Towards Optimal Constructions of Towards Optimal Constructions of - - PowerPoint PPT Presentation

Second International Conference on Quantum Error Correction

University of Southern California, Dec. 5-9 2011

Towards Optimal Constructions of Towards Optimal Constructions of Dynamically Corrected Quantum Gates Dynamically Corrected Quantum Gates

Lorenza Lorenza Viola Viola

• Dept. Physics & Astronomy

Dartmouth College

The premise: DQEC The premise: DQEC

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control c

Theory challenges for practical DQEC-I Theory challenges for practical DQEC-I

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Theory challenges for practical DQEC-2 Theory challenges for practical DQEC-2

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H H

• I. Analytical
• I. Analytical

DCG Framework DCG Framework

Control-theoretic setting Control-theoretic setting

4/21 4/21 S B

S B HSB

a S a

B a {Sa} S0 = IS Sa

H = [HS, + HS, ]

IB + IB

HB + HSB

Pure-system Pure-bath

{Ba}

B

m H m

I B hm t

H (t) H + H (t), H (t) = Texp i

t

ds

Control propagator

U (t) H (s)

Control inputs

S HS HS t

Error model assumptions Error model assumptions

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= H

i

B

i

B

i

U T Q Texp i

T

ds U ctrl s H err U ctrl s

HS, = 0, HS HS,

H SB

i 1 n a x , y , z a i

B a

i ,

H

= IS

HB + HSB

U ctrl T Q Texp i

T

ds H ctrl s , T U T Texp i

T

ds H ctrl s H S , g H err Q exp i E Q U

0 T

Q I B Texp i

T

ds H ctrl s H S , g I B T

Error action

• perator

Control assumptions Control assumptions

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mod B E Q

• p

S S 1

mod B E Q

• p

Non-pure-bath component

h x t

x i , h y t y i , hzz t z i z j

, i , j 1, ,n ha t h > 0

Q

Trapezoidal Rectangular Actual ldeal

Control objective Control objective

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U0 U1 Uj ... ... Q1 Q2 QN Qj ... ...

3 2

H

E Q 1

1

Q N

N

E Q 1

1

P 1 E Q 2

2 P 1

P N

1 E Q N

N P N

1

C

2

, P N

1

Q N

1

Q 1

i

mod B E Q i

i

. N mod B E Q mod B E Q

1 1

Dynamically correcting NOOP Dynamically correcting NOOP

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Q I {l } , l=1,...,L,

G = {gi }, i=1,...,G GDD

E EDD

l 1 L i 1 G

U g i E

l U g i

E EDD

2

, mod B E EDD mod B E EDD

2

O

2

G =

2 2

{I

, X , Y , Z } {X, Y}

X

all

X 1 X n exp i

0 hx s x 1 x n

ds

EDD

L G

DCGs beyond NOOP DCGs beyond NOOP

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Q

Q exp i E Q , I Q exp i E Q Q* E DCG E EDD

i 1 G

U g i E Q U g i E DCG

2

E l EQ I Q

IQ Q* mod B E DCG mod B E DCG

2

O

2

Finding gates with same error: Balance pairs Finding gates with same error: Balance pairs

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Q Texp i

• hQ t H Q dt

Q s Texp i s

s

hQ t s H Q dt I Q I Q Q

1 Q 2 ,

Q Q Q Q

1 Q ,

mod B E I Q mod B E Q O

2

I Q Q ' Q , Q Q 2 , 2 mod B E I Q mod B E Q O

2

0 2 E Q E ' Q s s E Q

Beyond first-order DCGs... Beyond first-order DCGs...

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m m Q Q-1

I Q

m

Q

1, m Q m 2 1 m 1 ,

Q

m

Q

m Q 1, m Q m ,

m Q Q

1

Q

m

1 m mod B E I Q mod B E Q O

m 2

m = 0

2 3 4

m Q Q m

m

m m

G m

m Q m+1

m

m

G m

Gm

Lm

G m

m

1

CDCGs: Performance analysis CDCGs: Performance analysis

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EQ m O

m+1

m = m c O m

1

G m L m 3 G m 1 1 2

1 m 1

• m

1

G m L m 3

m m m

Euler path Balance pairs

mod B E DCG

m

c4

m m m

2

m

H err

m 1

mopt 1 2 log 4 H err 1 ,

• pt

CDCGs: Illustrative results CDCGs: Illustrative results

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H error I S

k 1 N

Dk l I k I l

k 1 N

A k I k , N 5

m

m

a

B a ,a x , y , z Q

m 1

Q

m X m Y m X m Y m Y m I Q X m I Q Y m I Q X m

G m G

2 2

{I, X,Y ,Z}

{X, Y}

m

4 5 20

m

Q exp i 2 3

x S

1 2 1

B

I B 2

N

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CDCGs in the lab?... CDCGs in the lab?...

Ø Ø

U Q t exp S N t a t a Q , Q exp i t S N

2

t 2

t

exp i s ds tg j2 i E Q t S N t a t a

X X X

Uncorrected DCG[1] DCG[2]

X Q exp i E Q X Q exp i E Q Q

2 X exp

i E Q X exp i E Q Q

2

Q2

• II. Progress towards
• II. Progress towards

Optimized Optimized DCG Framework DCG Framework

Recap thus far... Recap thus far...

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HS n n

G m G

2 n

G

n L

adv

4

n

5

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The CDCG framework revisited The CDCG framework revisited

U T Texp i

T

ds H ctrl s H S , g H err Q exp i E Q T Q e

i

Texp i

T

ds H ctrl s H S mod B E Q

• m

O

m 1

H

Gate synthesis Error cancellation

Boosting efficiency via parametric optimization Boosting efficiency via parametric optimization

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1 2

EQ

3

mod B E Q

2

• 3

O

4

z 1 mod B E Q

1

hl ,l z 2 mod B E Q

2

hl ,l z1 z2 Q hl

l

Texp i

T

ds H ctrl s

l 1 n

exp i hl H l

l ,

T

l 1 n l

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S

1 2 1

B

I B 2

N

H error I S

k 1 N

Dk l I k I l

z k 1 N

A k I k z , N 5

Illustrative results Illustrative results

2 14 3 2 20 365 2 21 m

1

G m L m 3 G m 1 1 2

1 m 1

• 2

14 3 2 8 146

Q exp i 8

x

Uncorrected Optimized CDCG, simplified CDCG, generic

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Accommodating drift via robust optimization Accommodating drift via robust optimization

H tot H S H err H ctrl t

z z

h t

x

h(t) H' H

z –

x z 1 hl ,l U ctrl T Q z 2 hl ,l E Q

1

z hl ,l z 1 hl ,l z 2 hl ,l

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Illustrative results Illustrative results

Q exp i 8

x

z1 z1 z1 z2 z 10

8 min , min T

O z 1

min z 2 min min

10

4

Conclusion Conclusion

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