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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 Viola Viola Lorenza Dept. Physics & Astronomy Dartmouth College

The premise: DQEC The premise: DQEC 1/21 1/21 control c

Theory challenges for practical DQEC-I Theory challenges for practical DQEC-I 2/21 2/21

Theory challenges for practical DQEC-2 Theory challenges for practical DQEC-2 3/21 3/21 H 0 H

I. Analytical I. Analytical DCG Framework DCG Framework

Control-theoretic setting Control-theoretic setting 4/21 4/21 S B S B H SB Pure-system Pure-bath H = [ H S, + H S, ] I B + I B H B + H SB a S a B a { S a } S 0 = I S S a 0 { B a } B H ( t ) H + H ( t ), H ( t ) = m H m I B h m t Control inputs t U ( t ) H Texp i ds ( s ) Control propagator 0 S H S H S t

Error model assumptions Error model assumptions 5/21 5/21 T 0 T T U Q I B Texp i ds H ctrl s H S , g I B 0 T T Error action U T Texp i ds H ctrl s H S , g H err Q exp i E Q 0 operator H S, = 0, H S H S, T T U ctrl T Q Texp i ds H ctrl s , U T Q Texp i ds U ctrl s H err U ctrl s 0 0 n i , i H SB B a H = I S H B + H SB a i 1 a x , y , z i i i = B B H

Control assumptions Control assumptions 6/21 6/21 0 mod B E Q mod B E Q op op S S 1 Non-pure-bath component Actual ldeal i , h y t i , h zz t i j h x t , i , j 1, ,n x y z z h a t > 0 h Q Rectangular Trapezoidal

Control objective Control objective 7/21 7/21 U 0 U 1 Q 1 Q 2 ... ... U j Q j ... ... Q N N 2 E Q 1 E Q 1 P 1 E Q 2 2 P 1 P N 1 E Q N N P N C , Q N 1 1 N 1 P N Q N Q 1 1 1 mod B E Q i . i i H 2 mod B E Q 0 3 mod B E Q 1 1

Dynamically correcting NOOP Dynamically correcting NOOP 8/21 8/21 Q I G = { g i }, i=1,...,G { � l } , l=1,...,L, G DD L G 2 2 2 E EDD U g i E l U g i E EDD , mod B E EDD mod B E EDD O � l 1 i 1 G = { I } { X, Y } , X , Y , Z 2 2 all 1 n X X 1 X n exp i 0 h x s ds x x L G EDD

DCGs beyond NOOP DCGs beyond NOOP 9/21 9/21 Q Q * Q exp i E Q , I Q exp i E Q I Q I Q Q * G 2 E DCG E EDD U g i E Q U g i E DCG i 1 E � l E Q 2 2 mod B E DCG mod B E DCG O

Finding gates with same error: Balance pairs Finding gates with same error: Balance pairs 10/21 10/21 i t � s � Q � Texp i h Q t H Q dt Q s � Texp h Q s H Q dt 0 s 0 E Q � E ' Q s � s E Q � I Q Q ' � Q � , Q Q 2 � , 2 2 0 0 2 mod B E I Q mod B E Q O � 1 � Q 2 � , 1 � Q � , 0 0 2 I Q I Q Q Q Q Q � Q mod B E I Q mod B E Q O �

Beyond first-order DCGs... Beyond first-order DCGs... 11/21 11/21 0 1 m Q Q Q 1 m Q -1 m m Q m 2 1, m � Q 1 � , m � Q 1, m � Q m � , m 1 m m I Q Q Q Q m 0 m 2 mod B E I Q mod B E Q O � 1 m = 0 Q m m m Q 2 G m m G m m m G m L m G m 3 Q m+1 m 4 m m

CDCGs: Performance analysis CDCGs: Performance analysis 12/21 12/21 E Q m m+1 O m = m m � m � 1 m 1 � m G m L m 3 G m 1 1 2 � � m G m L m 3 1 1 m Euler path Balance pairs c O m 1 2 m m m m mod B E DCG c 4 H err � m 1 m opt 2 log 4 � H err 1 , opt

CDCGs: Illustrative results CDCGs: Illustrative results 13/21 13/21 N N H error I S D k l I k I l A k I k , N 5 k 1 k 1 m B a ,a x , y , z m a G m G { I, X,Y ,Z } { X, Y } 4 5 20 m 2 2 1 0 1 S 2 m N I B 2 B m X m Y m X m Y m Y m I Q X m I Q Y m I Q X m 1 m Q Q i 2 Q exp x 3

CDCGs in the lab?... CDCGs in the lab?... 14/21 14/21 Ø Ø 2 U Q t exp S N t a t a Q , Q exp i t S N t t exp i s ds 2 0 t g j 2 i E Q t S N t a t a X DCG [1] X X DCG [2] X Q exp i E Q X Q exp i E Q 2 X exp 2 Q i E Q X exp i E Q Q Uncorrected Q 2

II. Progress towards II. Progress towards Optimized Optimized DCG Framework DCG Framework

Recap thus far... Recap thus far... 15/21 15/21 H S n n n L G m G n G n 4 5 2 adv

The CDCG framework revisited The CDCG framework revisited 16/21 16/21 T T U T Texp i ds H ctrl s H S , g H err Q exp i E Q 0 T i Q e Texp i ds H ctrl s H S 0 Gate synthesis 0 m m 1 mod B E Q O � Error cancellation � H

Boosting efficiency via parametric optimization Boosting efficiency via parametric optimization 17/21 17/21 1 T n n Q h l Texp i ds H ctrl s exp i h l H l l , T l l 0 l 1 l 1 E Q 2 3 2 3 4 mod B E Q O � � 1 z 1 mod B E Q h l , � l 0 2 z 2 mod B E Q h l , � l 0 z 1 z 2

Illustrative results Illustrative results 18/21 18/21 N N H error I S D k l I k I l A k I k z , N 5 z k 1 k 1 1 m 1 � m G m L m 3 G m 1 1 2 � 1 1 0 1 S 2 N I B 2 B Uncorrected � 2 14 3 2 20 � 365 � CDCG, simplified Optimized � 2 14 3 2 8 � 146 � CDCG, generic 21 � Q exp i 8 � 2 x

Accommodating drift via robust optimization Accommodating drift via robust optimization 19/21 19/21 H tot H S H err H ctrl t h t z z x h ( t ) H ' H z – x z 1 h l , � l U ctrl T Q 1 z 2 h l , � l E Q z h l , � l z 1 h l , � l z 2 h l , � l

Illustrative results Illustrative results 20/21 20/21 z 1 z 1 z 1 z 2 z Q exp i 8 x 8 0 10 min z 2 min 4 min , min T O z 1 10 min

Conclusion Conclusion 21/21 21/21