Combinating random and cyclic decoupling techniques

00000000000Combined Decoupling [4]ˆd 1 ˆd† 0ˆd 0ˆr 1ˆr † 0 ˆd † n c −1ˆd 2 ˆd† 1ˆd 1 ˆd† 0ˆd 0ˆr 0time T c + τ n c τ = T c 2τ τ 0Û(nT c ) = e −iˆr† n−1 (Ĥ1+Ĥ2+...)ˆr n−1 T c...e −iˆr† 1 (Ĥ1+Ĥ2+...)ˆr 1 Tc· e −iˆr† 0 (Ĥ1+Ĥ2+...)ˆr 0 T cError bound:ǫ(T) = max|Ψ 0 〉E ( 1 − |〈Ψ(T)|Ψ 0 〉| 2) ( ( ) ) 2≤ O ‖Ĥ0‖ 2 T c /2 TTc111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 T 0000000000011111111111 Q 0000000000011111111111 P1111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 1111111111100000000000[4] O. Kern and G. Alber, quant-ph/0506038Varenna, July 5th-15th, 2005 – p.5/7