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Quantum ControlSynthesizing Robust GatesT. S. Mahesh

Indian Institute of Science Education and Research, Pune

DiVincenzo CriteriaQuantum ControlSingle and Two-qubit controlControl via Time-dependent HamiltoniansProgressive OptimizationGradient AscentPractical AspectsBounding within hardware limitsRobustnessNonlinearitySummaryContentsCriteria for Physical Realization of QIPScalable physical system with mapping of qubitsA method to initialize the systemBig decoherence time to gate timeSufficient control of the system via time-dependent Hamiltonians (availability of a universal set of gates).5. Efficient measurement of qubitsDiVincenzo, Phys. Rev. A 1998Given a quantum system, how best can we control its dynamics?Quantum Control 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 is state independent: Example: NOT, CNOT, Hadamard, etc.General UnitaryUTG1Hilbert SpaceFidelity = Tr{UEXPUTG} / N 20UEXPobtained bysimulation orprocess tomographyA particular input state is transferred to a particular output stateEg. 000 ( 000 + 111 ) /2State to State TransferInitialTargetHilbert SpaceFinalFidelity = FinalTarget 2obtained bytomography6Universal Gates Local gates (eg. Ry(), Rz()) and CNOT gates together form a universal setExample: Error Correction CircuitChiaverini et al, Nature2004

Degree of control For fault tolerant computation: Fidelity ~ 0.999

- E. Knill et al, Science 1998.Quantum gates need not be perfect

Error correction can take care of imperfections

Fault-tolerant computationSingle Qubit (spin-1/2) Control

(up to a global phase)Bloch sphere~Sampleresonance at 0 =B0 RF coilPulse/DetectSuperconductingcoilB0B1cos(wrft)NMR spectrometer

Control Parameters~1 = B1 rfB0B1cos(wrft) RF duration 1 RF amplitude RF phase RF offset

RF offset = = rf - ref (kHz rad)Chemical Shift01 = 0 - refAll frequencies aremeasured w.r.t. ref

timeSingle Qubit (spin-1/2) Control

(up to a global phase)

Bloch sphere(in RF frame)

(in REF frame)

A general state:90-x90xyxySingle Qubit (spin-1/2) Control

(in RF frame)

(in REF frame)

Single Qubit (spin-1/2) Controlxyw01Turning OFF 0 : RefocusingXRefocus Chemical Shifttime

(in RF frame)

(in REF frame)Two Qubit Control

Local Gates

Qubit Selective Rotations - HomonuclearBand-width 1/

1212dibromothiophene

= 1non-selectiveselective = 1Not a good method: ignores the time evolution Qubit Selective Rotations - Heteronuclear Larmor frequencies are separated by MHz

Usually irradiated by different coils in the probe

No overlap in bandwidths at all

Easy to rotate selectively

13CHCl31H (500 MHz @ 11T)13C (125 MHz @ 11T)~~Two Qubit Control

Local Gates

CNOT Gate

Two Qubit Control

Chemical shiftCoupling constantChemical shiftXXRefocus Chemical Shifts12Refocussing:XRefocus 0 & J-coupling12ZRz(90)Rz(90)Rz(0) = 1/(4J)timetimeTwo Qubit Control

Chemical shiftCoupling constantChemical shiftZHH= 1/(4J)1/(4J)R-z(90)R-z(90)timeXXXR-y(90)R-y(90)=Control via Time-dependent HamiltoniansH = H (a (t), b (t) , g (t) , )

NOT EASY !! (exception: periodic dependence)a (t)t21Control via Piecewise Continuous Hamiltonians

a3b3g3H 3a1b1g1H 1a2b2g2H 2a4b4g4H 4Time22Gradient AscentNavin Khaneja et al, JMR 2005Numerical Approaches for Control Progressive OptimizationD. G. Cory & co-workers, JCP 2002

Mahesh & Suter, PRA 2006Generate piecewise continuous HamiltoniansStart from a random guess, iteratively proceedGood solution not guaranteedMultiple solutions may existNo global optimizationCommon features(t1,w11,f1,w1)(t2,w12,f2,w2)(t3,w13,f3,w3)Piecewise Continuous ControlD. G. Cory, JCP 2002Strongly Modulated Pulse (SMP)Progressive OptimizationD. G. Cory, JCP 2002Random GuessMaximize FidelitySplitMaximize FidelitySplitMaximize Fidelity

simplexsimplexsimplexExample

Fidelity : 0.99Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y1

Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y1

SMPs arenot limitedby bandwidthInitial stateIz1+Iz2

Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y1

SMPs arenot limitedby bandwidthInitial stateIz1+Iz2Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y129

123

Time (ms)Amp (kHz)Pha (deg)Amp (kHz)Pha (deg)Amp (kHz)Pha (deg)0.990.990.99CH3CCNH3+O-OH31213C AlanineAB123456789101112AB-14231346.61-138745235.24.12.01.85.3214442.27411.54.411.52.24.43-968853.61476.14020111.52.24.4582335.369983.64.36.77-9988442116.25.39427916.25.3102455221.811175612-3878

Shifts and J-couplings Benchmarking circuitAA1234567891011Time Qubits

AA1234567981011Benchmarking 12-qubitsPRL, 2006Fidelity: 0.8

Quantum Algorithm for NGE (QNGE) :

PRA, 2006in liquid crystalQuantum Algorithm for NGE (QNGE) : Quantum Algorithm for NGE (QNGE) :

Crob: 0.98PRA, 2006Progressive OptimizationD. G. Cory, JCP 2002Works well for small number of qubits ( < 5 )Can be combined with other optimizations (genetic algorithm etc)Solutions consist of small number of segments easy to analyzeAdvantagesDisadvantage1. Maximization is usually via Simplex algorithms Takes a long time SMPs : Calculation Time2 x 2Single : Heff =4 x 4Two spins : Heff = 210 x 210 ~ Million10 spins : Heff =...During SMP calculation: U = exp(-iHeff t) calculated typically over 103 timesQubits Calc. time1 - 3 minutes4 - 6 Hours > 7 Days (estimation)Matrix Exponentiationis a difficult job

- Several dubious ways !!Gradient AscentNavin Khaneja et al, JMR 2005

Control parameters

Liouville von-Neuman eqn

Final density matrix:

Gradient AscentNavin Khaneja et al, JMR 2005

Correlation:

Backward propagated opeartor at t = jtForward propagated opeartor at t = jt

Gradient AscentNavin Khaneja et al, JMR 2005

?

= t

(up to 1st order in t)

Gradient AscentNavin Khaneja et al, JMR 2005

Step-sizeGradient AscentNavin Khaneja et al, JMR 2005

Guess uk

NoYesStopCorrelation > 0.99?GRAPE AlgorithmPractical AspectsBounding within hardware limits

Robustness

Nonlinearity

Bounding the control parametersQuality factor = Fidelity + Penalty functionShoots-up if any control parameter exceeds the limitTo be maximizedPractical AspectsBounding within hardware limits

Robustness

Nonlinearity

Spatial inhomogeneities in RF / Static fieldInitialFinalHilbert SpaceIncoherent ProcessesUEXPk(t)FinalFinalCoherent control in the presence of incoherence:Robust ControlInitialHilbert SpaceTargetUEXPk(t)

InhomogeneitiesSFI Analysis of spectral line shapes

RFI Analysis of nutation decayf f IdealSFIxyzxyzIdealRFIRFI: Spatialnonuniformityin RF powerRF PowerDesired RF Power01In practiceIdealProbabilityof distributionRF inhomogeneityRF inhomogeneity

Bruker PAQXI probe (500 MHz) Example

Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y1

Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y1

Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y1

Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y1Initial stateIz1+Iz2

Shifts: 500 Hz, - 500 HzCoupling: 20 HzTarget Operator : (/2)y1Initial stateIz1+Iz2

Robust ControlEg. Two-qubit systemShifts: 500 Hz, -500 HzJ = 50 HzFidelity = 0.99Target Operator : ()y1-Initial stateIx1+Ix2

Robust ControlEg. Two-qubit systemShifts: 500 Hz, -500 HzJ = 50 HzFidelity = 0.99-Target Operator : ()y1Initial stateIx1+Ix256Practical AspectsBounding within hardware limits

Robustness

Nonlinearity

Spectrometer non-linearities

Computer:This is what I sentSpectrometer non-linearities

Computer:This is what I sentSpins: This is what we got ~Multi-channel probes:Target coilSpy coil- D. G. Cory et al, PRA 2003.Spectrometer non-linearitiesF Feedback correctionFF-1F- D. G. Cory et al, PRA 2003. hardwarehardwareFeedback correction:

Spins: This is what we gotComputer:This is what I sentCompensatedShape- D. G. Cory et al, PRA 2003.SummaryDiVincenzo CriteriaQuantum ControlSingle and Two-qubit controlControl via Time-dependent HamiltoniansProgressive OptimizationGradient AscentPractical AspectsBounding within hardware limitsRobustnessNonlinearity