Quantum Error Correction by Optimal Control Basic Systems Concepts, - PowerPoint PPT Presentation

Decoupling Open Systems CNOT plus Decoupling J. Phys. B 44 (2011) 154013 Typical : system drives outside protected subspace Basic Systems no relaxation with relaxation ( T 2 , T 1 ) Theory 1 1 DYNAMO Platform relax.−opt. Applications I: trace fidelity ( F tr ) 0.9 0.9 Error Correction NV Centres • Markovian 0.8 0.8 • Non-Markovian • Sum-Up 0.7 0.7 Applications II: Fixed-Point Engineering time−opt. 0.6 0.6 Applications III: 0.4 0.6 0.8 1 0.4 0.6 0.8 1 Noise Switching time [1/J iso ] time [1/J iso ] Conclusions mean of 15 time-optimised pulse sequences � � dissipation affects sequences differently � relaxation-optimised: systematic substantial gain

Control of Non-Markovian Open Systems Qubit Coupled via Two-Level Fluctuator to Spin Bath with P . Rebentrost and F . Wilhelm −1 10 κ =0 κ =0.0001 κ =0.001 Basic Systems −2 10 Theory κ =0.005 κ =0.02 Gate error DYNAMO Platform κ =0.2 −3 10 Applications I: Error Correction NV Centres −4 • Markovian 10 • Non-Markovian • Sum-Up Applications II: −5 10 Fixed-Point Engineering −3 10 T1 Limit Applications III: 2 T1 Limit Noise Switching κ =0.005 −4 10 Conclusions ← R ABI pulse −2 10 � ← cut error by factor ≤ 10 � with optimal control −3 10 Penalty Rabi opt. � 2 4 6 8 10 12 14 16 18 Time [1/ ∆ ] PRL 102 090401 (2009)

Control of Non-Markovian Open Systems PRL 102 090401 (2009) � Principle: embed to Markovian and project Basic Systems Theory DYNAMO Platform Applications I: Error Correction Ad W ( t ) ρ 0 = ρ SE ( 0 ) ⊗ ρ B ( 0 ) ρ ( t ) = W ( t ) ρ 0 W † ( t ) NV Centres − − − − − − − − → • Markovian   • Non-Markovian   • Sum-Up � tr B � tr B Π SE Π SE Applications II: Fixed-Point F SE ( t ) Engineering ρ SE ( 0 ) − − − − − − − − → ρ SE ( t ) Applications III: Markovian   Noise Switching   � tr E � tr E Π S Π S Conclusions � F S ( t ) ρ S ( 0 ) − − − − − − − − − → ρ S ( t ) � non − Markovian �

Control of Open Systems Sum-Up J. Phys. B 44 154013 (2011) Basic Systems Theory � Gain: relax.-optimised control vs. time-opt. control DYNAMO Platform Applications I: Error Correction NV Centres • Markovian • Non-Markovian category Markovian non-Markovian • Sum-Up Applications II: Fixed-Point Engineering no encoding: Applications III: full Liouville space small–medium medium–big Noise Switching encoding: Conclusions difficult 1 protected subspace big � � � 1 problem roots in finding a viable protected subspace

Markovian Fixed-Point Engineering Algorithm Devise { V k } such that ρ ∞ is unique global fixed point of ρ = − Γ ρ = � Basic Systems k − 1 V k ρ V † 2 { V † ˙ k V k , ρ } + Theory k DYNAMO Platform Applications I: Error Correction 1 characterize target fixed-point ρ ∞ by its symmetries: Applications II: centraliser cent ( ρ ∞ ) := { s | [ s , ρ ∞ ] = 0 } Fixed-Point Engineering Ex.: Graph States 2 determine max. abelian subalgebra a of cent ( ρ ∞ ) System Algebra Lie Structure Applications III: 3 pick translations τ according to a Noise Switching Conclusions 4 translate into Lindblad terms { V k := σ ( k ) + i · σ ( k ) q } � p with τ m �→ σ m = i σ p ◦ σ q or m = p ⋆ q � � 5 ensure uniqueness of ρ ∞

Markovian Fixed-Point Engineering Algorithm Devise { V k } such that ρ ∞ is unique global fixed point of ρ = − Γ ρ = � Basic Systems k − 1 V k ρ V † 2 { V † ˙ k V k , ρ } + Theory k DYNAMO Platform Applications I: Error Correction 1 characterize target fixed-point ρ ∞ by its symmetries: Applications II: centraliser cent ( ρ ∞ ) := { s | [ s , ρ ∞ ] = 0 } Fixed-Point Engineering Ex.: Graph States 2 determine max. abelian subalgebra a of cent ( ρ ∞ ) System Algebra Lie Structure Applications III: 3 pick translations τ according to a Noise Switching Conclusions 4 translate into Lindblad terms { V k := σ ( k ) + i · σ ( k ) q } � p with τ m �→ σ m = i σ p ◦ σ q or m = p ⋆ q � � 5 ensure uniqueness of ρ ∞

Markovian Fixed-Point Engineering Algorithm Devise { V k } such that ρ ∞ is unique global fixed point of ρ = − Γ ρ = � Basic Systems k − 1 V k ρ V † 2 { V † ˙ k V k , ρ } + Theory k DYNAMO Platform Applications I: Error Correction 1 characterize target fixed-point ρ ∞ by its symmetries: Applications II: centraliser cent ( ρ ∞ ) := { s | [ s , ρ ∞ ] = 0 } Fixed-Point Engineering Ex.: Graph States 2 determine max. abelian subalgebra a of cent ( ρ ∞ ) System Algebra Lie Structure Applications III: 3 pick translations τ according to a Noise Switching Conclusions 4 translate into Lindblad terms { V k := σ ( k ) + i · σ ( k ) q } � p with τ m �→ σ m = i σ p ◦ σ q or m = p ⋆ q � � 5 ensure uniqueness of ρ ∞

Markovian Fixed-Point Engineering Algorithm Devise { V k } such that ρ ∞ is unique global fixed point of ρ = − Γ ρ = � Basic Systems k − 1 V k ρ V † 2 { V † ˙ k V k , ρ } + Theory k DYNAMO Platform Applications I: Error Correction 1 characterize target fixed-point ρ ∞ by its symmetries: Applications II: centraliser cent ( ρ ∞ ) := { s | [ s , ρ ∞ ] = 0 } Fixed-Point Engineering Ex.: Graph States 2 determine max. abelian subalgebra a of cent ( ρ ∞ ) System Algebra Lie Structure Applications III: 3 pick translations τ according to a Noise Switching Conclusions 4 translate into Lindblad terms { V k := σ ( k ) + i · σ ( k ) q } � p with τ m �→ σ m = i σ p ◦ σ q or m = p ⋆ q � � 5 ensure uniqueness of ρ ∞

Markovian Fixed-Point Engineering Algorithm Devise { V k } such that ρ ∞ is unique global fixed point of ρ = − Γ ρ = � Basic Systems k − 1 V k ρ V † 2 { V † ˙ k V k , ρ } + Theory k DYNAMO Platform Applications I: Error Correction 1 characterize target fixed-point ρ ∞ by its symmetries: Applications II: centraliser cent ( ρ ∞ ) := { s | [ s , ρ ∞ ] = 0 } Fixed-Point Engineering Ex.: Graph States 2 determine max. abelian subalgebra a of cent ( ρ ∞ ) System Algebra Lie Structure Applications III: 3 pick translations τ according to a Noise Switching Conclusions 4 translate into Lindblad terms { V k := σ ( k ) + i · σ ( k ) q } � p with τ m �→ σ m = i σ p ◦ σ q or m = p ⋆ q � � 5 ensure uniqueness of ρ ∞

Fixed-Points I Graph States, Topol. States Graph abelian subalgebra a { τ m } { V k } V 1 = y 1 + i · zz � xz , zx � τ xz Basic Systems Theory V 2 = 1 y + i · zz τ zx DYNAMO Platform � xz 1 , zxz , 1 zx � V 1 = y 11 + i · zz 1 τ xz 1 Applications I: Error Correction V 2 = 1 y 1 + i · zzz τ zxz Applications II: V 3 = 11 y + i · 1 zz τ 1 zx Fixed-Point Engineering Ex.: Graph States V 1 = y 11 + i · zzz � xzz , zxz , zzx � τ xzz System Algebra Lie Structure V 2 = 1 y 1 + i · zzz τ zxz Applications III: V 3 = 11 y + i · zzz Noise Switching τ zzx Conclusions � xz 1 z , zxz 1 , 1 zxz , z 1 zx � V 1 = y 111 + i · zz 1 z � τ xz 1 z � V 2 = 1 y 11 + i · zzz 1 τ zxz 1 � V 3 = 11 y 1 + i · 1 zzz τ 1 zxz V 4 = 111 y + i · z 1 zz τ z 1 zx

Fixed-Points II More States Target FP { τ m } { V k } ground state V 1 = σ + 11 .. 1 Basic Systems τ z 11 .. 1 Theory V 2 = 1 σ + 1..1 & perms. τ 1 z 1 .. 1 DYNAMO Platform · · · · · · Applications I: GHZ state V 1 = y 1 .. 1 + i · zx .. x τ xx .. x Error Correction V 2 = x 11 .. 1 + i · yz 1 .. 1 τ zz 1 .. 1 Applications II: Fixed-Point V 3 = 1 x 1 .. 1 + i · 1 yz .. 1 τ 1 zz 1 .. 1 Engineering · · · · · · Ex.: Graph States System Algebra W state V 1 = y 11 + i · zzz − τ zz .. z Lie Structure V 2 = σ + 11 .. 1 − 1 σ + 1 .. 1 Applications III: τ z 1 .. 1 − τ 1 z .. 1 Noise Switching V 3 = 1 σ + 11 .. 1 − 11 σ + 1 .. 1 τ 1 z 1 .. 1 − τ 11 z .. 1 Conclusions · · · · · · � Dicke state V 1 = y 11 .. 1 + i · zzz .. z − τ zz .. z � V 2 = σ + σ + 11 .. 1 − σ + 1 σ + 1 .. 1 τ zz 11 .. 1 − τ 1 z 1 z .. 1 � V 3 = 1 σ + σ + 11 .. 1 − 1 σ + 1 σ + 1 .. 1 τ 11 zz 1 .. 1 − τ 11 z 1 z .. 1 · · · · · ·

System Algebra of Controlled Markov Maps Relation to Lie Wedges Rep. Math. Phys. 64 (2009) 93 Consider the Lindblad control system Σ � � ( i � H 0 + ˆ Γ 0 ) + i � ρ ( 0 ) := ρ 0 ρ = − ˙ H u ρ Basic Systems Theory H u := � Γ 0 ( ρ ) := � V k ρ V † k − 1 2 { V † with � u j ( t ) � H j and � k V k , ρ } + . DYNAMO Platform j k Applications I: Error Correction Applications II: Fixed-Point Engineering Ex.: Graph States System Algebra Lie Structure Applications III: Noise Switching Conclusions � � �

System Algebra of Controlled Markov Maps Relation to Lie Wedges Rep. Math. Phys. 64 (2009) 93 Consider the Lindblad control system Σ � � ( i � Γ 0 ) + i � H 0 + ˆ ρ ( 0 ) := ρ 0 ρ = − ˙ ρ Basic Systems H u Theory H u := � Γ 0 ( ρ ) := � V k ρ V † k − 1 2 { V † DYNAMO Platform with � u j ( t ) � H j and � k V k , ρ } + . j k Applications I: Error Correction Applications II: Embedding I Fixed-Point Engineering Ex.: Graph States The system Lie algebra g Σ ⊆ g LK given as Lie closure System Algebra Lie Structure Applications III: g Σ := � ( iH 0 + Γ 0 ) , iH j | j = 1 , . . . , m � Lie Noise Switching Conclusions comprises the Lie wedge w Σ ⊆ g Σ . � � �

System Algebra of Controlled Markov Maps Relation to Lie Wedges Rep. Math. Phys. 64 (2009) 93 Consider the Lindblad control system Σ � � Basic Systems ( i � H 0 + ˆ Γ 0 ) + i � ρ ( 0 ) := ρ 0 ρ = − ˙ H u ρ Theory DYNAMO Platform H u := � Γ 0 ( ρ ) := � V k ρ V † k − 1 2 { V † with � u j ( t ) � H j and � k V k , ρ } + . Applications I: j k Error Correction Embedding II Applications II: Fixed-Point Engineering The Lindblad-Kossakowski Lie algebra g LK reads Ex.: Graph States System Algebra Lie Structure g LK := gl ( her N 2 ) ⊕ s i 0 Applications III: Noise Switching Conclusions with i 0 ≃ R N 2 . It generates a group of affine maps � � G := GL ( her N 2 ) ⊗ s I 0 ⊇ T � embracing the Lie-semigroup of LK-quantum maps T .

Algebraic Structure: 2-Qubit Examples I Lie Wedges and Embedding in System Algebras Noise Lindblad-V Control-H Drift-H dim ( g Σ ) dim ( w Σ – w Σ ) Basic Systems unital ( y , z ) 1 x1,1x z 1+1 z + zz 225 11 Theory DYNAMO Platform Applications I: deph. z 1 x 1 –”– 22 6 Error Correction –”– –”– 1 x –”– 5 4 Applications II: bit-flip x 1 x 1 –”– 16 4 Fixed-Point Engineering –”– –”– 1 x –”– 52 4 Ex.: Graph States System Algebra Lie Structure unital ( y , z ) 1 x1,1x z 1+1 z + H xxx 225 12 Applications III: Noise Switching Conclusions deph. z 1 x 1 –”– 225 6 � –”– –”– 1 x –”– 225 4 � bit-flip x 1 x 1 –”– 124 4 � –”– –”– 1 x –”– 225 4

Algebraic Structure: 2-Qubit Examples II Lie Wedges and Embedding in System Algebras Basic Systems Noise Lindblad-V Control-H Drift-H dim ( w Σ – w Σ ) g Σ Theory DYNAMO Platform deph. z 1 , 1 z su ( 4 ) z 1+1 z + zz g LK 135 0 Applications I: –”– z 1 , 1 z su ( 2 ) ⊕ su ( 2 ) –”– 21 Error Correction g LK 0 –”– z 1 , 1 z , zz su ( 2 ) ⊕ su ( 2 ) –”– g LK 27 Applications II: 0 Fixed-Point deph. z 1 , 1 z x 1 , 1 x –”– g LK 14 Engineering 0 Ex.: Graph States System Algebra depol. iso 2 su ( 4 ) –”– su ( 4 ) + R Γ 16 � Lie Structure –”– iso 1 : 1 su ( 2 ) ⊕ su ( 2 ) –”– su ( 2 ) ⊕ � su ( 2 ) + R Γ 7 Applications III: � Noise Switching Conclusions amp. +1,1+ su ( 4 ) –”– g LK . . . � damp. +1,1+ su ( 2 ) ⊕ su ( 2 ) –”– g LK . . . � �

Bilinear Control Systems Unified Approach PRA 84 022305 (2011) � � � ˙ X ( t ) = − A + u j ( t ) B j X ( t ) Basic Systems j Theory X ( t ) : ‘state’; A : drift; B j : control Hamiltonians; u j : control amplitudes DYNAMO Platform Applications I: Error Correction Setting and Task ‘State’ Drift Controls Applications II: X ( t ) A B j Fixed-Point Engineering closed systems: Applications III: pure-state transfer X ( t ) = | ψ ( t ) � iH 0 iH j Noise Switching gate synthesis (fixed global phase) X ( t ) = U ( t ) iH 0 iH j DYNAMO Extension state transfer i � i � X ( t ) = ρ ( t ) H 0 H j New Reachability Theorems gate synthesis (free global phase) X ( t ) = � i � i � U ( t ) H 0 H j Examples Open vs Closed Loop open systems: Conclusions state transfer I i � i � X ( t ) = ρ ( t ) H 0 + Γ H j � quantum-map synthesis i � i � X ( t ) = F ( t ) H 0 + Γ H j � � � H is Hamiltonian commutator superoperator (generating � U := U ( · ) U † ) in Liouville space.

Bilinear Control Systems Unified Approach PRA 84 022305 (2011) � � � ˙ X ( t ) = − A + u j ( t ) B j X ( t ) Basic Systems j Theory X ( t ) : ‘state’; A : drift; B j : control Hamiltonians; u j : control amplitudes DYNAMO Platform Applications I: Error Correction Setting and Task ‘State’ Drift Controls Applications II: X ( t ) A B j Fixed-Point Engineering closed systems: Applications III: pure-state transfer X ( t ) = | ψ ( t ) � iH 0 iH j Noise Switching gate synthesis (fixed global phase) X ( t ) = U ( t ) iH 0 iH j DYNAMO Extension state transfer i � i � X ( t ) = ρ ( t ) H 0 H j New Reachability Theorems gate synthesis (free global phase) X ( t ) = � i � i � U ( t ) H 0 H j Examples Open vs Closed Loop open systems: Conclusions state transfer I i � i � X ( t ) = ρ ( t ) H 0 + Γ H j � quantum-map synthesis i � i � X ( t ) = F ( t ) H 0 + Γ H j � � � H is Hamiltonian commutator superoperator (generating � U := U ( · ) U † ) in Liouville space.

Bilinear Control Systems Unified Approach � � � ˙ X ( t ) = − A + u j ( t ) B j X ( t ) Basic Systems j Theory X ( t ) : ‘state’; A : drift; B j : control Hamiltonians; u j : control amplitudes DYNAMO Platform Applications I: Error Correction Setting and Task ‘State’ Drift Controls Applications II: X ( t ) A B j Fixed-Point Engineering closed systems: Applications III: pure-state transfer X ( t ) = | ψ ( t ) � iH 0 iH j Noise Switching gate synthesis (fixed global phase) X ( t ) = U ( t ) iH 0 iH j DYNAMO Extension state transfer i � i � X ( t ) = ρ ( t ) H 0 H j New Reachability Theorems gate synthesis (free global phase) X ( t ) = � i � i � U ( t ) H 0 H j Examples Open vs Closed Loop open systems: Conclusions state transfer I i � i � X ( t ) = ρ ( t ) H 0 + Γ H j � quantum-map synthesis i � i � X ( t ) = F ( t ) H 0 + Γ H j state transfer II i � i � X ( t ) = ρ ( t ) H 0 H j , Γ j � � U := U ( · ) U † ) in Liouville space. � H is Hamiltonian commutator superoperator (generating �

Noise Switching as Control Extension to DYNAMO arXiv:1206.4945 Basic Systems Theory � add switchable noise amplitudes as further controls DYNAMO Platform Applications I: Error Correction Applications II: Fixed-Point Engineering Applications III: Noise Switching DYNAMO Extension New Reachability Theorems Examples Open vs Closed Loop Conclusions � � �

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise quant-ph/1206.4945 � 0 1 � switchable amp-damp noise: γ ( t ) · Γ L with V a := 1 in l ⊗ 0 0 Basic Systems Γ L ( ρ ) = 1 2 { V † a V a , ρ } + − V a ρ V † Theory a DYNAMO Platform Applications I: Error Correction Theorem (’woodcut’ version) Applications II: Fixed-Point Let Σ a be an n-spin- 1 2 ZZ-coupled unitarily controllable Engineering system. Applications III: Noise Switching Adding bang-bang switchable ( γ ( t ) ∈ [ 0 , 1 ] ) amp-damp DYNAMO Extension New Reachability Theorems noise on 1 spin allows that any target state can be Examples Open vs Closed Loop reached from any initial state Conclusions � Reach Σ a ( ρ 0 ) = { all density ops. } for all ρ 0 . � �

Reachable Sets II: Unital Controlled Bit Flip Noise quant-ph/1206.4945 switchable bit-flip noise: γ ( t ) · Γ L with V b := 1 l ⊗ σ x / 2 in Basic Systems Theory Γ L ( ρ ) = 1 2 { V † b V b , ρ } + − V b ρ V † DYNAMO Platform b Applications I: Error Correction Theorem (’woodcut’) Applications II: Fixed-Point Engineering Let Σ a be an n-spin- 1 2 ZZ-coupled unitarily controllable Applications III: system. Noise Switching DYNAMO Extension Adding bang-bang switchable bit-flip noise on 1 spin New Reachability Theorems Examples allows that any target state majorised by the initial state Open vs Closed Loop Conclusions can be reached � Reach Σ b ( ρ 0 ) = { ρ | ρ ≺ ρ 0 } for all ρ 0 . � �

Noise-Driven State Transfer I & II Transfer between Pairs of Random States arXiv:1206.4945 Example Basic Systems system: 3-qubit Ising- ZZ chain, x , y -controls, Theory controllable noise on terminal qubit DYNAMO Platform Applications I: task I: rand ρ 0 → ρ tar by amp-damp Error Correction Applications II: task II: rand ρ 0 → ρ tar ≺ ρ 0 by bit flip Fixed-Point Engineering Applications III: 0 Noise Switching 10 DYNAMO Extension New Reachability Theorems −1 10 Examples residual error δ F Open vs Closed Loop −2 Conclusions 10 � −3 10 � � −4 10 0 1000 2000 3000 4000 wall time [s]

Noise-Driven State Transfer III: Ion Traps Transfer to GHZ State arXiv:1206.4945 Basic Systems Theory DYNAMO Platform Example Applications I: Error Correction system: 4-ion system, individual z -controls, joint Applications II: Fixed-Point F x , F y -controls, joint ( F x ) 2 , ( F y ) 2 -controls, and Engineering controllable amp-damp noise on terminal qubit Applications III: Noise Switching task III: ρ 0 ≃ 1 l → ρ | GHZ 4 � by amp-damp DYNAMO Extension New Reachability Theorems Examples Open vs Closed Loop Conclusions � � �

Noise-Driven State Transfer III: Ion Traps Transfer to GHZ State arXiv:1206.4945 Fx control amplitudes [a] Fy 5 Fx 2 0 Fy 2 Basic Systems Theory z1 −5 z2 DYNAMO Platform z3 −10 z4 Applications I: 0 1 2 3 4 5 6 7 8 a1 Error Correction time [1/a] 1 Applications II: Fixed-Point 0.8 Engineering eigenvalues 0.6 Applications III: 0.4 Noise Switching DYNAMO Extension 0.2 New Reachability Theorems 0 Examples 0 1 2 3 4 5 6 7 8 Open vs Closed Loop time [1/a] Conclusions error × 100 � 0.4 � 0.3 0.2 0.2 0000 � 0010 0000 0.1 0100 0010 0.1 0110 0100 0 1000 0110 0 1000 0000 0010 0100 0110 1000 1010 1100 1110 1010 1010 1100 0000 0010 0100 0110 1000 1010 1100 1110 1100 1110 1110

Noise-Driven State Transfer III: Ion Traps Transfer to GHZ State arXiv:1206.4945 � open-loop noise control Fx control amplitudes [a] Fy 5 Basic Systems Fx 2 Theory 0 Fy 2 DYNAMO Platform z1 −5 z2 Applications I: z3 −10 Error Correction z4 0 1 2 3 4 5 6 7 8 a1 Applications II: time [1/a] Fixed-Point Engineering Applications III: Noise Switching � may replace measurement-based closed loop DYNAMO Extension feedback New Reachability Theorems Examples Open vs Closed Loop Conclusions � � � Barreiro,. . . , Blatt, Nature 470 , 486 (2011) Schindler,. . . , Nature Physics 9 , 361 (2013)

Noise-Driven State Transfer Open Loop as Strong Closed Loop arXiv:1206.4945 Basic Systems Theory DYNAMO Platform Applications I: Error Correction Markovian vs. non-Markovian State Transfer Applications II: Fixed-Point For state transfer, Markovian quantum maps are as Engineering powerful as non-Markovian maps, i.e. closed-loop control Applications III: Noise Switching can be replaced by open-loop control. DYNAMO Extension New Reachability Theorems Examples Open vs Closed Loop Conclusions � � �

Conclusions HIFI quantum engineering for closed and open systems. Basic Systems Theory DYNAMO Platform � DYNAMO platform Applications I: Error Correction • optimised gates: enabling HIFI error correction Applications II: Fixed-Point Engineering � symmetry principles of fixed-point engineering Applications III: Noise Switching • centraliser (stabiliser) Conclusions � � Is open-loop coherent control + switchable Markov noise � as strong as closed-loop control ? � • yes for state transfer • no for gate/map synthesis

Conclusions HIFI quantum engineering for closed and open systems. Basic Systems Theory DYNAMO Platform � DYNAMO platform Applications I: Error Correction • optimised gates: enabling HIFI error correction Applications II: Fixed-Point Engineering � symmetry principles of fixed-point engineering Applications III: Noise Switching • centraliser (stabiliser) Conclusions � � Is open-loop coherent control + switchable Markov noise � as strong as closed-loop control ? � • yes for state transfer • no for gate/map synthesis

Conclusions HIFI quantum engineering for closed and open systems. Basic Systems Theory DYNAMO Platform � DYNAMO platform Applications I: Error Correction • optimised gates: enabling HIFI error correction Applications II: Fixed-Point Engineering � symmetry principles of fixed-point engineering Applications III: Noise Switching • centraliser (stabiliser) Conclusions � � Is open-loop coherent control + switchable Markov noise � as strong as closed-loop control ? � • yes for state transfer • no for gate/map synthesis

Conclusions HIFI quantum engineering for closed and open systems. Basic Systems Theory DYNAMO Platform � DYNAMO platform Applications I: Error Correction • optimised gates: enabling HIFI error correction Applications II: Fixed-Point Engineering � symmetry principles of fixed-point engineering Applications III: Noise Switching • centraliser (stabiliser) Conclusions � � Is open-loop coherent control + switchable Markov noise � as strong as closed-loop control ? � • yes for state transfer • no for gate/map synthesis

Conclusions HIFI quantum engineering for closed and open systems. Basic Systems Theory DYNAMO Platform � DYNAMO platform Applications I: Error Correction • optimised gates: enabling HIFI error correction Applications II: Fixed-Point Engineering � symmetry principles of fixed-point engineering Applications III: Noise Switching • centraliser (stabiliser) Conclusions � � Is open-loop coherent control + switchable Markov noise � as strong as closed-loop control ? � • yes for state transfer • no for gate/map synthesis

Conclusions HIFI quantum engineering for closed and open systems. Basic Systems Theory DYNAMO Platform � DYNAMO platform Applications I: Error Correction • optimised gates: enabling HIFI error correction Applications II: Fixed-Point Engineering � symmetry principles of fixed-point engineering Applications III: Noise Switching • centraliser (stabiliser) Conclusions � � Is open-loop coherent control + switchable Markov noise � as strong as closed-loop control ? � • yes for state transfer • no for gate/map synthesis

Acknowledgements Thanks go to ETH Zurich and Richard R. Ernst and: Basic Systems Ville Bergholm, Corey O’Meara, Gunther Dirr, Theory F. Dolde, P . Neumann, F. Jelezko, J. Wrachtrup DYNAMO Platform Applications I: integrated EU programmes; excellence networks; DFG research group Error Correction Applications II: Fixed-Point Engineering Quantum Computing, Control & Communication Applications III: References: Noise Switching J. Magn. Reson. 172 , 296 (2005), PRA 72 , 043221 (2005), PRA 84 , 022305 (2011) Conclusions PRA 75 , 012302 (2007); PRL 102 090401 (2009), JPB 44 , 154013 (2011) � Rev. Math. Phys. 22 , 597 (2010), Rep. Math. Phys. 64 , 93 (2009); � PRA 81 , 032319 (2010); PRB 81 , 085328 (2010); � arXiv:0904.4654, IEEE Proc. ISCCSP 2010 23.2, Proc. MTNS, 2341 (2010), J. Math. Phys. 52 , 113510 (2011); Eur.Phys.J.:Quant.Technol. 1 , 11 (2014); New J. Phys. 16 , 065010 (2014) IEEE TAC 57 , 2050 (2012); arXiv:1206.4945; Nature 506 , 204 (2014), Nature Comm. 5 3371 (2014)

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise quant-ph/1206.4945 � 0 1 � new control term: γ ( t ) · Γ L with V a := 1 in l ⊗ 0 0 Basic Systems Theory Γ L ( ρ ) = 1 2 { V † a V a , ρ } + − V a ρ V † a DYNAMO Platform Applications I: Error Correction Theorem Applications II: Fixed-Point Let Σ a be an n-qubit bilinear control system satisfying Engineering (WH) for γ = 0 . Suppose the amp-damp noise amplitude Applications III: Noise Switching can be switched γ ( t ) ∈ { 0 , γ ∗ } with γ ∗ > 0 . If H d is Conclusions diagonal (Ising-ZZ type) and the only drift term, then Σ a � Reachability acts transitively on the set of all density operators pos 1 � � Reach Σ a ( ρ 0 ) = pos 1 for all ρ 0 ∈ pos 1 where the closure is understood as the limit T γ ∗ → ∞ .

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise Proof. choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � 1 �� Error Correction 1 − ǫ l ⊗ ( n − 1 ) r 0 with ǫ := e − t γ ∗ r ( t ) = 1 ⊗ 2 0 Applications II: ǫ Fixed-Point Engineering can obtain any state Applications III: Noise Switching Conclusions ρ ( t ) = diag ( . . . , [ ρ ii + ρ jj · ( 1 − ǫ )] ii , . . . , [ ρ jj · ǫ ] jj , . . . ); � Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators ∆ ⊂ pos 1 ; � by unitary controllability get all unitary orbits U (∆) = pos 1 .

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise Proof. choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � 1 �� Error Correction 1 − ǫ l ⊗ ( n − 1 ) r 0 with ǫ := e − t γ ∗ r ( t ) = 1 ⊗ 2 0 Applications II: ǫ Fixed-Point Engineering can obtain any state Applications III: Noise Switching Conclusions ρ ( t ) = diag ( . . . , [ ρ ii + ρ jj · ( 1 − ǫ )] ii , . . . , [ ρ jj · ǫ ] jj , . . . ); � Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators ∆ ⊂ pos 1 ; � by unitary controllability get all unitary orbits U (∆) = pos 1 .

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise Proof. choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � 1 �� Error Correction 1 − ǫ l ⊗ ( n − 1 ) r 0 with ǫ := e − t γ ∗ r ( t ) = 1 ⊗ 2 0 Applications II: ǫ Fixed-Point Engineering undo any unwanted transfer ρ ii ↔ ρ jj lasting a total of τ by Applications III: � � Noise Switching ρ ii e + γ ∗ τ + ρ jj 1 permuting ρ ii and ρ jj after τ ij := γ ∗ ln and Conclusions ρ ii + ρ jj evolve under noise for remaining τ − τ ij ; � Reachability with 2 n − 1 − 1 switches all but one desired transfer remain; � can obtain any state � ρ ( t ) = diag ( . . . , [ ρ ii + ρ jj · ( 1 − ǫ )] ii , . . . , [ ρ jj · ǫ ] jj , . . . ); in limit T γ → ∞ obtain set of all diagonal density

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise Proof. choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � 1 �� Error Correction 1 − ǫ l ⊗ ( n − 1 ) r 0 with ǫ := e − t γ ∗ r ( t ) = 1 ⊗ 2 0 Applications II: ǫ Fixed-Point Engineering undo any unwanted transfer ρ ii ↔ ρ jj lasting a total of τ by Applications III: � � Noise Switching ρ ii e + γ ∗ τ + ρ jj 1 permuting ρ ii and ρ jj after τ ij := γ ∗ ln and Conclusions ρ ii + ρ jj evolve under noise for remaining τ − τ ij ; � Reachability with 2 n − 1 − 1 switches all but one desired transfer remain; � can obtain any state � ρ ( t ) = diag ( . . . , [ ρ ii + ρ jj · ( 1 − ǫ )] ii , . . . , [ ρ jj · ǫ ] jj , . . . ); in limit T γ → ∞ obtain set of all diagonal density

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise Proof. choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � 1 �� Error Correction 1 − ǫ l ⊗ ( n − 1 ) r 0 with ǫ := e − t γ ∗ r ( t ) = 1 ⊗ 2 0 Applications II: ǫ Fixed-Point Engineering can obtain any state Applications III: Noise Switching Conclusions ρ ( t ) = diag ( . . . , [ ρ ii + ρ jj · ( 1 − ǫ )] ii , . . . , [ ρ jj · ǫ ] jj , . . . ); � Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators ∆ ⊂ pos 1 ; � by unitary controllability get all unitary orbits U (∆) = pos 1 .

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise Proof. choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � 1 �� Error Correction 1 − ǫ l ⊗ ( n − 1 ) r 0 with ǫ := e − t γ ∗ r ( t ) = 1 ⊗ 2 0 Applications II: ǫ Fixed-Point Engineering can obtain any state Applications III: Noise Switching Conclusions ρ ( t ) = diag ( . . . , [ ρ ii + ρ jj · ( 1 − ǫ )] ii , . . . , [ ρ jj · ǫ ] jj , . . . ); � Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators ∆ ⊂ pos 1 ; � by unitary controllability get all unitary orbits U (∆) = pos 1 .

Reachable Sets I: Non-Unital Controlled Amplitude Damping Noise Proof. choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � 1 �� Error Correction 1 − ǫ l ⊗ ( n − 1 ) r 0 with ǫ := e − t γ ∗ r ( t ) = 1 ⊗ 2 0 Applications II: ǫ Fixed-Point Engineering can obtain any state Applications III: Noise Switching Conclusions ρ ( t ) = diag ( . . . , [ ρ ii + ρ jj · ( 1 − ǫ )] ii , . . . , [ ρ jj · ǫ ] jj , . . . ); � Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators ∆ ⊂ pos 1 ; � by unitary controllability get all unitary orbits U (∆) = pos 1 .

Reachable Sets II: Unital Controlled Bit Flip Noise quant-ph/1206.4945 new control term: γ ( t ) · Γ L with V b := 1 l ⊗ σ x / 2 in Basic Systems Theory Γ L ( ρ ) = 1 2 { V † b V b , ρ } + − V b ρ V † b DYNAMO Platform Applications I: Error Correction Theorem Applications II: Fixed-Point Let Σ b be an n-qubit bilinear control system satisfying Engineering (WH) for γ = 0 . Suppose the bit-flip noise amplitude can Applications III: Noise Switching be switched γ ( t ) ∈ { 0 , γ ∗ } with γ ∗ > 0 . If all drift Conclusions components of H d are diagonal (Ising-ZZ), then Σ b � explores all states majorised by ρ 0 Reachability � Reach Σ b ( ρ 0 ) = { ρ | ρ ≺ ρ 0 } � for any ρ 0 ∈ pos 1 where the closure is understood as the limit T γ ∗ → ∞ .

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering Applications III: Noise Switching NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D Conclusions product of at most N − 1 such T -transforms � (e.g., Thm. B.6 in M ARSHALL -O LKIN or Thm. II.1.10 in B HATIA ) Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators diag ( r ) ≺ diag ( r 0 ) � by unitary controllability get all density operators ρ ≺ ρ 0 .

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering Applications III: Noise Switching NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D Conclusions product of at most N − 1 such T -transforms � (e.g., Thm. B.6 in M ARSHALL -O LKIN or Thm. II.1.10 in B HATIA ) Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators diag ( r ) ≺ diag ( r 0 ) � by unitary controllability get all density operators ρ ≺ ρ 0 .

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering to limit relaxative averaging to first two eigenvalues, Applications III: Noise Switching � 1 � ⊕ 2 n − 1 − 1 − 1 1 Conclusions conjugate ρ 0 with U 12 := 1 l 2 ⊕ √ 1 1 2 � Reachability gives protected state ρ ′ 0 := U 12 ρ 0 U † 12 � � ρ 11 � � ρ 33 + ρ 44 � 0 ρ 33 − ρ 44 � ⊕ 1 ρ ′ 0 = ⊕ · · · 2 o ρ 22 ρ 33 − ρ 44 ρ 33 + ρ 44 now relaxation acts as T -transform on ρ ′ 0 NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering to limit relaxative averaging to first two eigenvalues, Applications III: Noise Switching � 1 � ⊕ 2 n − 1 − 1 − 1 1 Conclusions conjugate ρ 0 with U 12 := 1 l 2 ⊕ √ 1 1 2 � Reachability gives protected state ρ ′ 0 := U 12 ρ 0 U † 12 � � ρ 11 � � ρ 33 + ρ 44 � 0 ρ 33 − ρ 44 � ⊕ 1 ρ ′ 0 = ⊕ · · · 2 o ρ 22 ρ 33 − ρ 44 ρ 33 + ρ 44 now relaxation acts as T -transform on ρ ′ 0 NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering to limit relaxative averaging to first two eigenvalues, Applications III: Noise Switching � 1 � ⊕ 2 n − 1 − 1 − 1 1 Conclusions conjugate ρ 0 with U 12 := 1 l 2 ⊕ √ 1 1 2 � Reachability gives protected state ρ ′ 0 := U 12 ρ 0 U † 12 � � ρ 11 � � ρ 33 + ρ 44 � 0 ρ 33 − ρ 44 � ⊕ 1 ρ ′ 0 = ⊕ · · · 2 o ρ 22 ρ 33 − ρ 44 ρ 33 + ρ 44 now relaxation acts as T -transform on ρ ′ 0 NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering by permutation of such T -transforms, one can obtain any Applications III: Noise Switching state Conclusions � 2 [ ρ ii + ρ jj + ( ρ ii − ρ jj ) · e − t . . . , 1 ρ ( t ) = diag 2 γ ∗ ] ii , . . . � � 2 [ ρ ii + ρ jj + ( ρ jj − ρ ii ) · e − t Reachability 1 2 γ ∗ ] jj , . . . � NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D � product of at most N − 1 such T -transforms (e.g., Thm. B.6 in M ARSHALL -O LKIN or Thm. II.1.10 in B HATIA ) in limit T γ ∗ → ∞ obtain set of all diagonal density

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering Applications III: Noise Switching NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D Conclusions product of at most N − 1 such T -transforms � (e.g., Thm. B.6 in M ARSHALL -O LKIN or Thm. II.1.10 in B HATIA ) Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators diag ( r ) ≺ diag ( r 0 ) � by unitary controllability get all density operators ρ ≺ ρ 0 .

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering Applications III: Noise Switching NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D Conclusions product of at most N − 1 such T -transforms � (e.g., Thm. B.6 in M ARSHALL -O LKIN or Thm. II.1.10 in B HATIA ) Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators diag ( r ) ≺ diag ( r 0 ) � by unitary controllability get all density operators ρ ≺ ρ 0 .

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof. again choose diagonal ρ 0 =: diag ( r 0 ) Basic Systems Theory with H d diagonal (Ising- ZZ ), evolution remains diagonal DYNAMO Platform Applications I: � � ( 1 + ǫ ) �� Error Correction ( 1 − ǫ ) r 0 with ǫ := e − t l ⊗ ( n − 1 ) ⊗ 1 2 γ ∗ r ( t ) = 1 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Applications II: Fixed-Point Engineering Applications III: Noise Switching NB: ρ tar ≺ ρ 0 iff ρ tar = D ρ 0 with doubly stochastic D Conclusions product of at most N − 1 such T -transforms � (e.g., Thm. B.6 in M ARSHALL -O LKIN or Thm. II.1.10 in B HATIA ) Reachability in limit T γ ∗ → ∞ obtain set of all diagonal density � operators diag ( r ) ≺ diag ( r 0 ) � by unitary controllability get all density operators ρ ≺ ρ 0 .

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof: further details. Basic Systems decouple protected states ρ ′ 0 from Hamiltonian H 0 Theory DYNAMO Platform to this end, observe Applications I: Error Correction e i π H 1 x e − t (Γ+ iH zz ) e − i π H 1 x = e − t (Γ − iH zz ) Applications II: Fixed-Point Engineering Applications III: Noise Switching so decoupling obtained in Trotter limit Conclusions � k →∞ ( e − t 2 k (Γ+ iH zz ) e − t 2 k (Γ − iH zz ) ) k = e − t Γ . lim Reachability � �

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof: further details. Basic Systems decouple protected states ρ ′ 0 from Hamiltonian H 0 Theory DYNAMO Platform to this end, observe Applications I: Error Correction e i π H 1 x e − t (Γ+ iH zz ) e − i π H 1 x = e − t (Γ − iH zz ) Applications II: Fixed-Point Engineering Applications III: Noise Switching so decoupling obtained in Trotter limit Conclusions � k →∞ ( e − t 2 k (Γ+ iH zz ) e − t 2 k (Γ − iH zz ) ) k = e − t Γ . lim Reachability � �

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof: further details. Basic Systems decouple protected states ρ ′ 0 from Hamiltonian H 0 Theory DYNAMO Platform to this end, observe Applications I: Error Correction e i π H 1 x e − t (Γ+ iH zz ) e − i π H 1 x = e − t (Γ − iH zz ) Applications II: Fixed-Point Engineering Applications III: Noise Switching so decoupling obtained in Trotter limit Conclusions � k →∞ ( e − t 2 k (Γ+ iH zz ) e − t 2 k (Γ − iH zz ) ) k = e − t Γ . lim Reachability � �

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof: further details. Basic Systems Theory T -transformation is convex combination DYNAMO Platform l + ( 1 − λ ) Q with pair transposition Q and λ ∈ [ 0 , 1 ] Applications I: λ 1 Error Correction Applications II: � � ( 1 + ǫ ) �� Fixed-Point ( 1 − ǫ ) Engineering l ⊗ ( n − 1 ) ⊗ 1 So R b ( t ) := 1 Applications III: 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Noise Switching covers λ ∈ [ 1 2 , 1 ] , while Conclusions � � 0 1 � � l ⊗ ( n − 1 ) � captures R ′ b ( t ) := R b ( t ) ◦ ⊗ 1 Reachability 2 1 0 λ ∈ [ 0 , 1 2 ] , and λ = 1 � 2 is obtained in the limit ǫ → 0 �

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof: further details. Basic Systems Theory T -transformation is convex combination DYNAMO Platform l + ( 1 − λ ) Q with pair transposition Q and λ ∈ [ 0 , 1 ] Applications I: λ 1 Error Correction Applications II: � � ( 1 + ǫ ) �� Fixed-Point ( 1 − ǫ ) Engineering l ⊗ ( n − 1 ) ⊗ 1 So R b ( t ) := 1 Applications III: 2 2 ( 1 − ǫ ) ( 1 + ǫ ) Noise Switching covers λ ∈ [ 1 2 , 1 ] , while Conclusions � � 0 1 � � l ⊗ ( n − 1 ) � captures R ′ b ( t ) := R b ( t ) ◦ ⊗ 1 Reachability 2 1 0 λ ∈ [ 0 , 1 2 ] , and λ = 1 � 2 is obtained in the limit ǫ → 0 �

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof: further details. Basic Systems Theory one cannot go beyond states majorised by ρ 0 : DYNAMO Platform bit-flip superoperator: doubly-stochastic Applications I: Error Correction   ( 1 + ǫ ) 0 0 ( 1 − ǫ ) Applications II: Fixed-Point l ⊗ ( n − 1 ) 0 ( 1 + ǫ ) ( 1 − ǫ ) 0 ⊗ 1 e − t Γ b = 1   Engineering 4 2 0 ( 1 − ǫ ) ( 1 + ǫ ) 0 Applications III: ( 1 − ǫ ) 0 0 ( 1 + ǫ ) Noise Switching Conclusions bit-flip plus unitary control: cpt unital map hence also � Reachability generalised doubly-stochastic linear map Φ in sense of � A NDO , Lin. Alg. Appl. 118 (1989) p 235 Thm. 7.1 saying � that for any hermitian A : Φ( A ) ≺ A .

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof: further details. Basic Systems Theory one cannot go beyond states majorised by ρ 0 : DYNAMO Platform bit-flip superoperator: doubly-stochastic Applications I: Error Correction   ( 1 + ǫ ) 0 0 ( 1 − ǫ ) Applications II: Fixed-Point l ⊗ ( n − 1 ) 0 ( 1 + ǫ ) ( 1 − ǫ ) 0 ⊗ 1 e − t Γ b = 1   Engineering 4 2 0 ( 1 − ǫ ) ( 1 + ǫ ) 0 Applications III: ( 1 − ǫ ) 0 0 ( 1 + ǫ ) Noise Switching Conclusions bit-flip plus unitary control: cpt unital map hence also � Reachability generalised doubly-stochastic linear map Φ in sense of � A NDO , Lin. Alg. Appl. 118 (1989) p 235 Thm. 7.1 saying � that for any hermitian A : Φ( A ) ≺ A .

Reachable Sets II: Unital Controlled Bit-Flip Noise Proof: further details. Basic Systems Theory one cannot go beyond states majorised by ρ 0 : DYNAMO Platform bit-flip superoperator: doubly-stochastic Applications I: Error Correction   ( 1 + ǫ ) 0 0 ( 1 − ǫ ) Applications II: Fixed-Point l ⊗ ( n − 1 ) 0 ( 1 + ǫ ) ( 1 − ǫ ) 0 ⊗ 1 e − t Γ b = 1   Engineering 4 2 0 ( 1 − ǫ ) ( 1 + ǫ ) 0 Applications III: ( 1 − ǫ ) 0 0 ( 1 + ǫ ) Noise Switching Conclusions bit-flip plus unitary control: cpt unital map hence also � Reachability generalised doubly-stochastic linear map Φ in sense of � A NDO , Lin. Alg. Appl. 118 (1989) p 235 Thm. 7.1 saying � that for any hermitian A : Φ( A ) ≺ A .

Reachable Sets III: Generalised Controlled Noise quant-ph/1206.4945 � � 0 ( 1 − θ ) new control term: γ ( t ) · Γ L with V θ := , θ ∈ [ 0 , 1 ] in 0 θ Basic Systems Γ L ( ρ ) = 1 2 { V † θ V θ , ρ } + − V θ ρ V † Theory θ DYNAMO Platform Applications I: Error Correction fixed point (single qubit) � ¯ � Applications II: θ 2 0 Fixed-Point 1 with ¯ θ := 1 − θ ρ ∞ ( θ ) = Engineering θ 2 θ 2 + θ 2 ¯ 0 Applications III: compare with canonical density operator at temperature β Noise Switching � e β/ 2 � 0 Conclusions 1 ρ β := 2 cosh ( β/ 2 ) e − β/ 2 0 � Reachability 1 so θ relates to inverse temperature β ( θ ) := k B T θ by � � ¯ � θ 2 − θ 2 � β ( θ ) = 2 artanh ¯ θ 2 + θ 2 switching condition θ 2 θ 2 ¯ θ 2 ≤ ρ ii ρ jj ≤ ¯ θ 2

Reachable Sets III: Generalised Controlled Noise quant-ph/1206.4945 � � 0 ( 1 − θ ) new control term: γ ( t ) · Γ L with V θ := , θ ∈ [ 0 , 1 ] in 0 θ Basic Systems Γ L ( ρ ) = 1 2 { V † θ V θ , ρ } + − V θ ρ V † Theory θ DYNAMO Platform Applications I: Error Correction fixed point (single qubit) � ¯ � Applications II: θ 2 0 Fixed-Point 1 with ¯ θ := 1 − θ ρ ∞ ( θ ) = Engineering θ 2 θ 2 + θ 2 ¯ 0 Applications III: compare with canonical density operator at temperature β Noise Switching � e β/ 2 � 0 Conclusions 1 ρ β := 2 cosh ( β/ 2 ) e − β/ 2 0 � Reachability 1 so θ relates to inverse temperature β ( θ ) := k B T θ by � � ¯ � θ 2 − θ 2 � β ( θ ) = 2 artanh ¯ θ 2 + θ 2 switching condition θ 2 θ 2 ¯ θ 2 ≤ ρ ii ρ jj ≤ ¯ θ 2

Reachable Sets III: Generalised Controlled Noise quant-ph/1206.4945 � � 0 ( 1 − θ ) new control term: γ ( t ) · Γ L with V θ := , θ ∈ [ 0 , 1 ] in 0 θ Basic Systems Γ L ( ρ ) = 1 2 { V † θ V θ , ρ } + − V θ ρ V † Theory θ DYNAMO Platform Applications I: Error Correction fixed point (single qubit) � ¯ � Applications II: θ 2 0 Fixed-Point 1 with ¯ θ := 1 − θ ρ ∞ ( θ ) = Engineering θ 2 θ 2 + θ 2 ¯ 0 Applications III: compare with canonical density operator at temperature β Noise Switching � e β/ 2 � 0 Conclusions 1 ρ β := 2 cosh ( β/ 2 ) e − β/ 2 0 � Reachability 1 so θ relates to inverse temperature β ( θ ) := k B T θ by � � ¯ � θ 2 − θ 2 � β ( θ ) = 2 artanh ¯ θ 2 + θ 2 switching condition θ 2 θ 2 ¯ θ 2 ≤ ρ ii ρ jj ≤ ¯ θ 2

Reachable Sets III: Generalised Controlled Noise quant-ph/1206.4945 � � 0 ( 1 − θ ) new control term: γ ( t ) · Γ L with V θ := , θ ∈ [ 0 , 1 ] in 0 θ Basic Systems Γ L ( ρ ) = 1 2 { V † θ V θ , ρ } + − V θ ρ V † Theory θ DYNAMO Platform Applications I: Error Correction fixed point (single qubit) � ¯ � Applications II: θ 2 0 Fixed-Point 1 with ¯ θ := 1 − θ ρ ∞ ( θ ) = Engineering θ 2 θ 2 + θ 2 ¯ 0 Applications III: compare with canonical density operator at temperature β Noise Switching � e β/ 2 � 0 Conclusions 1 ρ β := 2 cosh ( β/ 2 ) e − β/ 2 0 � Reachability 1 so θ relates to inverse temperature β ( θ ) := k B T θ by � � ¯ � θ 2 − θ 2 � β ( θ ) = 2 artanh ¯ θ 2 + θ 2 switching condition θ 2 θ 2 ¯ θ 2 ≤ ρ ii ρ jj ≤ ¯ θ 2

Reachable Sets III: Generalised Controlled Noise quant-ph/1206.4945 � � 0 ( 1 − θ ) new control term: γ ( t ) · Γ L with V θ := , θ ∈ [ 0 , 1 ] in 0 θ Γ L ( ρ ) = 1 2 { V † θ V θ , ρ } + − V θ ρ V † Basic Systems Theory θ DYNAMO Platform Applications I: Theorem Error Correction Applications II: Let Σ θ be an n-qubit bilinear control system satisfying Fixed-Point Engineering (WH) for γ = 0 . Suppose the V θ noise amplitude can be Applications III: switched γ ( t ) ∈ { 0 , γ ∗ } . If all drift components of H d are Noise Switching diagonal (Ising-ZZ), then Σ θ gives for the thermal state Conclusions ρ 0 = 1 2 n 1 l � Reachability Reach Σ θ ( 1 l ) ⊇ { ρ | ρ ≺ ρ δ } 2 n 1 � � where ρ δ is the purest state obtainable by partner-pairing θ 2 − θ 2 algorithmic cooling with bias δ := ¯ θ 2 + θ 2 (again closure by ¯ T γ ∗ → ∞ ).

Pontryagin’s Maximum Principle Theorem (Pontryagin) Basic Systems Theory Consider a system governed by ˙ X ( t ) = F ( X , u , t ) . DYNAMO Platform For u ∗ ( t ) to be an optimal control steering X ( 0 ) into X ( T ) so Applications I: � T Error Correction that J [ X ( t )] = L ( t ) dt assumes its critical points over (almost) Applications II: 0 Fixed-Point all times, it suffices there is Engineering an adjoint system λ ( t ) satisfying ˙ Applications III: λ = − ∂ h by virtue of Noise Switching ∂ X Conclusions a scalar Hamiltonian function (so ˙ ∂ h ∂λ † ), X ( t ) ≡ F ( X , u , t ) = h ( P , X , u , t ) := L ( X , u , t ) + � λ ( t ) | F ( X , u , t ) � where � � ◦ h attains its critical points for optimal controls u ∗ ( t ) , � ∂ u ∗ ( t ) = 0 at almost all 0 ≤ t ≤ T; ∂ h i.e., ◦ X ( T ) unspecified implies λ ( T ) = 0 .

Pontryagin’s Maximum Principle Theorem (Pontryagin) Basic Systems Theory Consider a system governed by ˙ X ( t ) = F ( X , u , t ) . DYNAMO Platform For u ∗ ( t ) to be an optimal control steering X ( 0 ) into X ( T ) so Applications I: � T Error Correction that J [ X ( t )] = L ( t ) dt assumes its critical points over (almost) Applications II: 0 Fixed-Point all times, it suffices there is Engineering an adjoint system λ ( t ) satisfying ˙ Applications III: λ = − ∂ h by virtue of Noise Switching ∂ X Conclusions a scalar Hamiltonian function (so ˙ ∂ h ∂λ † ), X ( t ) ≡ F ( X , u , t ) = h ( P , X , u , t ) := L ( X , u , t ) + � λ ( t ) | F ( X , u , t ) � where � � ◦ h attains its critical points for optimal controls u ∗ ( t ) , � ∂ u ∗ ( t ) = 0 at almost all 0 ≤ t ≤ T; ∂ h i.e., ◦ X ( T ) unspecified implies λ ( T ) = 0 .

Pontryagin’s Maximum Principle Theorem (Pontryagin) Basic Systems Theory Consider a system governed by ˙ X ( t ) = F ( X , u , t ) . DYNAMO Platform For u ∗ ( t ) to be an optimal control steering X ( 0 ) into X ( T ) so Applications I: � T Error Correction that J [ X ( t )] = L ( t ) dt assumes its critical points over (almost) Applications II: 0 Fixed-Point all times, it suffices there is Engineering an adjoint system λ ( t ) satisfying ˙ Applications III: λ = − ∂ h by virtue of Noise Switching ∂ X Conclusions a scalar Hamiltonian function (so ˙ ∂ h ∂λ † ), X ( t ) ≡ F ( X , u , t ) = h ( P , X , u , t ) := L ( X , u , t ) + � λ ( t ) | F ( X , u , t ) � where � � ◦ h attains its critical points for optimal controls u ∗ ( t ) , � ∂ u ∗ ( t ) = 0 at almost all 0 ≤ t ≤ T; ∂ h i.e., ◦ X ( T ) unspecified implies λ ( T ) = 0 .

Pontryagin’s Maximum Principle Theorem (Pontryagin) Basic Systems Theory Consider a system governed by ˙ X ( t ) = F ( X , u , t ) . DYNAMO Platform For u ∗ ( t ) to be an optimal control steering X ( 0 ) into X ( T ) so Applications I: � T Error Correction that J [ X ( t )] = L ( t ) dt assumes its critical points over (almost) Applications II: 0 Fixed-Point all times, it suffices there is Engineering an adjoint system λ ( t ) satisfying ˙ Applications III: λ = − ∂ h by virtue of Noise Switching ∂ X Conclusions a scalar Hamiltonian function (so ˙ ∂ h ∂λ † ), X ( t ) ≡ F ( X , u , t ) = h ( P , X , u , t ) := L ( X , u , t ) + � λ ( t ) | F ( X , u , t ) � where � � ◦ h attains its critical points for optimal controls u ∗ ( t ) , � ∂ u ∗ ( t ) = 0 at almost all 0 ≤ t ≤ T; ∂ h i.e., ◦ X ( T ) unspecified implies λ ( T ) = 0 .

Pontryagin’s Maximum Principle Theorem (Pontryagin) Basic Systems Theory Consider a system governed by ˙ X ( t ) = F ( X , u , t ) . DYNAMO Platform For u ∗ ( t ) to be an optimal control steering X ( 0 ) into X ( T ) so Applications I: � T Error Correction that J [ X ( t )] = L ( t ) dt assumes its critical points over (almost) Applications II: 0 Fixed-Point all times, it suffices there is Engineering an adjoint system λ ( t ) satisfying ˙ Applications III: λ = − ∂ h by virtue of Noise Switching ∂ X Conclusions a scalar Hamiltonian function (so ˙ ∂ h ∂λ † ), X ( t ) ≡ F ( X , u , t ) = h ( P , X , u , t ) := L ( X , u , t ) + � λ ( t ) | F ( X , u , t ) � where � � ◦ h attains its critical points for optimal controls u ∗ ( t ) , � ∂ u ∗ ( t ) = 0 at almost all 0 ≤ t ≤ T; ∂ h i.e., ◦ X ( T ) unspecified implies λ ( T ) = 0 .

Maximum Principle, ctd Proof. F RÉCHET derivatives provide ∂ L ∈ Mat n ( C ) and ∂ L ∈ Mat n , 1 ( C ) . ∂ X ∂ u � T Basic Systems Thus for J [ X ( t )] = dtL ( X , u , t ) calculate first variation in X and u as Theory 0 DYNAMO Platform δ J 1 ◦ Applications I: = J ( X + δ X , u + δ u , t ) − J ( X , u , t ) Error Correction T � � Applications II: dt {� ∂ L ∂ X | δ X � + � ∂ L T � = ∂ u | δ u �} + L ( t ) δ t . Fixed-Point � 0 Engineering 0 Applications III: NB: δ X depends on variation of control δ u via ˙ Noise Switching X = F ( X , u , t ) . Conclusions Incorporate dependence of δ X on δ u as in eqn. of motion by � operator-valued L AGRANGE multiplier λ ( t ) associated with zero-cost � term � � T dt � λ ( t ) | F ( X , u , t ) − ˙ X � = 0 J λ := . 0

Maximum Principle, ctd Proof. F RÉCHET derivatives provide ∂ L ∈ Mat n ( C ) and ∂ L ∈ Mat n , 1 ( C ) . ∂ X ∂ u � T Basic Systems Thus for J [ X ( t )] = dtL ( X , u , t ) calculate first variation in X and u as Theory 0 DYNAMO Platform δ J 1 ◦ Applications I: = J ( X + δ X , u + δ u , t ) − J ( X , u , t ) Error Correction T � � Applications II: dt {� ∂ L ∂ X | δ X � + � ∂ L T � = ∂ u | δ u �} + L ( t ) δ t . Fixed-Point � 0 Engineering 0 Applications III: NB: δ X depends on variation of control δ u via ˙ Noise Switching X = F ( X , u , t ) . Conclusions Incorporate dependence of δ X on δ u as in eqn. of motion by � operator-valued L AGRANGE multiplier λ ( t ) associated with zero-cost � term � � T dt � λ ( t ) | F ( X , u , t ) − ˙ X � = 0 J λ := . 0

Maximum Principle, ctd Proof. F RÉCHET derivatives provide ∂ L ∈ Mat n ( C ) and ∂ L ∈ Mat n , 1 ( C ) . ∂ X ∂ u � T Basic Systems Thus for J [ X ( t )] = dtL ( X , u , t ) calculate first variation in X and u as Theory 0 DYNAMO Platform δ J 1 ◦ Applications I: = J ( X + δ X , u + δ u , t ) − J ( X , u , t ) Error Correction T � � Applications II: dt {� ∂ L ∂ X | δ X � + � ∂ L T � = ∂ u | δ u �} + L ( t ) δ t . Fixed-Point � 0 Engineering 0 Applications III: NB: δ X depends on variation of control δ u via ˙ Noise Switching X = F ( X , u , t ) . Conclusions Incorporate dependence of δ X on δ u as in eqn. of motion by � operator-valued L AGRANGE multiplier λ ( t ) associated with zero-cost � term � � T dt � λ ( t ) | F ( X , u , t ) − ˙ X � = 0 J λ := . 0

Maximum Principle, ctd Proof. F RÉCHET derivatives provide ∂ L ∈ Mat n ( C ) and ∂ L ∈ Mat n , 1 ( C ) . ∂ X ∂ u � T Basic Systems Thus for J [ X ( t )] = dtL ( X , u , t ) calculate first variation in X and u as Theory 0 DYNAMO Platform δ J 1 ◦ Applications I: = J ( X + δ X , u + δ u , t ) − J ( X , u , t ) Error Correction T � � Applications II: dt {� ∂ L ∂ X | δ X � + � ∂ L T � = ∂ u | δ u �} + L ( t ) δ t . Fixed-Point � 0 Engineering 0 Applications III: NB: δ X depends on variation of control δ u via ˙ Noise Switching X = F ( X , u , t ) . Conclusions Incorporate dependence of δ X on δ u as in eqn. of motion by � operator-valued L AGRANGE multiplier λ ( t ) associated with zero-cost � term � � T dt � λ ( t ) | F ( X , u , t ) − ˙ X � = 0 J λ := . 0

Maximum Principle, ctd First variation of J λ in X and u gives T � Basic Systems � � � ∂ � λ | F � | δ X � + � ∂ � λ | F � 1 ◦ ˙ Theory δ J λ = dt | δ u � − � λ | ( δ X ) � ∂ X ∂ u DYNAMO Platform 0 Applications I: � T � � � Error Correction � ∂ � λ | F � | δ X � + � ∂ � λ | F � T � | δ u � + � ˙ = dt λ | δ X � − � λ | δ X � 0 , � ∂ X ∂ u Applications II: 0 Fixed-Point Engineering � T T � � T � ˙ dt � ˙ Applications III: (for last two terms integrate by parts: − λ | δ X � ) dt � λ | ( δ X ) � = −� λ | δ X � � 0 + Noise Switching 0 0 Sort terms to get total of first variations Conclusions T � � � � � � ∂ L + ∂ � λ | F � ∂ L + ∂ � λ | F � � T � T + ˙ δ J + δ J λ = dt � λ | δ X � + � | δ u � + L ( t ) 0 δ t − � λ ( t ) | δ X ( t ) � . � � 0 � ∂ X ∂ u 0 � Last two terms simplify to: L ( T ) δ t + � λ ( T ) | F ( X , u , T ) � δ t , because (a) L ( 0 ) = 0 and δ X ( 0 ) = 0. (b) end condition X ( T + δ t ) + δ X ( T + δ t ) = X ( T ) entails in first order ˙ X ( T ) δ t + δ X ( T ) = 0 , so δ X ( T ) = − ˙ X ( T ) δ t = − F ( T ) δ t and −� λ ( T ) | δ X ( T ) � = � λ ( T ) | F ( T ) � δ t .

Maximum Principle, ctd First variation of J λ in X and u gives T � Basic Systems � � � ∂ � λ | F � | δ X � + � ∂ � λ | F � 1 ◦ ˙ Theory δ J λ = dt | δ u � − � λ | ( δ X ) � ∂ X ∂ u DYNAMO Platform 0 Applications I: � T � � � Error Correction � ∂ � λ | F � | δ X � + � ∂ � λ | F � T � | δ u � + � ˙ = dt λ | δ X � − � λ | δ X � 0 , � ∂ X ∂ u Applications II: 0 Fixed-Point Engineering � T T � � T � ˙ dt � ˙ Applications III: (for last two terms integrate by parts: − λ | δ X � ) dt � λ | ( δ X ) � = −� λ | δ X � � 0 + Noise Switching 0 0 Sort terms to get total of first variations Conclusions T � � � � � � ∂ L + ∂ � λ | F � ∂ L + ∂ � λ | F � � T � T + ˙ δ J + δ J λ = dt � λ | δ X � + � | δ u � + L ( t ) 0 δ t − � λ ( t ) | δ X ( t ) � . � � 0 � ∂ X ∂ u 0 � Last two terms simplify to: L ( T ) δ t + � λ ( T ) | F ( X , u , T ) � δ t , because (a) L ( 0 ) = 0 and δ X ( 0 ) = 0. (b) end condition X ( T + δ t ) + δ X ( T + δ t ) = X ( T ) entails in first order ˙ X ( T ) δ t + δ X ( T ) = 0 , so δ X ( T ) = − ˙ X ( T ) δ t = − F ( T ) δ t and −� λ ( T ) | δ X ( T ) � = � λ ( T ) | F ( T ) � δ t .

Maximum Principle, ctd First variation of J λ in X and u gives T � Basic Systems � � � ∂ � λ | F � | δ X � + � ∂ � λ | F � 1 ◦ ˙ Theory δ J λ = dt | δ u � − � λ | ( δ X ) � ∂ X ∂ u DYNAMO Platform 0 Applications I: � T � � � Error Correction � ∂ � λ | F � | δ X � + � ∂ � λ | F � T � | δ u � + � ˙ = dt λ | δ X � − � λ | δ X � 0 , � ∂ X ∂ u Applications II: 0 Fixed-Point Engineering � T T � � T � ˙ dt � ˙ Applications III: (for last two terms integrate by parts: − λ | δ X � ) dt � λ | ( δ X ) � = −� λ | δ X � � 0 + Noise Switching 0 0 Sort terms to get total of first variations Conclusions T � � � � � � ∂ L + ∂ � λ | F � ∂ L + ∂ � λ | F � � T � T + ˙ δ J + δ J λ = dt � λ | δ X � + � | δ u � + L ( t ) 0 δ t − � λ ( t ) | δ X ( t ) � . � � 0 � ∂ X ∂ u 0 � Last two terms simplify to: L ( T ) δ t + � λ ( T ) | F ( X , u , T ) � δ t , because (a) L ( 0 ) = 0 and δ X ( 0 ) = 0. (b) end condition X ( T + δ t ) + δ X ( T + δ t ) = X ( T ) entails in first order ˙ X ( T ) δ t + δ X ( T ) = 0 , so δ X ( T ) = − ˙ X ( T ) δ t = − F ( T ) δ t and −� λ ( T ) | δ X ( T ) � = � λ ( T ) | F ( T ) � δ t .

Maximum Principle, ctd First variation of J λ in X and u gives T � Basic Systems � � � ∂ � λ | F � | δ X � + � ∂ � λ | F � 1 ◦ ˙ Theory δ J λ = dt | δ u � − � λ | ( δ X ) � ∂ X ∂ u DYNAMO Platform 0 Applications I: � T � � � Error Correction � ∂ � λ | F � | δ X � + � ∂ � λ | F � T � | δ u � + � ˙ = dt λ | δ X � − � λ | δ X � 0 , � ∂ X ∂ u Applications II: 0 Fixed-Point Engineering � T T � � T � ˙ dt � ˙ Applications III: (for last two terms integrate by parts: − λ | δ X � ) dt � λ | ( δ X ) � = −� λ | δ X � � 0 + Noise Switching 0 0 Sort terms to get total of first variations Conclusions T � � � � � � ∂ L + ∂ � λ | F � ∂ L + ∂ � λ | F � � T � T + ˙ δ J + δ J λ = dt � λ | δ X � + � | δ u � + L ( t ) 0 δ t − � λ ( t ) | δ X ( t ) � . � � 0 � ∂ X ∂ u 0 � Last two terms simplify to: L ( T ) δ t + � λ ( T ) | F ( X , u , T ) � δ t , because (a) L ( 0 ) = 0 and δ X ( 0 ) = 0. (b) end condition X ( T + δ t ) + δ X ( T + δ t ) = X ( T ) entails in first order ˙ X ( T ) δ t + δ X ( T ) = 0 , so δ X ( T ) = − ˙ X ( T ) δ t = − F ( T ) δ t and −� λ ( T ) | δ X ( T ) � = � λ ( T ) | F ( T ) � δ t .

Maximum Principle, ctd First variation of J λ in X and u gives T � Basic Systems � � � ∂ � λ | F � | δ X � + � ∂ � λ | F � 1 ◦ ˙ Theory δ J λ = dt | δ u � − � λ | ( δ X ) � ∂ X ∂ u DYNAMO Platform 0 Applications I: � T � � � Error Correction � ∂ � λ | F � | δ X � + � ∂ � λ | F � T � | δ u � + � ˙ = dt λ | δ X � − � λ | δ X � 0 , � ∂ X ∂ u Applications II: 0 Fixed-Point Engineering � T T � � T � ˙ dt � ˙ Applications III: (for last two terms integrate by parts: − λ | δ X � ) dt � λ | ( δ X ) � = −� λ | δ X � � 0 + Noise Switching 0 0 Sort terms to get total of first variations Conclusions T � � � � � � ∂ L + ∂ � λ | F � ∂ L + ∂ � λ | F � � T � T + ˙ δ J + δ J λ = dt � λ | δ X � + � | δ u � + L ( t ) 0 δ t − � λ ( t ) | δ X ( t ) � . � � 0 � ∂ X ∂ u 0 � Last two terms simplify to: L ( T ) δ t + � λ ( T ) | F ( X , u , T ) � δ t , because (a) L ( 0 ) = 0 and δ X ( 0 ) = 0. (b) end condition X ( T + δ t ) + δ X ( T + δ t ) = X ( T ) entails in first order ˙ X ( T ) δ t + δ X ( T ) = 0 , so δ X ( T ) = − ˙ X ( T ) δ t = − F ( T ) δ t and −� λ ( T ) | δ X ( T ) � = � λ ( T ) | F ( T ) � δ t .

Maximum Principle, ctd Introduce scalar-valued Hamiltonian function h ( X , λ, u , t ) := L ( X , u , t ) + � λ ( t ) | F ( X , u , t ) � = L + � λ | ˙ X � Basic Systems Theory to finally arrive at DYNAMO Platform � T � � Applications I: � ∂ h λ | δ X � + � ∂ h + ˙ δ J + δ J λ = dt ∂ u | δ u � + h ( X , λ, u , T ) δ t . Error Correction ∂ X 0 Applications II: Fixed-Point Therefore optimal controls u ∗ ( t ) leading to quality-optimising Engineering trajectories X ∗ ( t ) and their adjoints λ ∗ ( t ) result if Applications III: Noise Switching − ∂ h ( X ∗ , λ ∗ , u ∗ , t ) Conclusions ˙ λ ∗ ( t ) = ∂ X ∗ � ∂ h ( X ∗ , λ ∗ , u ∗ , t ) ˙ � X ∗ ( t ) ≡ F ( X ∗ , u ∗ , t ) = ∂λ † ∗ � ∂ h ( X ∗ , λ ∗ , u ∗ , t ) 0 = ∂ u ∗ 0 h ( X ∗ , λ ∗ , u ∗ , T ) = , as stated in P ONTRYAGIN ’s Theorem. �

Maximum Principle, ctd Introduce scalar-valued Hamiltonian function h ( X , λ, u , t ) := L ( X , u , t ) + � λ ( t ) | F ( X , u , t ) � = L + � λ | ˙ X � Basic Systems Theory to finally arrive at DYNAMO Platform � T � � Applications I: � ∂ h λ | δ X � + � ∂ h + ˙ δ J + δ J λ = dt ∂ u | δ u � + h ( X , λ, u , T ) δ t . Error Correction ∂ X 0 Applications II: Fixed-Point Therefore optimal controls u ∗ ( t ) leading to quality-optimising Engineering trajectories X ∗ ( t ) and their adjoints λ ∗ ( t ) result if Applications III: Noise Switching − ∂ h ( X ∗ , λ ∗ , u ∗ , t ) Conclusions ˙ λ ∗ ( t ) = ∂ X ∗ � ∂ h ( X ∗ , λ ∗ , u ∗ , t ) ˙ � X ∗ ( t ) ≡ F ( X ∗ , u ∗ , t ) = ∂λ † ∗ � ∂ h ( X ∗ , λ ∗ , u ∗ , t ) 0 = ∂ u ∗ 0 h ( X ∗ , λ ∗ , u ∗ , T ) = , as stated in P ONTRYAGIN ’s Theorem. �

Getting Optimized Quantum Controls Gradient Flow on Control Amplitudes Gradient Assisted Pulse Engineering GRAPE Basic Systems Theory DYNAMO Platform Applications I: Error Correction Applications II: Fixed-Point Engineering J. Magn. Reson. 172 (2005), 296 and Phys. Rev. A 72 (2005), 042331 Applications III: Noise Switching Conclusions � � �