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GRAPE.m
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(* ::Package:: *)
(* ::Title:: *)
(*QuantumUtils for Mathematica*)
(*GRAPE Package*)
(* ::Subsection::Closed:: *)
(*Copyright and License Information*)
(* ::Text:: *)
(*This package is part of QuantumUtils for Mathematica.*)
(**)
(*Copyright (c) 2015 and later, Christopher J. Wood, Christopher E. Granade, Ian N. Hincks*)
(**)
(*Redistribution and use in source and binary forms, with or withoutmodification, are permitted provided that the following conditions are met:*)
(*1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.*)
(*2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.*)
(*3. Neither the name of quantum-utils-mathematica nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.*)
(**)
(*THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THEIMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE AREDISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLEFOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIALDAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS ORSERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVERCAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USEOF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.*)
(* ::Subsection::Closed:: *)
(*Preamble*)
BeginPackage["GRAPE`",{"QUDoc`"}];
(* ::Text:: *)
(*The following packages are needed, but their contexts should not be loaded globally.*)
Needs["QUDevTools`"];
Needs["Visualization`"];
Needs["QSim`"];
Needs["Tensor`"];
$GRAPEUsages = LoadUsages[FileNameJoin[{$QUDocumentationPath, "api-doc", "GRAPE.nb"}]];
(* ::Section:: *)
(*Usage Declaration*)
(* ::Subsection::Closed:: *)
(*Options*)
Unprotect[
Repetitions,ParameterDistribution,DistortionOperator,ForceDistortionDependence,
PulsePenalty,DerivativeMask,PulseLegalizer,
ControlLimitPolicy,MonitorFunction,InitialStepSize,MinimumStepSize,
LineSearchMethod,MinimumImprovement,MinimumIterations,MaximumIterations,
SkipChecks,VerboseAscent,
Ignore,ProjectGradient
];
AssignUsage[
{
Repetitions,ParameterDistribution,DistortionOperator,ForceDistortionDependence,
PulsePenalty,DerivativeMask,PulseLegalizer,
ControlLimitPolicy,MonitorFunction,InitialStepSize,MinimumStepSize,
LineSearchMethod,MinimumImprovement,MinimumIterations,MaximumIterations,
SkipChecks,VerboseAscent,
Ignore,ProjectGradient
},
$GRAPEUsages
];
(* ::Subsection::Closed:: *)
(*Pulses*)
Unprotect[
Pulse,TimeSteps,UtilityValue,PenaltyValue,Target,ControlHamiltonians,
InternalHamiltonian,AmplitudeRange,ExitMessage,
ToPulse,FromPulse,SimForm,AddTimeSteps,SplitPulse,
PulseRemoveKeys,PulseReplaceKey,PulseHasKey,
PulsePhaseRotate,PulsePhaseRamp,PulseDivide,PulseModulate,
RandomPulse,RandomSmoothPulse,GenerateAnnealedPulse,AnnealingGenerator,GaussianTailsPulse,
LegalizePulse,LegalizeQuadraturePulse,NormalizePulse
];
AssignUsage[
{
Pulse,TimeSteps,UtilityValue,PenaltyValue,Target,ControlHamiltonians,
InternalHamiltonian,AmplitudeRange,ExitMessage,
ToPulse,FromPulse,SimForm,AddTimeSteps,SplitPulse,
PulseRemoveKeys,PulseReplaceKey,PulseHasKey,
PulsePhaseRotate,PulsePhaseRamp,PulseDivide,PulseModulate,
RandomPulse,RandomSmoothPulse,GenerateAnnealedPulse,AnnealingGenerator,GaussianTailsPulse,
LegalizePulse,LegalizeQuadraturePulse,NormalizePulse
},
$GRAPEUsages
];
(* ::Subsection::Closed:: *)
(*Utility Function and Targets*)
Unprotect[
Utility,UtilityGradient,
PropagatorFromPulse,PropagatorListFromPulse,
CoherentSubspaces,DensityTransfer
];
AssignUsage[
{
Utility,UtilityGradient,
PropagatorFromPulse,PropagatorListFromPulse,
CoherentSubspaces,DensityTransfer
},
$GRAPEUsages
];
(* ::Subsection::Closed:: *)
(*Distortions*)
Unprotect[
ApplyInverseDistortion,
LiftDistortionRank,JoinDistortions,ComposeDistortions,PerturbateDistortion,ScaleDistortion,
IdentityDistortion,
TimeScaleDistortion,VariableChangeDistortion,
ConvolutionDistortion,ExponentialDistortion,
ListConvolveDistortion,FastExponentialDistortion,
IQDistortion,
NonlinearTransferDistortion,
DEDistortion,DESolver,DESolverArgs,
LinearDEDistortion,CompensationTimeSteps,DecayRatios,StateMetric,
FrequencySpaceDistortion,
CompositePulseDistortion
];
AssignUsage[
{
ApplyInverseDistortion,
LiftDistortionRank,JoinDistortions,ComposeDistortions,PerturbateDistortion,ScaleDistortion,
IdentityDistortion,
TimeScaleDistortion,VariableChangeDistortion,
ConvolutionDistortion,ExponentialDistortion,
ListConvolveDistortion,FastExponentialDistortion,
IQDistortion,
NonlinearTransferDistortion,
DEDistortion,DESolver,DESolverArgs,
LinearDEDistortion,CompensationTimeSteps,DecayRatios,StateMetric,
FrequencySpaceDistortion,
CompositePulseDistortion
},
$GRAPEUsages
];
(* ::Subsection::Closed:: *)
(*Distributions*)
Unprotect[
IdentityDistribution,
ParameterDistributionMean,
ProductParameterDistribution,
HistogramParameterDistribution,
RandomSampleParameterDistribution,RandomMultinormalParameterDistribution,RandomUniformParameterDistribution,
UniformParameterDistribution,
TargetSelectorDistribution
];
AssignUsage[
{
IdentityDistribution,
ParameterDistributionMean,
ProductParameterDistribution,
HistogramParameterDistribution,
RandomSampleParameterDistribution,RandomMultinormalParameterDistribution,RandomUniformParameterDistribution,
UniformParameterDistribution,
TargetSelectorDistribution
},
$GRAPEUsages
];
(* ::Subsection::Closed:: *)
(*Pulse Penalties*)
Unprotect[
ZeroPenalty,
DemandValuePenalty,
RingdownPenalty
];
AssignUsage[
{
ZeroPenalty,
DemandValuePenalty,
RingdownPenalty
},
$GRAPEUsages
];
(* ::Subsection::Closed:: *)
(*Plotting Tools*)
Unprotect[
PulsePlot,PulseFourierPlot,
ShowDistortedPulse,ChannelMapping,PulseScaling,PulseLayout,PulsePaddingMultiplier,
DistributeOption,
RobustnessPlot,LegendIsCell,DistortionOperatorSweep
];
AssignUsage[
{
PulsePlot,PulseFourierPlot,
ShowDistortedPulse,ChannelMapping,PulseScaling,PulseLayout,PulsePaddingMultiplier,
DistributeOption,
RobustnessPlot,LegendIsCell,DistortionOperatorSweep
},
$GRAPEUsages
];
RobustnessPlot::keys = "RobustnessPlot requires all input pulses to have at least the following keys: TimeSteps, Pulse, InternalHamiltonian, ControlHamiltonians, Target, and DistortionOperator";
(* ::Subsection::Closed:: *)
(*Monitor Functions*)
Unprotect[FidelityMonitor,HistogramMonitor,PulsePlotMonitor];
AssignUsage[{FidelityMonitor,HistogramMonitor,PulsePlotMonitor},$GRAPEUsages];
(* ::Subsection::Closed:: *)
(*Gradient Ascent Tools*)
Unprotect[
QuadraticFitLineSearch,InterpolatedLineSearch,
MinStepMul,StepMulStep,MaxStepMul
];
AssignUsage[
{
QuadraticFitLineSearch,InterpolatedLineSearch,
MinStepMul,StepMulStep,MaxStepMul
},
$GRAPEUsages
];
(* ::Subsection::Closed:: *)
(*FindPulse*)
Unprotect[
FindPulse
];
AssignUsage[{FindPulse},$GRAPEUsages];
FindPulse::badguess = "Error: the supplied initialGuess matrix was not understood.";
FindPulse::badrange = "Error: Your controlRange should be of the form {{\[Epsilon]1min,\[Epsilon]1max},{\[Epsilon]2min,\[Epsilon]2max},...}.";
FindPulse::baddistortion = "Error: Distorted output control count inconsistent with length of Hcontrol, or has the wrong shape.";
FindPulse::badjacobian = "Error: The distortion Jacobian should be a rank 4 tensor. Maybe you need to call LiftDistortionRank on your distortion?";
FindPulse::badjacdim1 = "Error: First dimension of your distortion Jacobian does not match length of your distortedPulse.";
FindPulse::badjacdim2 = "Error: Second dimension of your distortion Jacobian does not match number of Hamiltonians.";
FindPulse::badjacdim3 = "Error: Third dimension of your distortion Jacobian does not match the length of the pulse given by initialGuess.";
FindPulse::badjacdim4 = "Error: Fourth dimension of your distortion Jacobian does not match number of control knobs (taken from length of \[Epsilon]range).";
FindPulse::badpenalty = "Error: The penalty function did not return a real number for a penalty.";
FindPulse::badpenaltygrad = "Error: The penalty function's gradient does not have the required shape.";
FindPulse::baddistsum = "Warning: The probabilities in your ParameterDistribution don't sum to 1. Proceeding anyways...";
FindPulse::baddistreps = "Warning: Not all of your replacement rules assign numbers to all present symbols. Proceeding anyways...";
FindPulse::baddistlength = "Error: The number of probabilities and replacement rules must be the same.";
FindPulse::badderivmask = "Error: your derivative mask seems to have the wrong dimensions."
(* ::Subsection::Closed:: *)
(*Exporters*)
Unprotect[ExportJCAMP,ExportSHP];
AssignUsage[{ExportJCAMP,ExportSHP},$GRAPEUsages]
JCAMPTitle::usage = "JCAMPTitle is an ExportJCAMP option with default value Automatic, which sets the title to the filename.Otherwise it should be a string.";
JCAMPDX::usage = "JCAMPDX is an ExportJCAMP option with default value '5.00 Bruker JCAMP library'.";
JCAMPDataType::usage = "JCAMPDataType is an ExportJCAMP option with default value 'Shape Data'.";
JCAMPOrigin::usage = "JCAMPOrigin is an ExportJCAMP option with default value 'Quantum-Utils Mathematica GRAPE'.";
JCAMPUser::usage = "JCAMPUser is an ExportJCAMP option with default value 'nmruser'. Should be your username on the spectrometer.";
JCAMPDate::usage = "JCAMPDate is an ExportJCAMP option with default value Automatic, which sets the date to today.";
JCAMPTime::usage = "JCAMPTime is an ExportJCAMP option with default value Automatic, which uses the current time.";
JCAMPShapeExMode::usage = "JCAMPShapeExMode is an ExportJCAMP option with default value 'None'.";
JCAMPShapeToTrot::usage = "JCAMPShapeToTrot is an ExportJCAMP option with default value '9.000000e+01' (string, not number).";
JCAMPShapeBWFac::usage = "JCAMPShapeBWFac is an ExportJCAMP option with default value '1.000000e+00' (string, not number).";
JCAMPShapeIntegFac::usage = "JCAMPShapeIntegFac is an ExportJCAMP option with default value '1.000000e+00' (string, not number).";
JCAMPShapeMode::usage = "JCAMPShapeMode is an ExportJCAMP option with default value '1' (string, not number).";
JCAMPCalibrationFactor::usage = "JCAMPCalibrationFactor is an ExportJCAMP option with default being the integer 1. This is the number the amplitudes are divided by before writing to the JCAMP file.";
(* ::Section:: *)
(*Implementation*)
Begin["`Private`"];
(* ::Subsection::Closed:: *)
(*Pulses*)
(* ::Subsubsection::Closed:: *)
(*Pulse Object*)
Pulse/:Pulse[args___][key_]:=Association[args][key]
Unprotect@Pulse;
Pulse/:Format[Pulse[args__Rule]]:=Module[{modpulse},
modpulse=Pulse[args];
If[PulseHasKey[modpulse,Pulse],modpulse=PulseReplaceKey[modpulse,Pulse,MatrixForm[modpulse[Pulse]\[Transpose]]]];
If[PulseHasKey[modpulse,ControlHamiltonians],modpulse=PulseReplaceKey[modpulse,ControlHamiltonians,MatrixListForm[modpulse[ControlHamiltonians]]]];
If[PulseHasKey[modpulse,InternalHamiltonian],modpulse=PulseReplaceKey[modpulse,InternalHamiltonian,MatrixForm[modpulse[InternalHamiltonian]]]];
If[PulseHasKey[modpulse,Target]&&MatrixQ[modpulse[Target]],modpulse=PulseReplaceKey[modpulse,Target,MatrixForm[modpulse[Target]]]];
Grid[
Prepend[
{#,#/.List@@modpulse}&/@
{UtilityValue,PenaltyValue,ExitMessage,Target,InternalHamiltonian,ControlHamiltonians,AmplitudeRange,TimeSteps,Pulse,ParameterDistribution,DistortionOperator,PulsePenalty},
{Head,Value}
],
Alignment->{Left,Top},
Dividers->All,
ItemSize->{{13,32},Automatic},
Background->{{LightBlue,None},{LightGreen,None}},
ItemStyle->{Automatic,{"Subsubsubsection"}},
Spacings->{2,.8}
]
]
(* ::Subsubsection::Closed:: *)
(*Pulse Conversion and Transformation*)
Options[ToPulse]={
UtilityValue->None,
PenaltyValue->0,
Target->IdentityMatrix[2],
ControlHamiltonians->2\[Pi]{TP[X]/2,TP[Y]/2},
InternalHamiltonian->0*TP[Z],
DistortionOperator->IdentityDistortion[],
PulsePenalty->ZeroPenalty[],
ParameterDistribution->None,
AmplitudeRange->{{-1,1},{-1,1}},
ExitMessage->"Generated by ToPulse."
};
ToPulse[pulsemat_,opt:OptionsPattern[]]:=Pulse[
TimeSteps -> pulsemat[[All,1]],
Pulse -> pulsemat[[All,2;;]],
Sequence@@DeleteDuplicates[Join[{opt},Options[ToPulse]],First[#1]===First[#2]&]
]
FromPulse[pulse_Pulse]:=AddTimeSteps[pulse[TimeSteps],pulse[Pulse]];
SimForm[pulse_Pulse, distort_:True]:=
Module[{p, Hc=pulse[ControlHamiltonians]},
p=pulse[Pulse];
If[Length@Dimensions@p==1,p=List/@p];
p=AddTimeSteps[pulse[TimeSteps], p];
If[distort, p = pulse[DistortionOperator][p, False]];
{p, Hc}
]
AddTimeSteps[dts_,pulse_]:=If[ListQ@dts,Prepend[pulse\[Transpose],dts]\[Transpose],Prepend[pulse\[Transpose],ConstantArray[dts,Length@pulse]]\[Transpose]]
SplitPulse[pulse_]:={pulse[[All,1]],pulse[[All,2;;-1]]}
(* ::Subsubsection::Closed:: *)
(*Pulse Key Modification*)
PulseRemoveKeys[pulse_Pulse,headers__]:=Select[pulse,Not[MemberQ[{headers},#[[1]]]]&]
PulseReplaceKey[pulse_Pulse,header_,newval_]:=Append[PulseRemoveKeys[pulse,header],header->newval]
PulseHasKey[pulse_Pulse,key_]:=MemberQ[pulse[[All,1]],key]
(* ::Subsubsection::Closed:: *)
(*Pulse Manipulation*)
PulseDivide[pulse_Pulse,n_]:=
Module[{p=pulse[Pulse],dt=pulse[TimeSteps]},
dt=Flatten[ConstantArray[dt/n,n]\[Transpose]];
p=p[[Flatten@Table[ConstantArray[k,n],{k,Length@p}]]];
PulseReplaceKey[PulseReplaceKey[pulse,Pulse,p],TimeSteps,dt]
]
PulsePhaseRotate[pulse_,\[Phi]_]:=
Module[{xy=pulse[Pulse],a\[Theta]},
a\[Theta]={Norm/@xy,\[Phi]+(If[First@#==0&&Last@#==0,0,ArcTan[First@#,Last@#]]&/@xy)}\[Transpose];
xy={First[#]Cos[Last@#],First[#]Sin[Last@#]}&/@a\[Theta];
PulseReplaceKey[pulse,Pulse,xy]
]
PulsePhaseRamp[pulse_,f_]:=
Module[{dt,xy=pulse[Pulse],a\[Theta]},
dt=pulse[TimeSteps];
a\[Theta]={Norm/@xy,2\[Pi]*f*(Accumulate[dt]-dt/2)+(If[#=={0,0},0,ArcTan@@#]&/@xy)}\[Transpose];
xy={First[#]Cos[Last@#],First[#]Sin[Last@#]}&/@a\[Theta];
PulseReplaceKey[pulse,Pulse,xy]
]
PulseModulate[pulse_,f_,\[Phi]_:0]:=
Module[{dt,xy=pulse[Pulse],a\[Theta]p,a\[Theta]m,a\[Theta]},
dt=pulse[TimeSteps];
a\[Theta]p={Norm/@xy,2\[Pi]*f*(Accumulate[dt]-dt/2)+(If[#=={0,0},0,ArcTan@@#]&/@xy)}\[Transpose];
a\[Theta]m={Norm/@xy,-2\[Pi]*f*(Accumulate[dt]-dt/2)+(If[#=={0,0},0,ArcTan@@#]&/@xy)}\[Transpose];
a\[Theta]p=#1*Exp[I*#2]&@@@a\[Theta]p;
a\[Theta]m=#1*Exp[I*#2]&@@@a\[Theta]m;
a\[Theta]=(Exp[-I*\[Phi]]a\[Theta]p+Exp[I*\[Phi]]a\[Theta]m)/2;
xy={Re[#],Im[\[Phi]]}&/@a\[Theta];
PulseReplaceKey[pulse,Pulse,xy]
]
(* ::Subsubsection::Closed:: *)
(*Pulse Matrix Generation*)
RandomPulse[dts_,controlRange_] := AddTimeSteps[dts, (RandomReal[#,Length@dts]&/@controlRange)\[Transpose]]
RandomPulse[dt_?NumericQ,n_Integer,controlRange_] := RandomPulse[ConstantArray[dt,n],controlRange]
Options[RandomSmoothPulse]={InterpolationPoints->10};
RandomSmoothPulse[dts_,controlRange_,OptionsPattern[]]:=
Module[
{randomEntries, interpFunctions, numRandomPoints},
numRandomPoints = Max[Floor[Length@dts/OptionValue[InterpolationPoints]],5];
randomEntries = RandomReal[#,numRandomPoints]&/@controlRange;
interpFunctions = Interpolation[#,Method->"Spline"]&/@randomEntries;
AddTimeSteps[dts, ((#/@Range[1, numRandomPoints, (numRandomPoints-1)/(Length@dts-1)])&/@interpFunctions)\[Transpose]]
];
RandomSmoothPulse[dt_?NumericQ,n_Integer,controlRange_,opt:OptionsPattern[]] := RandomSmoothPulse[ConstantArray[dt,n],controlRange,opt]
Options[GenerateAnnealedPulse] = {AnnealingGenerator -> RandomPulse};
GenerateAnnealedPulse[original_, controlRange_, OptionsPattern[]] := original + OptionValue[AnnealingGenerator][ConstantArray[0,Length@original], controlRange]
GaussianTailsPulse[dt_,T_,riseTime_,Area->area_]:=
Module[{\[Sigma],NI,fun,pulse},
\[Sigma]=riseTime/3;
fun[t_]:=Piecewise[{{1,riseTime<t<T-riseTime},{Exp[-(t-riseTime)^2/(2\[Sigma]^2)],t<=riseTime},{Exp[-(t-T+riseTime)^2/(2\[Sigma]^2)],t>=T-riseTime}}];
If[area!=0,
NI=NIntegrate[fun[t],{t,0,T}];
pulse=Table[{area*fun[t-dt/2]/NI,0},{t,0,T,dt}];
(*Correct the area now that we have a descent shape*)
pulse=area*pulse/(dt*Total@Flatten@pulse);,
pulse=Table[{0,0},{t,0,T,dt}];
];
AddTimeSteps[dt, pulse]
];
GaussianTailsPulse[dt_,T_,riseTime_,Max->max_]:=Module[
{apulse=GaussianTailsPulse[dt,T,riseTime,Area->1]},
AddTimeSteps[apulse[[All,1]],max*apulse[[All,2;;]]/Max[Flatten[apulse[[All,2;;]]]]]
]
(* ::Subsubsection::Closed:: *)
(*Legalization and Normalization*)
LegalizePulse[pulse_,controlRange_]:={#1, Sequence@@MapThread[Clip, {{##2},controlRange}, 1]}& @@@ pulse
LegalizePulse[profile_][pulse_,controlRange_]:=Table[
{
pulse[[n,1]],
Sequence@@MapThread[Clip, {pulse[[n,2;;]],profile[[n]]*controlRange}, 1]
},
{n,Length@pulse}
]
LegalizeQuadraturePulse[Rmax_][pulse_,controlRange_]:=Module[
{dts=pulse[[All,1]],r,amps=pulse[[All,2;;]]},
r=Norm/@amps;
amps=amps*Clip[Rmax/r,{0,1}];
AddTimeSteps[dts,amps]
]
NormalizePulse[pulse_, controlRange_] :=
With[{tp = Transpose[pulse[[All, -2;;]]]},
AddTimeSteps[pulse[[All,1]],Transpose[(tp - controlRange[[All, 1]]) / (controlRange[[All, 2]] - controlRange[[All, 1]])]]
];
(* ::Subsection::Closed:: *)
(*Utility Function and Targets*)
(* ::Subsubsection::Closed:: *)
(*Unitary Propagators*)
(* ::Text:: *)
(*Takes a pulse and finds the overall unitary.*)
PropagatorFromPulse[pulse_,Hint_,Hcontrol_]:=
Module[{step=1,dt,dts,amps},
{dts,amps} = SplitPulse[pulse];
(* Notice the delay equal on dt! Not a bug. *)
dt := dts[[step++]];
Fold[
MatrixExp[-I(Hint+#2.Hcontrol) dt].#1&,
IdentityMatrix[Length[Hint]],
amps
]
]
PropagatorListFromPulse[pulse_,Hint_,Hcontrol_]:=
Module[{dts,dt,step=1,amps},
{dts,amps} = SplitPulse[pulse];
dt := dts[[step++]];
Map[
MatrixExp[-I(Hint+#.Hcontrol)dt]&,
amps
]
]
(* ::Subsubsection::Closed:: *)
(*Target Unitary*)
(* ::Text:: *)
(*We divide by the dimension to make the maximum value of the objective function equal to 1.*)
Utility[Ucalc_,Utarget_List]:=Abs[Tr[Ucalc\[ConjugateTranspose].Utarget]/Length[Ucalc]]^2;
UtilityGradient[pulse_,Hint_,Hcontrol_,Utarget_List]:=
Module[
{
dim=Length[Hint],
dts,amps,Uforw,Uback,gradient,utility,unitaries
},
unitaries=PropagatorListFromPulse[pulse,Hint,Hcontrol];
{dts, amps} = SplitPulse[pulse];
Uforw=Rest[FoldList[#2.#1&,IdentityMatrix[dim],unitaries]];
Uback=Reverse[FoldList[#2\[ConjugateTranspose].#1&,Utarget,Reverse[Rest[unitaries]]]];
gradient=Table[
-2 Re[Tr[Uback[[i]]\[ConjugateTranspose].(I dts[[i]] Hcontrol[[j]].Uforw[[i]])]*Tr[Uforw[[i]]\[ConjugateTranspose].Uback[[i]]]],
{i,Length[unitaries]},{j,Length[Hcontrol]}
]/dim^2;
utility=Utility[Last[Uforw],Utarget];
{utility,gradient}
];
(* ::Subsubsection::Closed:: *)
(*Coherent Subspaces*)
Utility[Ucalc_,target_CoherentSubspaces]:=
With[
{Xs=target[[1]],Ys=target[[2]]},
Sum[
Abs[Tr[Ucalc\[ConjugateTranspose].(Ys[[i]].Xs[[i]]\[ConjugateTranspose])]/Length[Xs[[i,1]]]]^2,
{i,Length[Xs]}
]/Length[Xs]
]
UtilityGradient[pulse_,Hint_,Hcontrol_,target_CoherentSubspaces]:=
Module[
{
dim=Length[Hint],
Uforw,Uback,derivs,performIndex,unitaries,
Xs=target[[1]],
Ys=target[[2]],
numSubspaces,
dts,amps
},
unitaries=PropagatorListFromPulse[pulse,Hint,Hcontrol];
{dts, amps} = SplitPulse[pulse];
numSubspaces=Length[Xs];
Uforw=Rest[FoldList[#2.#1&,IdentityMatrix[dim],unitaries]];
Uback=Reverse[FoldList[#2\[ConjugateTranspose].#1&,IdentityMatrix[dim],Reverse[Rest[unitaries]]]];
derivs=Table[
Sum[
With[{Ubackj=Uback[[i]].Ys[[ss]].Xs[[ss]]\[ConjugateTranspose], ssDim2=Length[Xs[[ss,1]]]^2},
-2 Re[Tr[Ubackj\[ConjugateTranspose].(I dts[[i]] Hcontrol[[j]].Uforw[[i]])]*Tr[Uforw[[i]]\[ConjugateTranspose].Ubackj]]/ssDim2
],
{ss,numSubspaces}
],
{i,Length[unitaries]},{j,Length[Hcontrol]}
]/numSubspaces;
performIndex=Utility[Last[Uforw],target];
{performIndex,derivs}
];
(* ::Subsubsection::Closed:: *)
(*Density Transfer*)
Utility[Ucalc_,target_DensityTransfer]:=
With[
{\[Rho]0=target[[1]],\[Rho]1=target[[2]]},
Re[Tr[\[Rho]1\[ConjugateTranspose].Ucalc.\[Rho]0.Ucalc\[ConjugateTranspose]]/Tr[\[Rho]1.\[Rho]1]]
]
UtilityGradient[pulse_,Hint_,Hcontrol_,target_DensityTransfer]:=
Module[
{
dim=Length[Hint],
\[Rho]forw,\[Rho]back,derivs,unitaries,
\[Rho]0=target[[1]],
\[Rho]1=target[[2]],
dts,amps
},
unitaries=PropagatorListFromPulse[pulse,Hint,Hcontrol];
{dts, amps} = SplitPulse[pulse];
\[Rho]forw=Rest[FoldList[(#2.#1.#2\[ConjugateTranspose])&,\[Rho]0,unitaries]];
\[Rho]back=Reverse[FoldList[(#2\[ConjugateTranspose].#1.#2)&,\[Rho]1,Reverse[Rest[unitaries]]]];
derivs=-Re[Table[
Tr[\[Rho]back[[i]]\[ConjugateTranspose].(I dts[[i]] Com[Hcontrol[[j]], \[Rho]forw[[i]]])],
{i,Length[unitaries]}, {j,Length[Hcontrol]}
]/Tr[\[Rho]1.\[Rho]1]];
{Re[Tr[\[Rho]1\[ConjugateTranspose].Last[\[Rho]forw]]/Tr[\[Rho]1.\[Rho]1]], derivs}
];
(* ::Subsection::Closed:: *)
(*Distortions*)
(* ::Subsubsection::Closed:: *)
(*Distortion Operator*)
DistortionOperator[function_,___][args__]:=function[args]
DistortionOperator/:Format[DistortionOperator[_,format_]]:=format
(* ::Subsubsection::Closed:: *)
(*Apply Inverse Distortion*)
ApplyInverseDistortion[distortion_,exampleInput_,pulseMat_]:=Module[{p,jac,dM,dL,dN,dK},
p=pulseMat[[All,2;;-1]];
jac=Last[distortion[exampleInput,True]];
{dM,dL,dN,dK}=Dimensions[jac];
(* The idea here is to cast the contraction MLNK.NK over two indeces into regular
matrix multiplication so that we can use LinearSolve to do all of the work *)
AddTimeSteps[
exampleInput[[All,1]],
ArrayReshape[LinearSolve[ArrayReshape[jac,{dM*dL,dN*dK}],ArrayReshape[p,{dM*dL,1}]],{dN,dK}]
]
]
(* ::Subsubsection::Closed:: *)
(*Distortion Operator Tools*)
LiftDistortionRank[distortion_DistortionOperator]:=DistortionOperator[
Function[{pulse,computeJac},
Which[computeJac===True,
Module[{d,jac,all,controlRank},
controlRank=Last@Dimensions@pulse-1;
(* Compute the Jacobian and distortion for each control channel*)
all = Table[distortion[pulse[[All,{1,n}]], True], {n,2,Last@Dimensions@pulse}];
(* Same as above *)
d = AddTimeSteps[all[[1,1,All,1]], all[[All,1,All,2]]\[Transpose]];
jac = all[[All,2]];
If[Length@Dimensions@jac===3,
(* The user supplied Jacobian is a matrix for each channel; lift to rank 4. *)
jac = Transpose[Table[KroneckerDelta[in,out]*jac[[in,All,All]],{in,controlRank},{out,controlRank}], {4,2,1,3}];
];
If[Length@Dimensions@jac===5,
(* The user supplied Jacobian is rank 4 for each channel; remove singleton dimensions and repeat above procedure. *)
jac = Transpose[Table[KroneckerDelta[in,out]*jac[[in,All,1,All,1]],{in,controlRank},{out,controlRank}], {4,2,1,3}];
];
{d, jac}
],
computeJac===False,
Module[{d},
(* Compute the distortion for each control channel *)
d = Table[distortion[pulse[[All,{1,n}]], False], {n,2,Last@Dimensions@pulse}];
AddTimeSteps[d[[1,All,1]], d[[All,All,2]]\[Transpose]]
],
computeJac===All,
Module[{d},
(* Compute the distortion for each control channel *)
d = Table[distortion[pulse[[All,{1,n}]], All], {n,2,Last@Dimensions@pulse}];
{
AddTimeSteps[d[[1,1,All,1]], d[[All,1,All,2]]\[Transpose]],
AddTimeSteps[d[[1,2,All,1]], d[[All,2,All,2]]\[Transpose]],
AddTimeSteps[d[[1,3,All,1]], d[[All,3,All,2]]\[Transpose]]
}
]
]
],
Format[HoldForm[Superscript[Format[distortion],"\[CircleTimes]numInputChannels"]]]
];
JoinDistortions[Distortions__DistortionOperator]:=DistortionOperator[
Function[{pulse,computeJac},
Which[computeJac===True,
Module[{d,jac,all,controlRank},
controlRank=Length@{Distortions};
(* Compute the Jacobian and distortion for each control channel*)
all = MapIndexed[#1[pulse[[All,{1, First@#2+1}]], True]&, {Distortions}];
(* Same as above *)
d = AddTimeSteps[all[[1,1,All,1]], all[[All,1,All,2]]\[Transpose]];
jac = all[[All,2]];
(* The user supplied Jacobian is rank 4 for each channel *)
jac = Transpose[ConstantArray[jac[[All,All,1,All,1]], controlRank], {2,4,1,3}];
{d, jac}
],
computeJac===False,
Module[{d},
(* Compute the distortion for each control channel *)
d = MapIndexed[#1[pulse[[All,{1, First@#2+1}]], False]&, {Distortions}];
AddTimeSteps[d[[1,All,1]], d[[All,All,2]]\[Transpose]]
],
computeJac===All,
Module[{d},
(* Compute the distortion for each control channel *)
d = MapIndexed[#1[pulse[[All,{1, First@#2+1}]], All]&, {Distortions}];
{
AddTimeSteps[d[[1,1,All,1]], d[[All,1,All,2]]\[Transpose]],
AddTimeSteps[d[[1,2,All,1]], d[[All,2,All,2]]\[Transpose]],
AddTimeSteps[d[[1,3,All,1]], d[[All,3,All,2]]\[Transpose]]
}
]
]
],
With[{format=MatrixForm[List/@Format/@{Distortions}]},Format[HoldForm[format]]]
];
ComposeDistortions[Distortions__DistortionOperator]:=DistortionOperator[
Function[{pulse,computeJac},
Which[computeJac===True,
Fold[
With[{result=#2[First@#1,True]},{
First@result,
If[Last@#1===None,
Last@result,
TensorPairContract[Last@result,Last@#1,{{3,1},{4,2}}]
]
}]&,
{pulse,None},
{Distortions}
],
computeJac===False,
Fold[#2[#1,False]&,pulse,{Distortions}],
computeJac===All,
Fold[{#1[[1]],Sequence@@((#2[#1[[3]],All])[[{2,3}]])}&,{pulse,pulse,pulse},{Distortions}]
]
],
With[{format=Row[Riffle[Format/@{Distortions},"\[EmptyCircle]"]]},Format[HoldForm[format]]]
]
DistortionOperator /: Composition[distortions__DistortionOperator] := ComposeDistortions[distortions];
PerturbateDistortion[Distortion_DistortionOperator, h_:10^-8]:=Module[{jacobian},
jacobian[dts_,numControlKnobs_]:=jacobian[dts,numControlKnobs]=Transpose[
Table[
Distortion[AddTimeSteps[dts, SparseArray[{{n,k}->h},{Length@dts,numControlKnobs}]], False][[All,2;;-1]]/h,
{n,Length[dts]}, {k,numControlKnobs}
],
{3,4,1,2}
];
DistortionOperator[
Function[{pulse,computeJac},
If[computeJac===True,
{Distortion[pulse,False], jacobian[pulse[[All,1]],Last@Dimensions@pulse-1]},
Distortion[pulse,computeJac]
]
],
Format@HoldForm[PerturbateDistortion[Format[Distortion]]]
]
]
ScaleDistortion[distortion_DistortionOperator,timeScale_,amplitudeScale_?NumericQ]:=DistortionOperator[
Function[{pulse,computeJac},
Which[
computeJac===False,
Transpose[Prepend[ConstantArray[amplitudeScale,Length[#]-1],timeScale]*#]&[Transpose@distortion[pulse,False]],
computeJac===True,
Module[{outpulse,jac,scale},
{outpulse,jac}=distortion[pulse,True];
{
Transpose[Prepend[ConstantArray[amplitudeScale,Length[#]-1],timeScale]*#]&[Transpose@outpulse],
amplitudeScale*jac
}
],
computeJac===All,
{#1,#2,Transpose[Prepend[ConstantArray[amplitudeScale,Length[First@#3]-1],timeScale]*Transpose[#3]]}&@@distortion[pulse,All]
]
],
HoldForm@Format[ScaleDistortion[distortion,Row[{"t\[Rule]",timeScale,"t"}],Row[{"amps\[Rule]",amplitudeScale,"amps"}]]]
];
ScaleDistortion[distortion_DistortionOperator,timeScale_,amplitudeScale_List]:=DistortionOperator[
Function[{pulse,computeJac},
Which[
computeJac===False,
Transpose[Prepend[amplitudeScale,timeScale]*Transpose[distortion[pulse,False]]],
computeJac===True,
Module[{outpulse,jac,scale},
{outpulse,jac}=distortion[pulse,True];
{
Transpose[Prepend[amplitudeScale,timeScale]*Transpose[outpulse]],
Transpose[amplitudeScale*Transpose[jac,{2,1,3,4}],{2,1,3,4}]
}
],
computeJac===All,
{#1,#2,Transpose[Prepend[amplitudeScale,timeScale]*Transpose[#3]]}&@@@distortion[pulse,All]
]
],
With[{amps=Row[Flatten@Riffle[Table[{"amp",n,"\[Rule]",amplitudeScale[[n]],"amp",n},{n,Length@amplitudeScale}],", "]]},
HoldForm@Format[ScaleDistortion[distortion,Row[{"t\[Rule]",timeScale,"t"}],amps]]
]
]
(* ::Subsubsection::Closed:: *)
(*Identity Distortion*)
IdentityDistortion[]:=Module[{jac},
jac[{n_,k_}]:=(jac[{n,k}]=Transpose[Table[KroneckerDelta[kk,ll]*IdentityMatrix[n],{kk,k},{ll,k}], {2,4,1,3}]);
DistortionOperator[
Function[{pulse,computeJac},
Which[
computeJac===False,
pulse,
computeJac===True,
{pulse,jac[Dimensions@pulse-{0,1}]},
computeJac===All,
{pulse,pulse,pulse}
]
],
Format@HoldForm[IdentityDistortion[]]
]
]
(* ::Subsubsection::Closed:: *)
(*Variable Change Distortions*)
Options[VariableChangeDistortion]={
Simplify->True
};
VariableChangeDistortion[changeFn_, variableSymbol_Symbol, opts:OptionsPattern[]] := VariableChangeDistortion[{changeFn}, {variableSymbol}, opts]
VariableChangeDistortion[changeFn_List,{variableSymbols__Symbol},OptionsPattern[]] := Module[
{subJacobian},
(* Calculate the per-step Jacobian now, and we won't have to again *)
subJacobian = Outer[D, changeFn, {variableSymbols}];
If[OptionValue[Simplify],
subJacobian = Simplify@subJacobian;
];
With[{subjacVal=subJacobian},
DistortionOperator[
Function[{pulse,computeJac},
Which[computeJac===False,
AddTimeSteps[pulse[[All,1]],(changeFn/.Thread[{variableSymbols}->#])& /@ pulse[[All, 2;;-1]]],
computeJac===All,
{pulse,pulse,AddTimeSteps[pulse[[All,1]],(changeFn/.Thread[{variableSymbols}->#])& /@ pulse[[All, 2;;-1]]]},
computeJac===True,
{
AddTimeSteps[pulse[[All,1]],(changeFn/.Thread[{variableSymbols}->#])& /@ pulse[[All, 2;;-1]]],
Transpose[#,{1,3,2,4}]&@Table[
(* Whenever m!=n, we get no derivative, ie, different time steps do not affect each other. *)
If[m==n,
subjacVal /. Thread[{variableSymbols}->pulse[[n, 2;;-1]]],
ConstantArray[0,{Length@changeFn, Last@Dimensions@pulse-1}]
],
{m, Length@pulse}, {n, Length@pulse}
]
}
]
],
Format@HoldForm[VariableChangeDistortion[{variableSymbols}->changeFn]]
]
]
]
TimeScaleDistortion[multiplier_]:=DistortionOperator[
Function[{pulse,computeJac},
Which[computeJac===True,
{
AddTimeSteps[multiplier * pulse[[All,1]], pulse[[All,2;;-1]]],
(* The jacobian is the same as IdentityDistortion... *)
Transpose[ConstantArray[IdentityMatrix[Length@pulse],{Last@Dimensions@pulse-1, Last@Dimensions@pulse-1}], {2,4,1,3}]
},
computeJac===False,
AddTimeSteps[multiplier * pulse[[All,1]], pulse[[All,2;;-1]]],
computeJac===All,
{pulse,pulse,AddTimeSteps[multiplier * pulse[[All,1]], pulse[[All,2;;-1]]]}
]
],
Format@HoldForm[TimeScaleDistortion[multiplier]]
]
(* ::Subsubsection::Closed:: *)
(*Linear Distortions*)
ConvolutionDistortion[integral_,numInput_,numOutput_,dtOutput_]:=
Module[{discreteKernel},
discreteKernel = SparseArray@Table[
With[{int=integral[m,n]},If[MatrixQ[int],int,{{int}}]],
{m, numOutput}, {n, numInput}
];
(*Change from dimensions {M,N,L,K} to {M,L,N,K}*)
discreteKernel=Transpose[discreteKernel,{1,3,2,4}];
With[{jacval=discreteKernel},
DistortionOperator[
Function[{pulse,computeJac},
Module[{jac=jacval,outputPulse},
outputPulse=AddTimeSteps[dtOutput, Normal@TensorPairContract[jac,pulse[[All,2;;]],{{3,1},{4,2}}]];
Which[
computeJac===True,
{outputPulse,jac},
computeJac===False,
outputPulse,
computeJac===All,
{pulse,pulse,outputPulse}
]