Combinating random and cyclic decoupling techniques
Combinating random and cyclic decoupling techniques
Combinating random and cyclic decoupling techniques
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00000000000Combining <strong>r<strong>and</strong>om</strong> <strong>and</strong>deterministic <strong>decoupling</strong><strong>techniques</strong>Oliver Kernoliver.kern@physik.tu-darmstadt.deGernot AlberInstitut für Angew<strong>and</strong>te Physik64289 Darmstadt, GermanyIn collaboration with D. L. Shepelyansky (CNRS Toulouse)Supported by EU project EDIQIP111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 T 0000000000011111111111 Q 0000000000011111111111 P1111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 1111111111100000000000Varenna, July 5th-15th, 2005 – p.1/7
00000000000MotivationOur goal is to eliminate the action of a general two qubitHamiltonian:Ĥ 0 =∑0≤j
00000000000Deterministic Decoupling [1]ˆd 1 ˆd† 0ˆd 0 ˆd† n c −1ˆd 2 ˆd† 1ˆd 1 ˆd† 0ˆd 0time T c + τ n c τ = T c 2τ τ 0ˆd j ∈ {ˆ1, ˆX,Ŷ ,Ẑ}⊗n qj ∈ {0,...,n c − 1}111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 T 0000000000011111111111 Q 0000000000011111111111 P1111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 1111111111100000000000[1] e.g. L. Viola, E. Knill <strong>and</strong> S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999),M. Rötteler <strong>and</strong> P. Wocjan, quant-ph/0409135, ...Varenna, July 5th-15th, 2005 – p.3/7
00000000000Deterministic Decoupling [1]ˆd 1 ˆd† 0ˆd 0 ˆd† n c −1ˆd 2 ˆd† 1ˆd 1 ˆd† 0ˆd 0time T c + τ n c τ = T c 2τ τ 0Û(T c ) = e −i(Ĥ1+Ĥ2+...)T cError bound [2]:ǫ(T) = max|Ψ 0 〉(1 − |〈Ψ(T)|Ψ0 〉| 2) ( ) 2≤ O ‖Ĥ0‖ 2 TT c /2111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 T 0000000000011111111111 Q 0000000000011111111111 P1111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 1111111111100000000000[1] e.g. L. Viola, E. Knill <strong>and</strong> S. Lloyd, Phys. Rev. Lett. 82, 2417 (1999),M. Rötteler <strong>and</strong> P. Wocjan, quant-ph/0409135, ...[2] L. Viola <strong>and</strong> E. Knill, Phys. Rev. Lett. 94, 060502 (2005)Varenna, July 5th-15th, 2005 – p.3/7
00000000000R<strong>and</strong>om Decoupling [3]ˆr 5ˆr † 4ˆr 4ˆr † 3ˆr 3ˆr † 2ˆr 2ˆr † 1ˆr 1ˆr † 0ˆr 0time 5τ 4τ 3τ 2τ τ 0ˆr j ∈ {ˆ1, ˆX,Ŷ ,Ẑ}⊗n qÛ(nτ) = e −iˆr† n−1Ĥ0ˆr n−1 τ ...e −iˆr† 1Ĥ0ˆr 1τ · e −iˆr† 0Ĥ0ˆr 0 τ[3] O. Kern, G. Alber <strong>and</strong> D. L. Shepelyansky, Eur. Phys. J. D 32, 153 (2005)111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 T 0000000000011111111111 Q 0000000000011111111111 P1111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 1111111111100000000000Varenna, July 5th-15th, 2005 – p.4/7
00000000000R<strong>and</strong>om Decoupling [3]ˆr 5ˆr † 4ˆr 4ˆr † 3ˆr 3ˆr † 2ˆr 2ˆr † 1ˆr 1ˆr † 0ˆr 0time 5τ 4τ 3τ 2τ τ 0ˆr j ∈ {ˆ1, ˆX,Ŷ ,Ẑ}⊗n qÛ(nτ) = e −iˆr† n−1Ĥ0ˆr n−1 τ ...e −iˆr† 1Ĥ0ˆr 1τ · e −iˆr† 0Ĥ0ˆr 0 τError bound [2]:ǫ(T) = max|Ψ 0 〉 E( 1 − |〈Ψ(T)|Ψ 0 〉| 2) ≤ O( )‖Ĥ0‖ 2 Tτ111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 T 0000000000011111111111 Q 0000000000011111111111 P1111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 1111111111100000000000[3] O. Kern, G. Alber <strong>and</strong> D. L. Shepelyansky, Eur. Phys. J. D 32, 153 (2005)[2] L. Viola <strong>and</strong> E. Knill, Phys. Rev. Lett. 94, 060502 (2005)Varenna, July 5th-15th, 2005 – p.4/7
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 c111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 T 0000000000011111111111 Q 0000000000011111111111 P1111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 1111111111100000000000[4] O. Kern <strong>and</strong> G. Alber, quant-ph/0506038Varenna, July 5th-15th, 2005 – p.5/7
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 <strong>and</strong> G. Alber, quant-ph/0506038Varenna, July 5th-15th, 2005 – p.5/7
00000000000ReferencesDeterministic Decoupling Schemes [1]:e.g. L. Viola, E. Knill <strong>and</strong> S. Lloyd, Phys. Rev. Lett. 82, 2417(1999),M. Rötteler <strong>and</strong> P. Wocjan, quant-ph/0409135, ...R<strong>and</strong>om Decoupling [2]:O. Kern, G. Alber <strong>and</strong> D. L. Shepelyansky, Eur. Phys. J. D 32, 153(2005)Error Bounds for Deterministic <strong>and</strong> R<strong>and</strong>om Decoupling [3]:L. Viola <strong>and</strong> E. Knill, Phys. Rev. Lett. 94, 060502 (2005)111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 T 0000000000011111111111 Q 0000000000011111111111 P1111111111100000000000111111111110000000000011111111111000000000001111111111100000000000111111111110000000000011111111111000000000000000000000011111111111 1111111111100000000000Combined Decoupling [4]:O. Kern <strong>and</strong> G. Alber, quant-ph/0506038Thanks for your attention !Varenna, July 5th-15th, 2005 – p.7/7
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