= e
e πφπφ
m
p
2
2
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-1.0 -0.5 0.0 0.5 1.0-1
0
1
2
3
4
5
6
7
8
9
10β
L=0.5
U(φ
') [
EJ]
φ'/π
Single-junction interferometer (RF-SQUID)as nonlinear quartic oscillator
eΦ
=′)(φU
-1.0 -0.5 0.0 0.5 1.0
0
1
2
3
4
βL=1
U(φ
') [
EJ]
φ'/π
eΦ eΦ
-1.0 -0.5 0.0 0.5 1.0
-0.5
0.0
0.5
1.0
1.5
U(φ
') [
EJ]
φ'/π
βL=2
(JEU =)'(φ
0=′eφ
0
2Φ
== c
L
JL
LI
E
E πβ
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Classical bit
eφ′
JE
U
φπ 2π0
-1
0
1
2
3
4
-1
0
1
2
3
4
-1
0
1
2
3
4
Particle with mass ~ C in potential well
minU 01
Switching a classical bit from one state to another costs some energy. How much?
eφ′0
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Quantum computer
10 ba +=ψ
xcN
xxN ∑
−
=
=12
0
ψ
Paul Benioff: Miniaturization of logic gates will fi nally lead to requirement to describe them quantum mechanically. He theorizedabout creating a quantum Turing machine.
R. Feynman:
First ideas and visions in 1981
Qubit (quantum bit) – two level system is described by vector in two-dimensional complex space
Quantum state of N qubits is characterized by a ray in 2N
dimensional Hilbert space
0101......001010101=x
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Simulation of quantum computer
It’s possible by classical computer!
However, classical computer should compute with 2 N complex numbers
Simulation of 32 qubits is beyond capability of any c lassical computeravailable in present time.
R. Feynman: Quantum computer can effectively simula te quantum Objects.
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Qubit – spin in magnetic field
B
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Deutsch problem
Deutsch problem
x f(x)
input bit output bit
Logic table
x f(x)
0 0,1
1 0,1
Constant f(0)=f(1) or balanced f(0) ≠≠≠≠f(1) ?
On classical computer we should run computer two times.
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Deutsch algorithm
Input qubit can be in superposition of the states |0> a |1>
Qubit 1 Qubit 2
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Deutsch algoritm
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Deutsch algorithm
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Implementation of the Deutsch and Grover algorithmL. DiCarlo et al., Demonstration of two-qubit algorithms wi th a superconducting quantum processorNature 460, 240-244 (2009)
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Quantum Parallelism.Hilbert space is a big place. - Carlton Caves
0101......001010101=x
Output contain some information on f(x) for all pos sible classical inputs
Input N-qubit register
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Shor’s algorithm
Factorization of big numbers
classical algoritm
130 bit number - 1 month on network of hundreds classical computers
400 bit number - 10 10 years
RSA cryptography is save in classical word
Quantum computer
time ∝∝∝∝(ln n) 3If 130 bit number - 1 monththen400 bit number - 3 years
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Quantum error corrections
Qubits errors and error correction problems:
1) Qubit flip process
2) Dephasing
3) Error accumulation
4) Measurement problem – projection of the qubit afte r the measurement
5) Noncloning theorem of quantum mechanics
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Persistent current qubit -amonia molecule analogue
rightleft
U/E
J
θ
e
h
20 =Φ
Bµµ 1010=
m
eB 2
h=µ
nΦΦΦΦ0
B
N
H
H
H++
+
Oscillation of magnetic dipole in superconducting r ing Is an analogue of oscillation of electric dipole mo ment In amonia molecule.
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Quartic nonlinear quantum oscillator
φ′∂∂≡↔ hipQ
=′′ )(ˆ φφeH
0
2Φ
== c
L
JL
LI
E
E πβ
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Pure quartic quantum oscillator, ββββL=1, φφφφ’e=0
1=Lβ
=)(ˆ qH
1>Lβ
00112 EEE ≈−
0Eop
h>τ
For IPHT technology of Al Josephsonjunctions:
]m[
025.0 [K]
2µSEC ≈ ][5.2[K] 2mSEJ µ≈
2)m][(100
1
µα
S≈
3/10
1]GHz[
SE ≈
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Pseudospin Hamiltonian
IC, φφφφ2222
IC, φφφφ1111
γγγγIC
(0.5<γ<1)
Φx
1 um
−εaε a
E
↑↓
a∆2
( )↑−↓2
1
( )↑+↓2
1
↑ ↓
e
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How to measure?Projective measurement – dc squid
TU Delft
dc squid measures σσσσz at degeneracy point the qubit is in the eigenstate of observable σσσσxDestructive measurement- after measurement qubit is in the eigenstate of σσσσz
0610 Φ≈ -
Sensitivity
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Flux qubit coupled to oscillator
Φi
VTLT
L
CT
Ib
M
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-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
ε(fx) (GHz)
Adiabatic measurement away from degeneracy point
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-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
ε(fx) (GHz)
Adiabatic measurement at degeneracy point
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Lagrangian of the qubit-resonator system
Expanding into Taylor series up to the second order term
2
-
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Impedance Measurement,classical resonator
Φ
0.0 0.4 0.8 1.2 1.6 2.0-2
-1
0
1
2
ϑ, r
adω
VT
LTL CT
Ib
M
Ya. S. Greenberg et al., PRB 66, 214525 (2002)DC-Squid Josephson Inductance: A. Lupascu et al., PRL 93, 177006 (2004).
0.0 0.4 0.8 1.2 1.6 2.00
2
4
6
8
10
∆ω
Am
plitu
de
ω
TTT CL
1=ω
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Resonant frequency of the resonator
Y. Greenberg et al., PRB 66214525 (2002).
Fitting parameters
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Sisyphus work
As a punishment from the gods for his trickery, Sisyphus was compelledto roll a huge rock up a steep hill, but before he reached the top of the hill,the rock always escaped him and he had to begin again.
Greek mythology
Titian (1549) artist vision of Sisyphus work
Physical realization: For atomsD. J. Wineland, J. Dalibard and C. Cohen-Tannouji, J. Opt. Soc. B9, 3242 (1992).
For qubit M. Grajcar et al., „Sisyphus cooling and amplification by a superconducting qubit“ Nature Physics 4, 612-616 (2008).
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-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
ε(fx) (GHz)
Sisyphus cooling
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-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
E (
GH
z)
ε(fx) (GHz)
Sisyphus pumping
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Design for spectroscopic measurement
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Adiabatic vs. spectroscopic measurement
Solid line is theoretical curve for Parameters determined from adiabatic measurement
0.000 0.005 0.010 0.0152
4
6
8
10
12
14
16
18
20
f [G
Hz]
Φdc
(Φ0)
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More rigorous treatment of Sisyphus cooling/heating
A. Fedorov, A. Shnirman, Gerd Schön
fmw=14 GHz
M. Grajcar et al., „Nature Physics 4, 612-616 (2008).
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Strong microwave driving at f mw=4.5 GHz
Weak driving
Transition from weak to strong driving
Φdc (Φ0)
M. Sillanpää et al., PRL 96, 187002 (2006)
W.D. Oliver et al.,SCIENCE 310, 1653(2005)
Strong driving
A. Izmalkov et al., PRL 101, 017003 (2008)
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Spectral density of the voltage noise of the tank
fmw=12 GHz
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Spectral density of the voltage noise of the tank
fmw=8 GHz
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G. Oelsner et al., IPHTPhys. Rev. B 81 , 172505 2010
Quantum electrodynamics on the chipInteraction of artificial atom (flux qubit) with ca vity (CPW)
A. Wallraff et al., Nature 431, 162 (2004).
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Interaction of a flux qubit with a coplanar waveguide resonator in quantum regime
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Spectroscopy of the flux qubit coupled to a coplanar waveguide resonator
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Φ
island
Josephson junctionsgate
probeelectrode
to resonator
0.5 µm
qubit
Nb coplanar resonator
100 µm
External MW line
1 mm
Oleg Astafiev, Single Artificial-atom Lasing,Nature 449, 588-590 (4 October 2007)NEC Nano Electronics Research Lab. & RIKEN
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-0.2 0.0 0.2 0.40.0
0.2
0.4
0.6
abso
rptio
nsi
de
emis
sion
side
“hot spots” S (10
-22W
/Hz)0
1
0
- 2
- 1
1
2
23456
δω/2
π (M
Hz)
I(nA
)
ng
Emission
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Generating Single Microwave Photons in a Circuit
A. Houck et al., Yale University, USA Nature (London) 449, 328 (2007)
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Quantum metamaterials
Single photon detector with high efficiency in GHz r ange
G. Romero et al., Microwave Photon Detector in Circuit QED, arXiv:0811.3909v1
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Four qubit sample
MicrographLayout
q1
q2
q3
Iq2
Iq3 Iq1
Ib4 A1
A2
A3
M. Grajcar, et al., Phys. Rev. Lett. 96, 047006 (2006).
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Theoretical fits. Phys. Rev. Lett. 96, 047006 (2006)
Experiment Theory
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Quantum ground state of a mechanical resonator
A. D. O’Connell et al., Nature 464, 697 (2010)
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Nanobridge from IPHT Jena
Nanobridge from IPHT Jenacoupled to flux qubit
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Summary
1. Superconducting flux qubits are well described by two-level (pseudospin) Hamiltonian
2. Simple two and three qubits algorithms were demonstrated
3. The superconducting qubits can be used as an tunable artificial atoms for many applications as coolers of electrical and nanomechanical oscillator s (Sisyphus cooling), sensitive detectors, single pthotons and phonon sources, etc.
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CNOT-gate
Nevyhnutne musíme ma ť dva interagúce spiny.
To nie je problém. Ľubovo ľné dva spiny interagujé v ďaka dipólovej interakcií.
Musíme však vedie ť interakciu vypnú ť. To už problém je.
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Error corrections
error
Majority voting
Correction is possible if
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Binomic distrubution
nNn ppnNn
NnP −−
−= )1(
)!(!
!)(
∞→N ε−=2
1p
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Logic qubits
pcp <2
22)(cpc
Steane code by 7 physical qubits
7 physical qubits = 1 logical qubit
Steane code by logických qubits
After k-recursion
c
cpk2)(
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Fault tolerance issues:Oskin, Chong & Chuang, IEEE Jan. 2002• need transversal gates• can perform elementary operations in parallel• can couple any 2 qubits regardlessof distance
• source of fresh ancillas• larger problem size requires larger k
Steane [7,1,3] code:Error rate threshold ρth ~ 1/c=10-4
( ))(
2
nc
cpk
ξε≤
Quantum error corrections
Parameter 3-junctions flux qubit „transmon“ qubitQcalc 97000 Qmeas 40000 70000Error rate≈1/Q ≈ 10-4 - 10-5 ≈ 10-4 -10-5
Qubit Quality Q=ωqT2 , ρth~1/Q
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Recursion level for Shor and Grover algorithm
Oskin, Chong & Chuang, IEEE Jan. 2002
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Adiabatic quantum computing
1) Start with initial Hamiltonian HI with known ground state |I>2) Make adiabatic evolution of HI to final Hamiltonian of HP with uknown
ground state which is difficult to calculate3) According adiabatic theorem the system is with high probability in the
ground state |g> of HP4) Readout the ground state of HP
1) For flux qubits we choose initial Hamiltonian HI with trivial ground state |0>
∑=i
iziIH ,)0( σε
2/ii w−=ε2) Changing the bias of individual qubits adiabatically to thethe initial Hamiltonian Hi is transformed to HP.
Realization for superconducting flux qubits
jziz
N
i
N
jijiixiizi
N
iiP JfH ,,
1,,,
1
)( σσσσε ∑ ∑∑= <=
+∆+=
Farhi et al., quant-ph/0001106
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MAXCUT problem
•The MAXCUT problem is part of the core NP-complete problems•The MAXCUT problem has application in VLSI design•MAXCUT adiabatic quantum algorithm already demonstrated by NMRM. Stephen et al., quant-ph/0302057
Simple example for 4 nodes
w1
w2
w3w4
w12
w23
w34
w14
01
( ) ∑∑ −+=ji
jijiii
i wssswsP,
,)1(
Payoff function
w24
w13
S4=0 S3=0
S2=1
S1=1
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Hamiltonian of N inductively coupled flux qubits
∑∑<=
+=N
jijzizizi
N
iiP fH ,,,
1
)( σσσε
( ) ∑∑ −+=ji
jijiii
i wssswsP,
,)1(
Payoff function is encoded in Hamiltonian HP if ∆i<<Ji,j and
2/ii w−=ε 2/,, jiji wJ =
HP – The MAXCUT problem Hamiltonian
gEgH gP =
Time of adiabatic evolution ττττ>>EJ/∆∆∆∆2min
∆∆∆∆min – minimum gap betweeen ground and first excited leve l
M. Stephen et al., quant-ph/0302057
ix
N
ii ,
1
σ∑=
∆+
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Demonstration of adiabaticquantum algorithm MAXCUT
A. Izmalkov et al.,
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Ground of an informatics - Physics
Information is physical. - Rolf Landauer
Function is effectively calculable if its values ca n be found by some purely mechanical process.' We may take this literally, u nderstanding that by a purely mechanical process one which could be carri ed out by a machine - A. Turing
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Ground of an Informatics - Physics
The theory of computation has traditionally been st udied almost entirely in the abstract, as a topic in pure mathematics. This is t o miss the point of it. Computers are physical objects, and computations ar e physical processes. What computers can or cannot compute is determined by the laws of physics alone, and not by pure mathematics. - David Deutsch, pioneered the field of quantum computers by being the first person to formulate a specifically quantum computational algorithm.
Like mathematics, computer science will be somewhat different from the other sciences, in that it deals with artificial la ws that can be proved, instead of natural laws that are never known with c ertainty. - Donald Knuth,the "father" of the analysis of algorithms.
The opposite of a profound truth may well be anothe r profound truth. - Niels Bohr,Danish physicist who made foundational contributions to understanding quantum mechanics