Comenius

University

1

Superconducting Qubits and Quantum computing.

M. GrajcarComenius University, SlovakiaIPHT Jena, Germany

Comenius

University

2

Comenius

University

3

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

Comenius

University

4

Switching energy of bistable system

Pot

entia

l ene

rgy

1

Es

0

Comenius

University

5

Switching energy of bistable system

Pot

entia

l ene

rgy

1 0

Es

Comenius

University

6

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

Comenius

University

7

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

Comenius

University

8

Switching Energy in CMOS Logic

Saibal Mukhopadhyay et al., Switching Energy in CMOS Logic:How far are we from physical limit?

Comenius

University

9

Switching energy and interconnect (wiring) capacitance

GHz 100~4

3

2

sE

Tc∆α

Comenius

University

10

Superconducting computer

We can use superconductor instead of semiconductor

Frequency of superconducting, still irreversible co mputer based on RSFQ ~100 GHz

Comenius

University

11

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

Comenius

University

12

-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

Comenius

University

13

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

Comenius

University

15

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

Comenius

University

16

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.

Comenius

University

17

Qubit – spin in magnetic field

B

Comenius

University

18

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.

Comenius

University

19

Deutsch algorithm

Input qubit can be in superposition of the states |0> a |1>

Qubit 1 Qubit 2

Comenius

University

20

Deutsch algoritm

Comenius

University

21

Deutsch algorithm

Comenius

University

22

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)

Comenius

University

23

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

Comenius

University

24

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

Comenius

University

25

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

Comenius

University

26

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.

Comenius

University

27

Quartic nonlinear quantum oscillator

φ′∂∂≡↔ hipQ

=′′ )(ˆ φφeH

0

2Φ

== c

L

JL

LI

E

E πβ

Comenius

University

28

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 ≈

Comenius

University

29

Pseudospin Hamiltonian

IC, φφφφ2222

IC, φφφφ1111

γγγγIC

(0.5<γ<1)

Φx

1 um

−εaε a

E

↑↓

a∆2

( )↑−↓2

1

( )↑+↓2

1

↑ ↓

e

Comenius

University

30

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

Comenius

University

31

Flux qubit coupled to oscillator

Φi

VTLT

L

CT

Ib

M

Comenius

University

32

-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

Comenius

University

33

-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

Comenius

University

34

Lagrangian of the qubit-resonator system

Expanding into Taylor series up to the second order term

2

-

Comenius

University

35

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

Comenius

University

36

Resonant frequency of the resonator

Y. Greenberg et al., PRB 66214525 (2002).

Fitting parameters

Comenius

University

37

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).

Comenius

University

38

-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

Comenius

University

39

-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

Comenius

University

40

Design for spectroscopic measurement

Comenius

University

41

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)

Comenius

University

42

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).

Comenius

University

43

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)

Comenius

University

44

Spectral density of the voltage noise of the tank

fmw=12 GHz

Comenius

University

45

Spectral density of the voltage noise of the tank

fmw=8 GHz

Comenius

University

46

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).

Comenius

University

47

Interaction of a flux qubit with a coplanar waveguide resonator in quantum regime

Comenius

University

48

Spectroscopy of the flux qubit coupled to a coplanar waveguide resonator

Comenius

University

49

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

Comenius

University

50

-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

Comenius

University

51

Generating Single Microwave Photons in a Circuit

A. Houck et al., Yale University, USA Nature (London) 449, 328 (2007)

Comenius

University

52

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

Comenius

University

53

Four qubit sample

MicrographLayout

q1

q2

q3

Iq2

Iq3 Iq1

Ib4 A1

A2

A3

M. Grajcar, et al., Phys. Rev. Lett. 96, 047006 (2006).

Comenius

University

54

Theoretical fits. Phys. Rev. Lett. 96, 047006 (2006)

Experiment Theory

Comenius

University

55

Quantum ground state of a mechanical resonator

A. D. O’Connell et al., Nature 464, 697 (2010)

Comenius

University

56

Nanobridge from IPHT Jena

Nanobridge from IPHT Jenacoupled to flux qubit

Comenius

University

57

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.

Comenius

University

58

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.

Comenius

University

59

Error corrections

error

Majority voting

Correction is possible if

Comenius

University

60

Binomic distrubution

nNn ppnNn

NnP −−

−= )1(

)!(!

!)(

∞→N ε−=2

1p

Comenius

University

61

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)(

Comenius

University

62

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

Comenius

University

63

Recursion level for Shor and Grover algorithm

Oskin, Chong & Chuang, IEEE Jan. 2002

Comenius

University

64

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

Comenius

University

65

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

Comenius

University

66

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∑=

∆+

Comenius

University

67

Demonstration of adiabaticquantum algorithm MAXCUT

A. Izmalkov et al.,

Comenius

University

68

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

Comenius

University

69

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