Finding and optimizing gates Application to Josephson qubits Summary Optimal Control of Josephson qubits What can quantum control do for quantum computing? .K. Wilhelm 1 2 M.J. Storcz 2 J. Ferber 2 A. Spörl 3 F T. Schulte-Herbrüggen 3 S.J. Glaser 3 . Rebentrost 1 2 P 1 Institute for Quantum Computing (IQC) and Physics Department University of Waterloo, Canada 2 Physics Department, Arnold Sommerfeld Center, and CeNS Ludwig-Maximilians-Universität München, Germany 3 Chemistry Department Munich University of Technology, Germany Conference on Quantum Information and Quantum Control II, 2006 F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates Application to Josephson qubits Summary Outline Finding and optimizing gates 1 The challenge of finding the right pulse Control theory and GRAPE Application to Josephson qubits 2 Avoiding leakage in a single phase qubit Towards better pulses Optimizing two-qubit gates F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Basic problem setting Our physical system gives us a Hamiltonian � H ( t ) = H d + u j ( t ) H j (1) j with static drift H d , controls u j and control Hamiltonians H j . Our goal: Build a propagator � t � � − i dt ′ H ( t ′ ) U gate = U ( t , 0 ) = T exp (2) � 0 using physical u j ( t ) . F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Rotating wave and area theorem. Spin in static z plus rotating xy field � E λ ( t ) e i ω t � σ = 1 H ( t ) = − γ� B ( t ) · � (3) λ ( t ) e − i ω t − E 2 in co-rotating frame � E − ω H ′ ( t ) = 1 � λ ( t ) (4) 2 λ ( t ) − ( E − ω ) On resonance: E − ω = 0 [ H ′ ( t ) , H ′ ( t ′ )] = 0, thus � t � t � − i � � − i � dt ′ H ( t ′ ) dt ′ H ( t ′ ) T exp = exp = � � 0 0 � t φ ( t ) = 1 dt ′ λ ( t ′ ) = cos φ ( t ) − i σ x sin φ ( t ) (5) � 0 Area theorem F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Beyond the area theorem The area theorem does in general hold for [ H ′ ( t ) , H ′ ( t ′ )] � = 0 out of resonance for non-rotating wave Hamiltonians and strong driving (non-RWA) i.e. high pulses for multi-qubit systems Power of quantum computing comes from the global non-validity of the area theorem! F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Complementing quantum circuits Quantum circuit solution: Discretize into RWA steps with full control. Complemented by control theory even the single qubit gates may not be accessible by RWA decomposition into elementary gates may not be efficient F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Complex control sequences There are ingenious NMR solutions based on 50 years of quantum control ... do we have to do it again? Analogous situation: Steering / parallel parking F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Using control theory Established discipline in applied math / engineering Applied to quantum systems for state transfers e.g. in quantum chemistry Developed for NMR by N. Khaneja (Harvard), S.J. Glaser, T. Schulte-Herbrüggen . . . (TUM) You do not need to know molecular biology in order to fry an egg. (Donald E. Knuth) F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Basic idea. Take any dynamical system with variables x i and controls u j with EOM ˙ x = f ( x , u , t ) (6) Optimize a performance index at final time t f , F ( x ( t f ) , u ( t f )) using J = F ( x ( t f ) , u ( t f )) + (7) � t f dt λ T ( t )( ˙ x − f ( x , u , t )) t i with initial conditions x ( t i ) . F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Solution of the control problem Variation with constraints leads to initial value problem ˙ x = f ( x , u , t ) x ( t i ) = x i (8) final value problem for influence function λ � ∂ f � T � T � ∂ F ˙ λ = − λ λ ( t f ) = (9) ∂ x ∂ x and equation for the controls � ∂ f � T λ = 0 (10) ∂ u Solvable, typically hard (split conditions!) F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary From Rockets to Propagators Control problem for a quantum gate: U ( t i ) = ˆ x �→ U ( t ) 1 (11) � f �→ − i ( H d + u i ( t ) H i ) U (12) i � 2 = 2 N − 2 ReTr ( U † � � φ = � U gate − U ( t f ) gate U ( t f )) (13) So we need to maximize Tr ( U † gate U ( t f )) . Problem : Fixes global phase, too gate U ( t f )) | 2 instead. Solution : Maximize Φ = | Tr ( U † F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Numerical solution Numerical solution: Minimize J directly. Problem: Computationally hard optimization, numerical gradients ∂φ ∂ u i time-consuming ( ≈ hours on supercomputer). From A.O. Niskanen, J.J. Vartiainen and M.M. Salomaa, PRL 90 , 197901 (2003). F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Challenge In the discretized grid, how does Φ change when the control is changed in one point? F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Gradient Ascent Pulse Engineering (GRAPE) I Rewrite performance index 2 gate U ( t f )) | 2 = � � | Tr ( U † � Tr ( U † ( t j , t N ) U gate ) † U ( t j , t 1 ) Φ = � � � 2 � � � † � U † j + 1 . . . U † � � = N U gate U j . . . U 1 � Tr � � � Trotterized time-step propagators � � �� � U i = exp − i ∆ t H d + u k ( t i ) H k (14) Using � 1 � d dx e A + Bx = e A d τ e − A τ Be A τ � (15) � � 0 x = 0 F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates The challenge of finding the right pulse Application to Josephson qubits Control theory and GRAPE Summary Gradient Ascent Pulse Engineering (GRAPE) II we can derive ∂ Φ ∂ u k analytically. ∂ Φ �� � Tr U † = δ t Re gate U N . . . U j + 1 H k U j . . . U 1 ∂ u k ( t j ) � �� Tr U † gate U N . . . U j + 1 U j . . . U 1 N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbrüggen, S.J. Glaser, JMR 172 , 296 (2005). F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates Avoiding leakage in a single phase qubit Application to Josephson qubits Towards better pulses Summary Optimizing two-qubit gates The physical problem. Successful superconducting qubit with close leakage level δω = ω 12 − ω 23 ≃ 0 . 1 ω 12 (16) Drive resonantly on ω 12 . RWA-Hamiltonian √ − δω 2 λ ( t ) 0 √ H ′ = 2 λ ( t ) 0 λ ( t ) a 0 λ ( t ) 0 a Martinis and Simmonds (17) groups, UCSB and NIST How to avoid leakage to the higher level? F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates Avoiding leakage in a single phase qubit Application to Josephson qubits Towards better pulses Summary Optimizing two-qubit gates Properties of the problem. At low λ leakage is small ∝ λ/δω , area theorem o.k. - slow pulse At extremely high λ ≫ δω area theorem again. Can we at least push the limits at intermediate λ ≃ δω ? Populations of | 0 � , | 1 � , | 2 � � t 0 dt ′ λ ( t ′ ) φ ( t ) = F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates Avoiding leakage in a single phase qubit Application to Josephson qubits Towards better pulses Summary Optimizing two-qubit gates GRAPE for this problem. We want an X gate on the two levels, i.e. e i φ 2 0 0 U gate = e i φ 1 0 0 1 (18) 0 1 0 so we have two free phases. Performance index Φ d = 1 5 ( | M 22 | 2 + | M 00 + M 11 | 2 ) M = U † gate U ( t f ) . (19) F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates Avoiding leakage in a single phase qubit Application to Josephson qubits Towards better pulses Summary Optimizing two-qubit gates Overall performance Rectangular Rabi pulse GRAPE, fixed internal phase GRAPE, free internal phase F.K. Wilhelm et al. QC for SQubits
Finding and optimizing gates Avoiding leakage in a single phase qubit Application to Josephson qubits Towards better pulses Summary Optimizing two-qubit gates Optimum pulse shapes F.K. Wilhelm et al. QC for SQubits
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