Quantum Control Synthesizing Robust Gates T S Mahesh
Quantum Control Synthesizing Robust Gates T. S. Mahesh Indian Institute of Science Education and Research, Pune
Contents 1. Di. Vincenzo Criteria 2. Quantum Control 3. Single and Two-qubit control 4. Control via Time-dependent Hamiltonians • Progressive Optimization • Gradient Ascent 5. Practical Aspects • Bounding within hardware limits • Robustness • Nonlinearity 6. Summary
Criteria for Physical Realization of QIP 1. Scalable physical system with mapping of qubits 2. A method to initialize the system 3. Big decoherence time to gate time 4. Sufficient control of the system via time-dependent Hamiltonians (availability of a universal set of gates). 5. Efficient measurement of qubits Di. Vincenzo, Phys. Rev. A 1998
Quantum Control Given a quantum system, how best can we control its dynamics? • Control can be a general unitary or a state to state transfer (can also involve non-unitary processes: eg. changing purity) • Control parameters must be within the hardware limits • Control must be robust against the hardware errors • Fast enough to minimize decoherence effects or combined with dynamical decoupling to suppress decoherence
General Unitary General unitary is state independent: Example: NOT, CNOT, Hadamard, etc. Hilbert Space 1 UTG 0 UEXP Fidelity = Tr{UEXP·UTG} / N 2 obtained by simulation or process tomography
State to State Transfer A particular input state is transferred to a particular output state Eg. 000 ( 000 + 111 ) / 2 Hilbert Space Target Final Initial Fidelity = Final Target 2 obtained by tomography
Universal Gates • Local gates (eg. Ry( ), Rz( )) and CNOT gates together form a universal set Example: Error Correction Circuit Chiaverini et al, Nature 2004
Degree of control Fault-tolerant computation - E. Knill et al, Science 1998. Quantum gates need not be perfect Error correction can take care of imperfections For fault tolerant computation: Fidelity ~ 0. 999
Single Qubit (spin-1/2) Control (up to a global phase) Bloch sphere
NMR spectrometer RF coil Pulse/Detect Sample resonance at 0 = B 0 ~ B 0 Superconducting coil B 1 cos( rft)
Control Parameters All frequencies are measured w. r. t. ref Chemical Shift 01 = 0 - ref 1 = B 1 RF offset = rf - ref time (k. Hz rad) ~ B 0 RF duration 1 RF amplitude B 1 cos( rft) RF phase RF offset
Single Qubit (spin-1/2) Control (in RF frame) x (in REF frame) y 90 x y 90 -x A general state: Bloch sphere (up to a global phase)
Single Qubit (spin-1/2) Control (in RF frame) (in REF frame)
Single Qubit (spin-1/2) Control (in RF frame) (in REF frame) Turning OFF 0 : Refocusing Refocus Chemical Shift X y time x 01
Two Qubit Control Local Gates
Qubit Selective Rotations - Homonuclear Band-width 1/ = 1 non-selective 1 2 dibromothiophene = 1 selective Not a good method: ignores the time evolution
Qubit Selective Rotations - Heteronuclear 13 CHCl 3 1 H (500 MHz @ 11 T) 13 C (125 MHz @ 11 T) • Larmor frequencies are separated by MHz • Usually irradiated by different coils in the probe • No overlap in bandwidths at all • Easy to rotate selectively ~ ~
Two Qubit Control Local Gates CNOT Gate
Two Qubit Control Chemical shift Coupling constant Refocussing: Refocus Chemical Shifts 1 2 X X = 1/(4 J) Rz(90) time Refocus 0 & J-coupling 1 X 2 time Rz( 0 ) Z
Two Qubit Control Chemical shift Coupling constant = H Z H R-z(90) = X 1/(4 J) R-y(90) R-z(90) X 1/(4 J) X time R-y(90)
Control via Time-dependent Hamiltonians H = H (a (t), b (t) , g (t) , ) a (t) t NOT EASY !! (exception: periodic dependence)
Control via Piecewise Continuous Hamiltonians a 1 b 1 a 3 a 2 b 3 g 1 g 2 H 1 H 2 H 3 a 4 b 4 g 4 H 4 Time
Numerical Approaches for Control Progressive Optimization D. G. Cory & co-workers, JCP 2002 Gradient Ascent Navin Khaneja et al, JMR 2005 Mahesh & Suter, PRA 2006 Common features 1. Generate piecewise continuous Hamiltonians 2. Start from a random guess, iteratively proceed 3. Good solution not guaranteed 4. Multiple solutions may exist 5. No global optimization
Piecewise Continuous Control D. G. Cory, JCP 2002 Strongly Modulated Pulse (SMP) ( 1, 1, 1) ( 3, 13, 3, 3) ( 2, 12, 2, 2) …
Progressive Optimization D. G. Cory, JCP 2002 Random Guess Maximize Fidelity simplex Split Maximize Fidelity simplex
Example Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1 Fidelity : 0. 99
Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1
Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1 Initial state Iz 1+Iz 2 SMPs are not limited by bandwidth
Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1 Initial state Iz 1+Iz 2 SMPs are not limited by bandwidth
Amp (k. Hz) Pha (deg) 0. 99 Pha (deg) Amp (k. Hz) 0. 99 O Time (ms) 1 13 C Alanine -O C 1 H 2 C CH 3 3 NH 3+ 2 3
Shifts and J-couplings AB 1 2 3 4 5 6 7 8 9 10 11 12 AB 1 -1423 134 -13874 2 3 4 5 6 7 8 9 10 11 2. 2 4. 4 12 6. 6 52 35. 2 4. 1 1444 2. 2 74 -9688 2. 0 1. 8 5. 3 11. 5 4. 4 53. 6 0 11. 5 147 6. 1 201 8233 11. 5 2. 2 4. 3 6. 7 4. 4 5. 3 998 3. 6 -998 4421 4279 16. 2 5. 3 2455 221. 8 1756 -3878
Benchmarking 12 -qubits 8 11 Ø Benchmarking circuit 10 9 A A’ 2 1 4 3 5 7 6 AA’ 1 2 3 Qubits 4 5 6 7 8 9 10 11 Time Fidelity: 0. 8 PRL, 2006
Quantum Algorithm for NGE (QNGE) : in liquid crystal PRA, 2006
Quantum Algorithm for NGE (QNGE) : Crob: 0. 98 PRA, 2006
Progressive Optimization D. G. Cory, JCP 2002 Advantages 1. Works well for small number of qubits ( < 5 ) 2. Can be combined with other optimizations (genetic algorithm etc) 3. Solutions consist of small number of segments – easy to analyze Disadvantage 1. Maximization is usually via Simplex algorithms Takes a long time
SMPs : Calculation Time During SMP calculation: U = exp(-i. Heff t) calculated typically over 103 times Single ½ : Heff = Two spins : Heff = 2 x 2 Qubits Calc. time 1 -3 minutes 4 -6 Hours >7 4 x 4 Days (estimation) . . . Matrix Exponentiation is a difficult job 10 spins : Heff = 210 x 210 ~ Million - Several dubious ways !!
Gradient Ascent Liouville von-Neuman eqn Control parameters Final density matrix: Navin Khaneja et al, JMR 2005
Gradient Ascent Correlation: Backward propagated opeartor at t = j t Forward propagated opeartor at t = j t Navin Khaneja et al, JMR 2005
Gradient Ascent Navin Khaneja et al, JMR 2005 ’ ? = ’ t (up to 1 st order in t) ’ ’
Gradient Ascent Step-size Navin Khaneja et al, JMR 2005
Gradient Ascent Navin Khaneja et al, JMR 2005 GRAPE Algorithm Guess uk No Correlation > 0. 99? Yes Stop
Practical Aspects 1. Bounding within hardware limits 2. Robustness 3. Nonlinearity
Bounding the control parameters Quality factor = Fidelity + Penalty function To be maximized Shoots-up if any control parameter exceeds the limit
Practical Aspects 1. Bounding within hardware limits 2. Robustness 3. Nonlinearity
Incoherent Processes Spatial inhomogeneities in RF / Static field Hilbert Space Final Initial Final UEXPk( )
Robust Control Coherent control in the presence of incoherence: Hilbert Space Target Initial UEXPk( )
Inhomogeneities SFI Analysis of spectral line shapes SFI Ideal f RFI f Analysis of nutation decay z z RFI Ideal y y x x
RF inhomogeneity 1 Ideal Probability of distribution In practice RFI: Spatial nonuniformity in RF power 0 RF Power Desired RF Power
RF inhomogeneity Bruker PAQXI probe (500 MHz)
Example Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1
Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1
Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1
Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1 Initial state Iz 1+Iz 2
Shifts: 500 Hz, - 500 Hz Coupling: 20 Hz Target Operator : ( /2)y 1 Initial state Iz 1+Iz 2
Robust Control Eg. Two-qubit system Shifts: 500 Hz, -500 Hz J = 50 Hz Fidelity = 0. 99 Target Operator : ( )y 1 Initial state Ix 1+Ix 2 -
Robust Control Eg. Two-qubit system Shifts: 500 Hz, -500 Hz J = 50 Hz Fidelity = 0. 99 Target Operator : ( )y 1 - Initial state Ix 1+Ix 2
Practical Aspects 1. Bounding within hardware limits 2. Robustness 3. Nonlinearity
Spectrometer non-linearities Computer: “This is what I sent”
Spectrometer non-linearities Computer: “This is what I sent” Spins: “This is what we got”
Multi-channel probes: Target coil Spy coil ~ - D. G. Cory et al, PRA 2003.
Spectrometer non-linearities F
Feedback correction F hardware F-1 F hardware - D. G. Cory et al, PRA 2003.
Feedback correction: Computer: “This is what I sent” Spins: “This is what we got” Compensated Shape - D. G. Cory et al, PRA 2003.
Summary 1. Di. Vincenzo Criteria 2. Quantum Control 3. Single and Two-qubit control 4. Control via Time-dependent Hamiltonians • Progressive Optimization • Gradient Ascent 5. Practical Aspects • Bounding within hardware limits • Robustness • Nonlinearity
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