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Pulse Methods for Preserving Quantum Coherences T. S. Mahesh Indian Institute of Science Education and Research, Pune. Criteria for Physical Realization of QIP. Scalable physical system with mapping of qubits A method to initialize the system Big decoherence time to gate time

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slide1

Pulse Methods for Preserving Quantum Coherences

T. S. Mahesh

Indian Institute of Science Education and Research, Pune

slide2

Criteria for Physical Realization of QIP

  • Scalable physical system with mapping of qubits
  • A method to initialize the system
  • Big decoherence time to gate time
  • Sufficient control of the system via time-dependent Hamiltonians
  • (availability of universal set of gates).
  • 5. Efficient measurement of qubits

DiVincenzo, Phys. Rev. A 1998

slide3

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide4

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide5

Closed and Open Quantum System

Environment

Environment

Hypothetical

slide6

Coherent Superposition

An isolated 2-level quantum system

| = c0|0 + c1|1, with |c0|2 + |c1|2 = 1

Density Matrix

rs = || = c0c0*|0 0| + c1c1*|1 1|+

c0c1*|0 1| + c1c0*|1 0|

c0c0*c0c1*

c1c0* c1c1*

Coherence

=

Population

slide7

Effect of environment

Quantum System – Environment interaction Evolution U(t)

System  Environment

System  Environment

||E  = (c0|0 + c1|1)|E 

c0|0|E0 + c1|1|E1

Entangled

U(t)

|0|E  |0|E0

|1|E  |1|E1

U(t)

U(t)

System  Environment

slide8

Decoherence

  • = ||E |E|
  • = c0c0*|0 0||E0 E0| + c1c1*|1 1||E1 E1| +
  • c0c1*|0 1||E0 E1| + c1c0*|1 0||E1 E0|
  • rs = TraceE[r] = c0c0*|0 0| + c1c1*|1 1|+
  • E1|E0 c0c1*|0 1| + E0|E1 c1c0*|1 0|
  • c0c0*E1|E0 c0c1*
  • E0|E1 c1c0* c1c1*

Coherence

=

Population

Coherence decays irreversibly

|E1(t)|E0(t)| = eG(t)

Decoherence

slide9

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide10

Signal Decay

13-C signal of chloroform

in liquid

Signal

 x

Time

Frequency

slide11

Signal Decay

Decoherence

Incoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

slide12

Signal Decay

Decoherence

Incoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

slide13

Incoherence

Individual (30 Hz, 31 Hz)

Net signal – faster decay

Time

slide14

Hahn-echo or Spin-echo (1950)

Echo

Signal

y

/2-x

+ d

 d

y

Symmetric distribution of  pulses removes incoherence

slide15

Signal Decay

Decoherence

Incoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

slide16

Time to reach equilibrium, (energy of spin-system is not conserved)

T1

Lifetime of coherences, (energy of spin-system is conserved)

T2

Bloch’s Phenomenological Equations (1940s)

slide17

Bloch’s Phenomenological Equations (1940s)

Solutions in rotating frame:

0

0

slide18

Signal Decay

Decoherence

Incoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

slide19

Effect of environment

’ = E()

= ∑ Ek Ek†

(operator-sum representation)

k

slide20

Amplitude damping (T1 process, dissipative)

g(t) is net damping : eg., g(t) = 1  et/T1

1 0

0 (1g)1/2

p 0

0 1  p

E0 = p1/2

=

0g1/2

0 0

0 0

g1/20

E3 = (1  p)1/2

E1 = p1/2

(1g)1/20

0 1

E2 = (1  p)1/2

E()= ∑ Ek Ek†

k

Asymptotic state (t , g  1) :

In NMR,

p =

~ 0.5 + 104

1

1 + eE/kT

slide21

Amplitude damping (T1 process, dissipative)

Measurement of T1: Inversion Recovery

Equilibrium

Inversion

M() = 1 2exp( /T1)

slide22

Signal Decay

Incoherence

Decoherence

Amplitude

decay

Phase

decay

Depolarization

T1

process

T2

process

Relaxation

slide23

Phase damping (T2 process, non-dissipative)

g(t) is net damping : eg., g(t) = 1  et/T2

1 0

0 (1g)1/2

0 0

0 g1/2

a 0

0 1-a

a b

b* 1-a

E0 =

E1 =

=

(t) =

E()= ∑ Ek Ek†

k

Stationary state (t , g  1) :

slide24

Phase damping (T2 process, non-dissipative)

Spin-Spin

Relaxation

dMx(t)Mx(t)

dt T2

=

Signal envelop: s(t) = exp( t/T2)

Transverse magnetization: Mx(t) Re{01(t)}

FWHH = /T2

Bloch’s equation :

Solution : Mx(t) = Mx(0) exp( t/T2)

slide25

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide26

Carr-Purcell (CP) sequence (1954)

Signal

y

/2y

y

y

Shorter  is better (limited by duty-cycle of hardware)

H. Y. Carr and E. M. Purcell, Phys. Rev. 94, 630 (1954)

slide27

Meiboom-Gill (CPMG) sequence (1958)

Signal

x

/2y

x

x

Robust against errors in  pulse !!!

S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)

slide28

CPMG

Dynamical effects are minimized Dynamical decoupling

Sampling points

1

2

3

4

time

j = T(2j-1) / (2N)

Linear in j

Time

Signal

CP

CPMG

No

pulse

Hahn

Echo

S. Meiboom and D. Gill, Rev. Sci. Instrum. 29, 688 (1958)

slide29

Dynamical Decoupling (DD)

Götz S. Uhrig

PRL 98, 100504 (2007)

CPMG (1958):

Uniformly distributed  pulses

Uhrig 2007

(UDD):

Optimal distribution of p pulses for

a system with dephasing bath

j = T sin2 ( j /(2N+1) )

T = total time of the sequence

N = total number of pulses

slide30

Carr & Purcell, Phys. Rev (1954) .

Meiboom & Gill, Rev. Sci. Instru. (1958).

Carr Purcell Sequence

time

0

T

Was believed to be optimal for N  flips in duration T

1

2

3

4

5

6

7

j = T(2j-1) / (2N)

Linear in j

Uhrig Sequence

Uhrig, PRL (2007)

time

2

1

3

4

5

6

7

0

T

Proved to be optimal for N  flips in duration T

j = T sin2 ( j /(2N+1) )

slide31

Dynamical Decoupling (DD)

Hahn-echo (1950)

CPMG (1958)

PDD (XY-4)

(Viola et al, 1999)

UDD

(Uhrig, 2007)

CDDn = Cn = YCn−1XCn−1YCn−1XCn−1

C0 = 

(Lidar et al, 2005)

slide32

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide33

DD performance

ION-TRAP qubits

M. J. Biercuk et al,

Nature 458, 996 (2009)

slide34

DD performance

Electron Spin Resonance

(-irradiated malonic acid

single crystal)

J. Du et al, Nature

461, 1265 (2009)

Time (s)

Time (s)

slide35

DD performance

Solid State NMR

13C of Adamantane

Dieter et al,

PRA 82, 042306 (2010)

slide36

Dynamical Decoupling in Solids

13C of Adamantane

D. Suter et al,

PRL 106, 240501 (2011)

slide37

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide38

Sources of decoherence – dipole-dipole interaction

Randomly

fluctuating

local fields

Spin in a

coherent

state

slide39

Sources of decoherence – dipole-dipole interaction

Randomly

fluctuating

local fields

Spin looses

coherence

slide41

Redfield Theory: semi-classical

System - > Quantum, Lattice - > Classical

System

Completely reversible

No decoherence

System+

Random field

(coarse grain)

slide42

Auto-correlation

Local field X(t)

time

Auto-correlation

function



G() =  X(t) X*(t+)  = dx1 dx2 x1 x2 p(x1,t) p(x1,t | x2, )

Fluctuations have finite memory: G() = G(0) exp(||/ c)

c  Correlation Time

2c

1+ 2c2

Spectral density J() = G() exp(-i) d = G(0)

slide43

Spectral density

2c

1+ 2c2

J() =

G(0)

J()

c = 1

(after secular approximation)

slide44

Spectral density

2c

1+ 2c2

J() =

G(0)

J()

c = 1

c0c0*eGtc0c1*

eGtc1c0* c1c1*

2 J()

 2

 = d

3

8

15

4

3

8

1

T2

1

T1

 J(2) + J() + J(0)

 J(2) + J()

0

Dipolar Relaxation in Liquids

slide45

Effect of decoupling pulses

L. Cywinski et al, PRB 77, 174509 (2008).

M. J. Biercuk et al, Nature (London) 458, 996 (2009)

0

Time-dependent Hamiltonian

exp(-iH(t) dt)

Magnus expansion

slide46

Filter Functions

Cywiński, PRB 77, 174509 (2008)

M. J. Biercuk et al, Nature (London) 458, 996 (2009)

|x()|= e()

Fourier Transform of Pulse-train

F()



F()

2 J()

 2

 = F() d

0

slide47

Filter Functions

J()

Modified Spectral density:

J’() = J() F()

2 J()

 2

 = F() d

0

Residual area contributes

to decoherence

Cywiński, PRB 77, 174509 (2008)

M. J. Biercuk et al, Nature (London) 458, 996 (2009)

slide48

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide50

Two-qubit DD

Electron-nuclear entanglement

(Phosphorous donors in Silicon)

No DD

PDD

Wang et al,

PRL 106, 040501 (2011)

slide51

Two-qubit DD – in NMR

Levitt et al, PRL, 2004

Hamiltonian: H =h1Iz1 + h2Iz2+ hJ I1  I2

S. S. Roy & T. S. Mahesh,

JMR, 2010

Hz

HE

|01+|10

2

Eigenbasis of HE

Eigenbasis of Hz

|01−|10

|11

2

|11

90x , , , 90y ,

|10

|01

|00

Fidelity = 0.995

|00



1

2J



slide52

Two-qubit DD – in NMR

27s

j = N sin2 ( j /(2N+1) )

 = 4.027 ms

2 ms

2 ms

5-chlorothiophene-2-carbonitrile

slide53

UDD-7 on 2-qubits

Singlet

Fidelity

S. S. Roy, T. S. Mahesh, and G. S. Agarwal,

Phys. Rev. A 83, 062326 (2011)

slide54

UDD-7 on 2-qubits

Product state

0110

01+10

Entanglement

0011

00+11

S. S. Roy, T. S. Mahesh, and G. S. Agarwal,

Phys. Rev. A 83, 062326 (2011)

slide55

Dynamical Decoupling in Solids

CPMG

UDD

Uhrig, 2011

RUDD

Abhishek et al

slide56

Dynamical Decoupling in Solids

1H of Hexamethylbenzene

DD on single-quantum coherences

Abhishek et al

slide57

Dynamical Decoupling in Solids

RUDD

No DD

1H of Hexamethyl Benzene

Abhishek et al

slide58

Dynamical Decoupling in Solids

2q

4q

6q

8q

Abhishek et al

slide59

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide60

Noise Spectroscopy

Alvarez and D. Suter,

arXiv: 1106.3463 [quant-ph]

|x()|= e()

F()

2 J()

 2

() = F() d

0

slide61

Contents

  • Coherence and decoherence
  • Sources of signal decay
  • Dynamical decoupling (DD)
  • Performance of DD in practice
  • Understanding DD
  • DD on two-qubits and many qubits
  • Noise spectroscopy
  • Summary
slide62

Summary

Dynamical decoupling can greatly enhance the coherence times,

some times by orders of magnitude

Various types of pulsed DD sequences are available. Best DD depends

on the spectral density of the bath, the state to be preserved, robustness

to pulse errors, etc.

3. Filter-functions are useful tools to understand the performance of DD.

4. DD on large number of interacting qubits also shows improved performance.