Efficient Optimal Control for a Unitary Operation under Dissipative Evolution

Transcription

1 Effcent Optmal Control for a Untary Operaton under Dsspatve Evoluton Mchael Goerz, Danel Rech, Chrstane P. Koch Unverstät Kassel March 20, 2014 DPG Frühjahrstagung 2014, Berln Sesson Q 43 Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 1 / 20

2 Introducton: numercal optmal control Control Problem Fnd a tme-dependent control (e.g. laser pulse) that steers the system towards some desred goal (e.g. quantum gate) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 2 / 20

3 Introducton: numercal optmal control Control Problem Fnd a tme-dependent control (e.g. laser pulse) that steers the system towards some desred goal (e.g. quantum gate) defne optmzaton functonal for a guess pulse, solve the equaton of moton numercally modfy control pulse to mprove value of optmzaton functonal Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 2 / 20

4 Introducton: numercal optmal control Control Problem Fnd a tme-dependent control (e.g. laser pulse) that steers the system towards some desred goal (e.g. quantum gate) defne optmzaton functonal for a guess pulse, solve the equaton of moton numercally modfy control pulse to mprove value of optmzaton functonal ɛ ɛ new OCT teraton ɛ old Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 2 / 20

5 Introducton: numercal optmal control Control Problem Fnd a tme-dependent control (e.g. laser pulse) that steers the system towards some desred goal (e.g. quantum gate) defne optmzaton functonal for a guess pulse, solve the equaton of moton numercally modfy control pulse to mprove value of optmzaton functonal ɛ ɛ new OCT teraton ɛ old optmal : not lmted to smple ntutve schemes, operate at the quantum speed lmt Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 2 / 20

6 Gate optmzaton CPHASE = dag( 1, 1, 1, 1) CNOT = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 3 / 20

7 Gate optmzaton CPHASE = dag( 1, 1, 1, 1) CNOT = Goal: Maxmze F = 1 d d =1 Two-qubt gates: d = 4 Re Ψ Ô Û(T, 0, ɛ) Ψ Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 3 / 20

8 Gate optmzaton CPHASE = dag( 1, 1, 1, 1) CNOT = Goal: Maxmze F = 1 d d =1 Two-qubt gates: d = 4 ɛ(t) χ(t) ɛ Ĥ Ψ(t) 11 Ô Ô 10 ɛ new ɛ old 01 Ô Ô 00 t 0 t Re Ψ Ô Û(T, 0, ɛ) Ψ T Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 3 / 20

9 OCT for open quantum systems In the real world: decoherence Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 4 / 20

10 OCT for open quantum systems ˆρ(T ) = D(ˆρ(0)); for example ˆρ t = [Ĥ, ˆρ] + L D (ˆρ) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 5 / 20

11 OCT for open quantum systems ˆρ(T ) = D(ˆρ(0)); for example ˆρ t = [Ĥ, ˆρ] + L D (ˆρ) Lft F = 1 d d =1 Re Ψ Ô ˆPÛ(T, 0, ɛ)ˆp Ψ to Louvlle space. Kallush & Kosloff, Phys. Rev. A 73, (2006), Ohtsuk, New J. Phys. 12, (2010) Schulte-Herbrüggen et al., J. Phys. B 44, (2011),... F = 1 d 2 d 2 j=1 ] tr [Ôˆρj (0)Ô ˆρ j (T ) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 5 / 20

12 OCT for open quantum systems ˆρ(T ) = D(ˆρ(0)); for example ˆρ t = [Ĥ, ˆρ] + L D (ˆρ) Lft F = 1 d d =1 Re Ψ Ô ˆPÛ(T, 0, ɛ)ˆp Ψ to Louvlle space. Kallush & Kosloff, Phys. Rev. A 73, (2006), Ohtsuk, New J. Phys. 12, (2010) Schulte-Herbrüggen et al., J. Phys. B 44, (2011),... F = 1 d 2 d 2 j=1 ] tr [Ôˆρj (0)Ô ˆρ j (T ) ˆρ 1 = , ˆρ2 = , ˆρ3 = , d 2 matrces to propagate! (16 for two-qubt gate) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 5 / 20

13 OCT for open quantum systems ˆρ(T ) = D(ˆρ(0)); for example ˆρ t = [Ĥ, ˆρ] + L D (ˆρ) Lft F = 1 d d =1 Re Ψ Ô ˆPÛ(T, 0, ɛ)ˆp Ψ to Louvlle space. Kallush & Kosloff, Phys. Rev. A 73, (2006), Ohtsuk, New J. Phys. 12, (2010) Schulte-Herbrüggen et al., J. Phys. B 44, (2011),... Clam F = 1 d 2 d 2 j=1 ] tr [Ôˆρj (0)Ô ˆρ j (T ) We only need to propagate three matrces (ndependent of d), nstead of d 2. Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 5 / 20

14 A reduced set of densty matrces No need to characterze the full dynamcal map! much less nformaton requred to assess how well a desred untary s mplemented ˆρ 1 = , ˆρ 2 = , ˆρ 3 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 6 / 20

15 A reduced set of densty matrces No need to characterze the full dynamcal map! much less nformaton requred to assess how well a desred untary s mplemented 1 Do we stay n the logcal subspace? ˆρ 1 = , ˆρ 2 = , ˆρ 3 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 6 / 20

16 A reduced set of densty matrces No need to characterze the full dynamcal map! much less nformaton requred to assess how well a desred untary s mplemented 1 Do we stay n the logcal subspace? ˆρ 1 = , ˆρ 2 = , ˆρ 3 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 6 / 20

17 A reduced set of densty matrces No need to characterze the full dynamcal map! much less nformaton requred to assess how well a desred untary s mplemented 1 Do we stay n the logcal subspace? 2 Are we untary, and f yes, dd we mplement the rght gate? ˆρ 1 = , ˆρ 2 = , ˆρ 3 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 6 / 20

18 A reduced set of densty matrces No need to characterze the full dynamcal map! much less nformaton requred to assess how well a desred untary s mplemented 1 Do we stay n the logcal subspace? 2 Are we untary, and f yes, dd we mplement the rght gate? ˆρ 1 = , ˆρ 2 = , ˆρ 3 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 6 / 20

19 A reduced set of densty matrces No need to characterze the full dynamcal map! much less nformaton requred to assess how well a desred untary s mplemented 1 Do we stay n the logcal subspace? 2 Are we untary, and f yes, dd we mplement the rght gate? ˆρ 1 = , ˆρ 2 = , ˆρ 3 = E.g. Ô = dag( 1, 1, 1, 1); For Û = dag(e φ00, e φ01, e φ10, e φ11 ) usng just ˆρ 1 wll not dstngush Û from Ô. (Ûˆρ 1 Û = Ôˆρ 1 Ô = ˆρ 1 ) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 6 / 20

20 A reduced set of densty matrces No need to characterze the full dynamcal map! much less nformaton requred to assess how well a desred untary s mplemented 1 Do we stay n the logcal subspace? 2 Are we untary, and f yes, dd we mplement the rght gate? ˆρ 1 = , ˆρ 2 = , ˆρ 3 = E.g. Ô = dag( 1, 1, 1, 1); For Û = dag(e φ00, e φ01, e φ10, e φ11 ) usng just ˆρ 1 wll not dstngush Û from Ô. (Ûˆρ 1 Û = Ôˆρ 1 Ô = ˆρ 1 ) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 6 / 20

21 Effcent gate optmzaton n Louvlle space Optmzaton States ˆρ 1 = , ˆρ 2 = , ˆρ 3 = populatons phases subspace Functonal J T = 1 3 j=1 w ] j [Ôˆρj tr[ˆρ 2 j (0)] tr Ô D[ˆρ j ] Allow for dfferent weghts ( w j = 1) J T = 0 ff for all ˆρ j : D[ˆρ j ] target state mplemented untary gate Ô. Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 7 / 20

22 Example 1 Optmzaton of a Dagonal Gate usng Rydberg Atoms Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 8 / 20

23 Two trapped neutral atoms Sngle-qubt Hamltonan 1 τ = 25 ns r Ω B (t) Ω R (t) 0 1 In the RWA: Ω R(t) 0 Ĥ 1q = 0 E Ω 1 R(t) Ω B(t) Ω B(t) 0 Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 9 / 20

24 Two trapped neutral atoms Sngle-qubt Hamltonan 1 τ = 25 ns r Ω B (t) Ω R (t) 0 1 In the RWA: Ω R(t) 0 Ĥ 1q = 0 E Ω 1 R(t) Ω B(t) Ω B(t) 0 Two-qubt Hamltonan Ĥ 2q = Ĥ 1q Ĥ 1q U rr rr Dpole-dpole nteracton when both atoms n Rydberg state. Only dagonal gates! Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 9 / 20

25 Optmzaton of a Rydberg gate 0.20 τ = 25 ns, T = 75 ns full bass 0.15 gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 10 / 20

26 Optmzaton of a Rydberg gate 0.20 τ = 25 ns, T = 75 ns full bass 3 states 0.15 gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 10 / 20

27 Dagonal gates 1 τ = 25 ns r Ω B (t) Ω R (t) 0 1 no couplng between 0, 1 Û = dag(e φ00, e φ01, e φ10, e φ11 ) only dagonal gates are possble Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 11 / 20

28 Dagonal gates 1 τ = 25 ns r Ω B (t) Ω R (t) 0 1 no couplng between 0, 1 Û = dag(e φ00, e φ01, e φ10, e φ11 ) only dagonal gates are possble ˆρ 1 = , ˆρ 2 = , ˆρ 3 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 11 / 20

29 Dagonal gates 1 τ = 25 ns r Ω B (t) Ω R (t) 0 1 no couplng between 0, 1 Û = dag(e φ00, e φ01, e φ10, e φ11 ) only dagonal gates are possble ˆρ 1 = , ˆρ 2 = , ˆρ 3 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 11 / 20

30 Optmzaton of a Rydberg gate 0.20 τ = 25 ns, T = 75 ns full bass 3 states 0.15 gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 12 / 20

31 Optmzaton of a Rydberg gate 0.20 τ = 25 ns, T = 75 ns full bass 3 states 2 states 0.15 gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 12 / 20

32 Optmzaton of a Rydberg gate τ = 25 ns, T = 75 ns full bass 3 states 2 states 2 states (weghted) gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 12 / 20

33 Optmzaton of a Rydberg gate asymptotc behavor 10-1 τ = 25 ns, T = 75 ns full bass 3 states 2 states 2 states (weghted) gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 13 / 20

34 Example 2 Optmzaton of a non-dagonal gate usng transmon qubts Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 14 / 20

35 Two coupled transmon qubts A. Blas et al. PRA 75, (2007) Cavty medates drven exctaton of qubt nteracton between left and rght qubt left qubt g eff ɛ(t) 0 J eff j rght qubt gj eff ɛ(t) 0 Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 15 / 20

36 Two coupled transmon qubts A. Blas et al. PRA 75, (2007) Cavty medates drven exctaton of qubt nteracton between left and rght qubt left qubt g eff ɛ(t) 0 J eff j rght qubt gj eff ɛ(t) 0 Many gates possble, e.g. SWAP: Ô = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 15 / 20

37 Optmzaton of a transmon gate 10 0 full bass gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 16 / 20

38 Optmzaton of a transmon gate 10 0 full bass 3 states gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 16 / 20

39 Optmzaton of a transmon gate 10 0 full bass 3 states 3 states (weghted) gate error OCT teraton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 16 / 20

40 Optmzaton of a transmon gate CPU tme 10 0 full bass 3 states 3 states (weghted) gate error number of propagatons (equvalent to CPU tme) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 17 / 20

41 Optmzaton of a transmon gate CPU tme 10 0 full bass 3 states 3 states (weghted) asymptotc regon gate error number of propagatons (equvalent to CPU tme) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 17 / 20

42 Usng pure states only ˆρ 1 = , ˆρ 2 = , ˆρ 3 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 18 / 20

43 Usng pure states only ˆρ 1 = , ˆρ 2 = , ˆρ 3 = ˆρ 1 = , ˆρ 2 = , ˆρ 3 = , ˆρ 4 = , ˆρ 5 = Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 18 / 20

44 optmzaton of a transmon gate CPU tme 10 0 full bass 3 states 3 states (weghted) asymptotc regon gate error number of propagatons (equvalent to CPU tme) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 19 / 20

45 optmzaton of a transmon gate CPU tme 10 0 full bass 3 states 3 states (weghted) 5 states asymptotc regon gate error number of propagatons (equvalent to CPU tme) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 19 / 20

46 Concluson A set of three densty matrces s suffcent for gate optmzaton: (ndependent of dmenson of Hlbert space!) one to check dynamcal map on subspace one to check the bass one to check the phases Further reducton possble for restrcted systems States can (should!) be weghted accordng to physcal nterpretaton Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 20 / 20

47 Concluson A set of three densty matrces s suffcent for gate optmzaton: (ndependent of dmenson of Hlbert space!) one to check dynamcal map on subspace one to check the bass one to check the phases Further reducton possble for restrcted systems States can (should!) be weghted accordng to physcal nterpretaton Gate optmzaton n open quantum systems wth large Hlbert spaces have become sgnfcantly more feasble. Reference: M. H. Goerz, D. M. Rech, C. P. Koch. arxv: In press: New Journal of Physcs (specal ssue) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 20 / 20

48 Concluson A set of three densty matrces s suffcent for gate optmzaton: (ndependent of dmenson of Hlbert space!) one to check dynamcal map on subspace one to check the bass one to check the phases Further reducton possble for restrcted systems States can (should!) be weghted accordng to physcal nterpretaton Gate optmzaton n open quantum systems wth large Hlbert spaces have become sgnfcantly more feasble. Reference: M. H. Goerz, D. M. Rech, C. P. Koch. arxv: In press: New Journal of Physcs (specal ssue) Thank you Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 20 / 20

49 Optmzed dynamcs of the Rydberg gate populaton feld ( rel. unts ) tme (ns) 00 r0 0r rr nt Ω R (t) guess Ω B (t) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 21 / 20

50 populaton feld ( rel. unts ) populaton feld ( rel. unts ) wth dsspaton, full bass tme (ns) wthout dsspaton, full bass tme (ns) 00 r0 0r rr nt Ω R (t) guess Ω B (t) 00 r0 0r rr nt Ω R (t) guess Ω B (t) populaton feld ( rel. unts ) wth dsspaton, two states (weghted) 1 00 r r rr nt tme (ns) Ω R (t) guess Ω B (t) wthout dsspaton, two states (weghted) 1 00 r r rr nt 0 populaton feld ( rel. unts ) tme (ns) Ω R (t) guess Ω B (t) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 22 / 20

51 Two Coupled Transmon Qubts A. Blas et al. PRA 75, (2007) J. Koch et al. PRA 76, (2007) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 23 / 20

52 Two Coupled Transmon Qubts A. Blas et al. PRA 75, (2007) J. Koch et al. PRA 76, (2007) Full Hamltonan Ĥ = ω c â â + ω }{{} 1ˆb 1ˆb 1 + ω 2ˆb 2ˆb 2 1 }{{} 2 (α 1ˆb 1ˆb 1ˆb 1ˆb 1 + α 2ˆb 2ˆb 2ˆb 2ˆb 2 ) + 1 }{{} g 1 (ˆb 1â + ˆb 1 â ) + g 2 (ˆb 2â + ˆb 2 â ) + ɛ (t)â + ɛ(t)â }{{}}{{} 4 5 Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 23 / 20

53 Effectve Hamltonan Ĥ eff = q=1,2 + j N q 1 =0 (ω (q) Jj eff (Ĉ (1) Ĉ + (2) + χ (q) )ˆΠ (q) + N q 1 g q=1,2 j + Ĉ + (1) =0 Ĉ (2) j ). eff (q) ɛ(t)(ĉ + (q) + Ĉ (q) ) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 24 / 20

54 Effectve Hamltonan Ĥ eff = q=1,2 + j N q 1 =0 (ω (q) Jj eff (Ĉ (1) Ĉ + (2) + χ (q) )ˆΠ (q) + N q 1 g q=1,2 j + Ĉ + (1) =0 Ĉ (2) j ). eff (q) ɛ(t)(ĉ + (q) + Ĉ (q) ) wth ω (q) = ω q 1 2 ( 2 )α q, g (q) = g q ˆΠ (q) χ (q) = eff (q) g = = q, Ĉ + (q) (ω (q) (g (q) ) 2 ω (q) (ω (q) 1 ωc ) g (q) ω (q) 1 ωc ) = 1 q Jj eff = 1 eff (1) 2g g (2) j + 1 eff (2) 2gj g (1) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 24 / 20

55 IBM Qubt Poletto et al. PRL 109, (2012) qubt frequency ω GHz qubt frequency ω GHz drve frequency ω d GHz anharmoncty α MHz anharmoncty α MHz effectve qubt-qubt couplng J -2.3 MHz qubt 1,2 decay tme T µs, 32.0 µs qubt 1,2 dephasng tme T µs, 16.0 µs Effectve Hamltonan Ĥ eff = jq ( (ω (q) + χ (q) )ˆΠ (q) eff (q) + g ɛ(t)(ĉ + (q) + Ĉ (q) ) + J eff j (Ĉ (1) Ĉ + (2) ) j + c.c.) Master Equaton L D (ˆρ) = ( ) N 1 ] N 1 ] γ q D [ 1 q ˆρ + γ φ,q D [ q ˆρ, q=1,2 =1 [Â] wth D ˆρ = ˆρ ( ˆρ 1 + ˆρ Â) 2 =0 Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 25 / 20

56 IBM Qubt Poletto et al. PRL 109, (2012) qubt frequency ω GHz qubt frequency ω GHz drve frequency ω d GHz anharmoncty α MHz anharmoncty α MHz effectve qubt-qubt couplng J -2.3 MHz qubt 1,2 decay tme T µs, 32.0 µs qubt 1,2 dephasng tme T µs, 16.0 µs Effectve Hamltonan Ĥ eff = jq ( (ω (q) + χ (q) )ˆΠ (q) eff (q) + g ɛ(t)(ĉ + (q) + Ĉ (q) ) + J eff Near resonance of α 1 wth ω 1 ω 2 j (Ĉ (1) Ĉ + (2) ) j + c.c.) Master Equaton L D (ˆρ) = ( ) N 1 ] N 1 ] γ q D [ 1 q ˆρ + γ φ,q D [ q ˆρ, q=1,2 =1 [Â] wth D ˆρ = ˆρ ( ˆρ 1 + ˆρ Â) 2 =0 Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 25 / 20

57 IBM Qubt Poletto et al. PRL 109, (2012) qubt frequency ω GHz qubt frequency ω GHz drve frequency ω d GHz anharmoncty α MHz anharmoncty α MHz effectve qubt-qubt couplng J -2.3 MHz qubt 1,2 decay tme T µs, 32.0 µs qubt 1,2 dephasng tme T µs, 16.0 µs Effectve Hamltonan Ĥ eff = jq ( (ω (q) + χ (q) )ˆΠ (q) eff (q) + g ɛ(t)(ĉ + (q) + Ĉ (q) ) + J eff Near resonance of α 1 wth ω 1 ω 2 sngle frequency drve centered between two qubts j (Ĉ (1) Ĉ + (2) ) j + c.c.) Master Equaton L D (ˆρ) = ( ) N 1 ] N 1 ] γ q D [ 1 q ˆρ + γ φ,q D [ q ˆρ, q=1,2 =1 [Â] wth D ˆρ = ˆρ ( ˆρ 1 + ˆρ Â) 2 =0 Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 25 / 20

58 Transmon Optmzed Pulse feld (MHz) feld phase (π) spectrum (arb. unts) Ω guess arg(ω) arg(guess) tme (ns) frequency (MHz) Mchael Goerz Un Kassel Effcent OCT for a Untary under Dsspaton 26 / 20

59 populaton Transmon Populaton Dynamcs log tme (ns) 1.5 rght qubt σj 1 j left qubt σ rght qubt σj 1 j left qubt σ log. subsp log. populaton 0.5 qubt exctaton 1.5 qubt exctaton (b) (a) populaton tme (ns) tme (ns) (b) qubt exctaton ulaton Mchael Goerz FIG. rght 10: Populaton dynamcs for (t ˆ = 0) = 01h01 (a) and (t ˆ = 0) = 11h11 (b) und qubt σj Ψ(t = 0) = 01 each of the two propagated Ψ(t =value 0) = 11 states, the expectaton of the rght qubt exctaton quant j devaton n gray, the expectaton value for the correspondng qua panel, wth the standard s shown n the center panel, and the populaton dynamcs for all the logcal subspace sta (colored lnes), along wth the total populaton n the logcal subspace (black lne). left qubt σ APPENDIX A: THREE STATES ARE SUFFICIENT TO ASSESS WHETHE UNITARY IS IMPLEMENTED In the followng we 10 dscuss the functonal Jdst, Un Kassel Effcent OCT for a Untary under Dsspaton 27 / 20