Multi-parameter estimation along quantum trajectories with Sequential Monte Carlomethods

Jason F. Ralph,1, ∗ Simon Maskell,1, † and Kurt Jacobs2, 3, 4, ‡

1Department of Electrical Engineering and Electronics,University of Liverpool, Brownlow Hill, Liverpool, L69 3GJ, UK.

2U.S. Army Research Laboratory, Computational and Information Sciences Directorate, Adelphi, Maryland 20783, USA.3Department of Physics, University of Massachusetts at Boston, Boston, MA 02125, USA

4Hearne Institute for Theoretical Physics, Louisiana State University, Baton Rouge, LA 70803, USA(Dated: September 11, 2017)

This paper proposes an efficient method for the simultaneous estimation of the state of a quan-tum system and the classical parameters that govern its evolution. This hybrid approach benefitsfrom efficient numerical methods for the integration of stochastic master equations for the quantumsystem, and efficient parameter estimation methods from classical signal processing. The classicaltechniques use Sequential Monte Carlo (SMC) methods, which aim to optimize the selection ofpoints within the parameter space, conditioned by the measurement data obtained. We illustratethese methods using a specific example, an SMC sampler applied to a nonlinear system, the Duff-ing oscillator, where the evolution of the quantum state of the oscillator and three Hamiltonianparameters are estimated simultaneously.

PACS numbers: 05.45.Mt, 03.65.Ta, 05.45.PqKeywords: quantum state-estimation, continuous measurement, classical parameter estimation, sequentialMonte Carlo methods

I. INTRODUCTION

Stochastic master equations provide a model for the evo-lution of open quantum systems subject to continuousmeasurements [1–3]. The trajectories that the stochas-tic master equations generate represent the evolution ofthe state of an individual quantum system, conditionedon a particular measurement record. In theoretical stud-ies, the measurement record is a simulated sequence cor-responding to a particular realization of the evolution.However, recent experiments that implement continu-ous quantum measurements have demonstrated that theevolution of individual quantum systems can be recon-structed from experimental data [4–7]. The generationof such trajectories in real time during the measure-ment process will be an important step towards state-dependent feedback control of individual quantum sys-tems [1–3]. Feedback control has been demonstrated inquantum systems using the output measurement recordas an input signal to the control system in optical [8, 9],opto-mechanical [10–12], and mesoscopic superconduct-ing systems [13, 14]. In most of these examples, while thedirect use of the measurement record in the control sys-tem demonstrates the utility of quantum feedback con-trol, it is limited by the fact that the evolution of the un-derlying state of the system is not included in the genera-tion of the controls. State-dependent control is more flex-ible and can include quantities that are estimated from,

∗Electronic address: jfralph@liverpool.ac.uk†Electronic address: smaskell@liverpool.ac.uk‡Electronic address: kurt.jacobs@umb.edu

but are not directly measured in experiments.

Given a measurement record, a Stochastic MasterEquation (SME) provides an estimate of the quantumstate at each point in time, and – in the case of mixedstates – an indication of the uncertainty associated withthis state in terms of an estimate of its purity. The SMEis derived by taking a single quantum system and cou-pling it weakly to environmental degrees of freedom thatmediate a continuous measurement process. The con-tinuous measurement of the coupled system is realizedby continuously measuring the state of the environment,which can be modeled as a sequence of projective mea-surements on successive environmental degrees of free-dom. The simplest example of this process is the mea-surement of the electromagnetic radiation emitted by thesystem, in which case the environment is the electromag-netic field [2, 3]. In the most common form of the SME,a Markovian condition is applied, meaning that the en-vironment carries information away from the system butdoes not by itself feed this information back to affect thesystem at a later time. The resulting evolution of thesystem is continuous and stochastic, with the stochasticterm arising from the effect of the sequence of measure-ments on the combined system.

In simulations, an SME is used to analyze the proper-ties of an open system under the action of a continuous se-quence of measurement operators, using a realization forthe noise process and calculating the evolution of the sys-tem conditioned on this realization. When interpretingexperiments, the SME is used to reconstruct the estimateof the quantum state as a function of time from the givenmeasurement record provided by the experiment. In thisregard, the SME is very similar to classical state estima-tion techniques (often referred to as ‘target tracking’ or

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‘object tracking’ [32–34]), which are used to interpret se-quences of classical noisy sensor measurements to form acoherent picture of the world. These classical techniqueshave been developed to interpret sensor data (radar orsonar signals, and sequences of images) where objectsare moving against noisy backgrounds. The motion ofthe objects may be unpredictable or uncooperative, theiridentity may not be known from the measurements, theymay be occluded for periods of time, and individual ob-jects may not be fully resolved by the sensor. Classicalstate estimation techniques provide methods to solve allof these problems and ambiguities.

With a continuous quantum measurement, the mea-surement record contains information about the evolu-tion of the particular quantum state (a quantum trajec-tory) but the properties of this trajectory also containinformation about the classical parameters that governthe dynamics of the system: the classical parameters inthe Hamiltonian and the strength of the coupling to theenvironment. In this paper, we demonstrate how thestochastic master equation can be augmented with tech-niques drawn from classical state estimation, SequentialMonte Carlo (SMC) methods, to estimate several Hamil-tonian parameters efficiently alongside the quantum tra-jectories.

We begin our presentation by first reviewing other ap-proaches to Hamiltonian parameter estimation, and thedevelopment of a set of Hybrid SMEs for the quantumevolution and the Kushner-Stratonovich equation for theclassical parameters [15] in sections II and III, respec-tively. In section IV, we then discuss how efficient classi-cal parameter estimation techniques [16] can be appliedto the solution of the classical aspects of the Hybrid SME.Section V introduces an example system, the Duffing os-cillator, which contains a number of relevant experimen-tal parameters, and Section VI presents results for thesimultaneous estimation of the quantum trajectories forthe oscillator and up to three Hamiltonian parameters.Section VII discusses how such methods may be useful inpractical systems and draws conclusions from the resultspresented.

II. HAMILTONIAN PARAMETERESTIMATION

The problem of estimating the dynamical parametersof a quantum system has been studied previously by anumber of authors, including those who have adopted acontinuous measurement approach. This estimation pro-cess is often referred to as Hamiltonian parameter esti-mation. This is because the basic description of quantumdynamics is encapsulated by the Hamiltonian operator,which determines the equations of motion, and values ofthe classical parameters in the Hamiltonian determinethe specifics of the evolution.

A standard method for determining the dynamics of aquantum system is to prepare it many times in a range

of different initial states, allow it to evolve, and thenmeasure it before re-preparation. The results of themeasurements can then be combined in a tomographic-like process to obtain the equation of motion for linearSchrodinger evolution [17]. An alternative approach, andthe one in which we are interested here, is to prepare thesystem only once and to continually monitor its subse-quent evolution to build a picture of its dynamics. Afull description of the problem involves starting with aprior probability density for the parameters one wishesto determine and then using Bayes’ theorem to contin-ually update this probability density from the stream ofmeasurement results as they are obtained. A number ofauthors have considered this problem [15, 18–28]. Thisis of particular interest when the parameters of a systemchange slowly with time, and one wishes to be able totrack the variations in the parameters. It is also rele-vant to the problem of using quantum systems as probesto measure time-varying classical fields (such as gravitywaves [29] and magnetic fields [30]), as these fields appearas parameters in the Hamiltonian.

As discussed in the introduction, a dynamical equationreferred to as the stochastic master equation (SME) canbe used to track the evolution of a quantum system fromthe results of a continuous measurement so long as thedynamical parameters of the system are known. If theyare not known then the full estimation problem involvesboth the SME and a Kushner-Stratonovich equation thatevolves the probability density for the parameters of thesystem. The combined set of dynamical equations hasbeen referred to as a Hybrid SME [15]. The first paperson the subject of Hamiltonian parameter estimation viacontinuous measurements were concerned mainly withderiving the Hybrid SME and applying it to the estima-tion of a single parameter [18, 19]. Subsequently, Tsangand collaborators considered the more general problem ofsmoothing in which a time-varying parameter (a signalor wave-form) is estimated from all the measurement re-sults obtained, and determined the ultimate limits to thisprocedure [21–24, 26]. An alternative and interesting ap-proach to the problem was proposed recently by Bassa etal. [27]. While most of the related work on parameterestimation employs continuous measurements, this ap-proach considered a sequence of instantaneous measure-ments, and employed a discrete version of the HybridSME where several classical parameter values were en-coded in an expanded quantum state.

A major problem with the Hybrid SME is that it ishighly demanding from a computational point of view;in order to evolve the Kushner-Stratonovich equation forthe probability density describing the observer’s knowl-edge of the parameters, the SME must be evolved forevery value of the parameters for which this densityis appreciable. The grid of points for which the SMEmust be evolved becomes large very quickly as the num-ber of parameters increases. Two previous papers haveput forward methods aimed at addressing this difficulty.Ralph et al. [15] and Cortez et al. [28] considered the es-

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timation of a single frequency parameter, and presentedmethods to bypass the Kushner-Stratonovich equation.These papers estimate the natural oscillation frequencyof a qubit directly from the measurement record. Thisapproach has many benefits in terms of computationalefficiency, but it has the disadvantage of not providing asimultaneous estimate of the quantum state of the sys-tem – which would be provided by the full solution ofthe Hybrid SME. Here, we will explore the use of a po-tentially more powerful technique in which the proba-bility density is replaced with a finite set of samples ofthe parameters that are evolved instead. The examplesgiven below typically use 50-100 quantum states and theequivalent of thousands to millions of classical parame-ter values. The purpose is again to reduce the numberof copies of the quantum state that must be evolved inparallel using the SME, but we will apply this method tothe challenging problem of estimating multiple parame-ters simultaneously.

III. HYBRID STOCHASTIC MASTEREQUATIONS

The simultaneous estimation of the quantum state of asystem and the classical Hamiltonian parameters thatgovern its evolution was considered in Ref. [15], where anapproach was presented based on a set of parallel SMEs,each using a different set of parameter values containedin a vector λ, which have an associated probability. Thefinal mixed state is then constructed by averaging overthe probabilities for the classical parameters. The prob-abilities associated with the different parameter vectorsevolve via a Kushner-Stratonovich equation and are con-ditioned on the continuous measurement record [15].

For a quantum system subject to a continuous mea-surement, with a known set of Hamiltonian parameters,the evolution of the quantum state, ρc(t), conditioned onthe measurement record, y(t), is given by the stochas-tic master equation [1–3]. In general, the interactionwith the environment can be represented by a set of sys-tem operators which are coupled to environmental de-grees of freedom, some of which are not measured Vj(j = 1 . . .m) (‘unprobed’ operators), and some of whichare measured and generate the continuous weak measure-ment Lr (r = 1 . . .m′). In an ideal case, the measurementrecord is 100% efficient, with all of the available infor-mation being reflected in the measurement record. Un-fortunately, real measurements are rarely ideal and thecontinuous measurement record is often corrupted withextraneous (classical) noise sources. These extraneousdegrees of freedom can be characterized by an efficiencyparameter for the measurement operators, Lr has an effi-ciency ηr. Specifically, ηr is the fraction of the total noisepower due to the quantum measurement as opposed topower contained in the other extraneous noise sources.

For unprobed operators Vr and measurement operators

Lr, the general form for the SME is given by,

dρc = −i[H, ρc

]dt

+

m∑j=1

{VjρcV

†j −

1

2

(V †j Vjρc + ρcV

†j Vj

)}dt

+

m′∑r=1

{LrρcL

†r −

1

2

(L†rLrρc + ρcL

†rLr

)}dt

+

m′∑r=1

√ηr

(Lrρc + ρcL

†r − Tr(Lrρc + ρcL

†r))dWr

(1)

where H is the Hamiltonian of the system, dt is an in-finitesimal time increment, and the measurement recordfor each of the measurement operators Lr during a timestep t → t + dt is given by, y(t + dt) − y(t) = dyr(t) =√ηjTr(Lrρc + ρcL

†r)dt + dWr. We will take dWr to

be a real Wiener increment such that dWr = 0 anddWrdWr′ = δrr′dt for simplicity, but this is not strictlynecessary. More general forms of complex incrementsmay also be used [35].

Where the evolution of a quantum system is governedby a set of Hamiltonian parameters that are not knownexactly, we can describe the parameters in terms of aclassical probability density, P (λ), where λ = (λ1, λ2, ...).The system is then described by a set of SMEs, one foreach set of possible parameter values,

dρc,λ = −i[H(λ), ρc,λ

]dt

+

m∑j=1

{Vjρc,λV

†j −

1

2

(V †j Vjρc,λ + ρc,λV

†j Vj

)}dt

+

m′∑r=1

{Lrρc,λL

†r −

1

2

(L†rLrρc,λ + ρc,λL

†rLr

)}dt

+

m′∑r=1

√ηr

(Lrρc,λ + ρc,λL

†r

−Tr(Lrρc,λ + ρc,λL†r)dWr

)(2)

The evolution of the probability density P (λ) is gov-erned by a Kushner-Stratonovich stochastic differentialequation, derived in [15] for a single efficient measure-ment operator and in the absence of additional unprobedenvironmental operators. Any unprobed environmentaloperators affect the evolution of the individual SMEs butthey do not play a role in the evolution of P (λ). How-ever, the equation given in [15] generalizes naturally toinclude measurement inefficiencies and is given by,

dPr(λ) =√ηr(Tr(Lrρc,λ + ρc,λL

†r)− Tr(Lrρc + ρcL

†r))

×(dyr(t)−√ηrTr(Lrρc + ρcL

†r)dt)P (λ) (3)

where dPr(λ) is the update to the probability density dueto a measurement increment dyr(t) corresponding to the

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measured operator Lr, such that

P (dyr(t)|λ) =e(−(dyr(t)−

√ηTr(Lrρc,λ+ρc,λL

†r))

2/dt)

√2πdt

(4)

and the full conditional density matrix ρc is given by,

ρc =

∫λ

P (λ)ρc,λdλ (5)

IV. SEQUENTIAL MONTE CARLO METHODS

Sequential Monte Carlo methods originate in the field ofmulti-target tracking [31]), but have been adopted andgeneralized to form a set of very efficient methods for pa-rameter and state estimation in classical signal process-ing and nonlinear filtering. SMC methods are sometimesreferred to as particle filters, but particle filters are a spe-cial case of the general approach. An SMC method relieson the approximation of a continuous probability distri-bution by a finite set of points (or particles) which sam-ple the parameter space. The importance of each sam-ple point changes in response to (is conditioned by) themeasurements associated with the parameters being esti-mated, and the sample points can be periodically resam-pled to concentrate sampling towards regions of higherrelative probability.

In particle filters, the sample points are allowed toevolve according to some dynamical process, generatinga time dependent history or a track within the param-eter space. In the example presented in this paper, theparameters are selected to be constant and another SMCmethod is more suitable. We adopt an approach usedrecently for parameter estimation in classical differen-tial equations [16], which is an example of a Sequen-tial Monte Carlo sampler. This approach is particularlywell-suited to the estimation of fixed parameters; how-ever, the SMC sampler used here still embodies all of thekey features of a general SMC method: sampling, con-ditioning/updating, and resampling. A number of veryapproachable tutorials and introductions to particle fil-ters and general SMC methods have been published. Forexample, a comprehensive guide to SMC methods andtheir applications is available in [36], a mathematical in-troduction is given in [37], and a widely cited tutorial toparticle filters and SMC methods is contained in [38].

Formally, an SMC method approximates a (classical)expectation for a function h(x) over a probability distri-bution p(x) defined on some parameter space Λ, x ∈ Λ,given by

h =

∫h(x)p(x)dx

using a finite sum of a set of points x(i) (i = 1 . . . N)drawn from p(x), which is known as the target distribu-tion. The expectation value for an arbitrary function can

be approximated by,

h ' 1

N

N∑i=1

h(x(i))

The larger the number of sample points, the better theapproximation – in fact, under reasonable assumptions,the variance of the error in h can be shown to scale as1/N in any number of dimensions [39]. The problem isthat, in most practical cases, the probability distributionis unknown. It needs to be estimated from a sequenceof measurements. To do this, another distribution, theproposal distribution q(x), is introduced such that [37],

h =

∫h(x)q(x)

p(x)

q(x)dx =

∫h(x)q(x)w(x)dx

'N∑i=1

w(i)∑Nj=1 w

(j)h(x(i)) (6)

where w(x) = p(x)/q(x) and the w(i)’s are (unnormal-ized) weights associated with each sample point. In ourcase, each sample point is associated with a parametervalue or a vector of values for each of the parameters be-

ing sought, w(i) ↔ λ(i), where λ(i) ∈ Λ. Initially, thesample points are randomly selected from a prior dis-tribution, which covers the entire range of possible pa-rameter values, and are given a uniform weight. As newmeasurements are added, the accuracy of the estimatedquantity h is improved by updating the weights associ-ated with the particles to reflect the new information thatthe measurement contains. Some weights are increasedand some weights are reduced when the measurementsupports or contradicts the corresponding sample point,respectively.

The values w(i) are referred to as ‘weights’ rather thanprobabilities because, although they are related to prob-abilities, they are not necessarily normalized after eachtime step and do not necessarily sum to one. In prac-tice, it is convenient to normalize the weights after eachtime step. Here, we denote the normalized weights byw(i) [16]. When sample points have very low weight,and hence very low probability, they can be removed andreplaced with alternative particles, but this resamplingprocess must be done carefully so as to ensure that thestatistical quantities remain unbiased and will convergeefficiently to the desired values.

The proposal distribution should be simple to calculateand different choices of q(x) are used in different variantsof the SMC approach [36–38]. The secret to working withSMC methods is to pick a suitable proposal distributionto solve the problem in a robust manner using limitedcomputational resources. In particular, a good choice ofproposal distribution allows classical parameters to beestimated significantly more efficiently than when usingan enumerative or grid based method [38], as was usedfor a one parameter Hybrid SME problem in [15, 27].

For our Hybrid SME problem, we start by selectingan initial set of sample points in the parameter space

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using a prior distribution and initialize an SME (2) foreach of the sample points. For the examples shown be-low, the quantum state of the system is initialized to bea thermal mixed state, and the prior distribution is cho-sen to be uniform over some finite range within whichthe true parameter values are known to lie. An accu-rate initial prior distribution can significantly reduce thenumber of particles required by the SMC sampler, butin many situations the prior is not well defined. Oncethe points have been selected, the weights are initialized

with w(i)0 = w

(i)0 = 1/N .

For each time step, the individual SMEs are integratedusing the increment (2) found using the parameter value

λ(i) associated with the particle. A corresponding mea-surement probability is found from (4) and used to up-

date the (unnormalized) particle weight w(i)k−1,r → w

(i)k,r

using [16]

w(i)k,r = p(dyr(tk)|λ(i))w(i)

k−1,r (7)

for the k’th measurement from measurement operator Lrat time tk. All particles are updated after the measure-ment increment and then weights are normalized.

Resampling to generate new particles only occurs whenthe distribution of weights amongst the particles is suchthat the effective sample size (or the effective number ofparticles) Neff = 1/(

∑i(w

(i))2) falls below some thresh-old value – indicating that the weight is being concen-trated in a relatively small number of particles and a sig-nificant number of particles have low weight and do notcontribute to the estimates; a problem known in the SMCliterature as sample impoverishment or weight degeneracy[37]. It is known that the variance of the weight distri-butions across different realizations of the SMC sampleris guaranteed to grow with each time step [38]. However,since a given realization (i.e. one run of the algorithm)does not have access to the ensemble of all possible real-izations, it is convenient to monitor something that canbe computed from a single realization. The effective sam-ple size is well established as such a quantity [40] and itcan be considered to be a noisy measurement of the (in-verse of the) variance. Between resampling events, thevariance of the weight distributions will (on average) in-crease and the effective sample size will decrease. Whilethe precise threshold value used is a somewhat arbitrarychoice for the algorithm designer, it is common (acrossthe vast range of applications of SMC samplers and par-ticle filters) to consider threshold values between N/10and N/2. In the cases shown below the threshold valuefor Neff was set to be N/2 [16].

When the particles are resampled, the new candi-date values λ are sampled from the distribution formedfrom the current particle weights. The particles withthe highest weights are more likely to be selected, al-though the particles with relatively low weights stillhave a chance of being selected. The new particle pa-rameter values are then selected using the distribution

q(λ|λ(i)) = N (λ;λ(i),Σ), where N (x;µx, S) is a normal

distribution with mean µx and covariance S, and Σ isrelated to the covariance matrix associated with the cur-rent particle weights, Σk. The role of q(λ|λ(i)) is to select

new points, λ(i)

, around the current particles with largeweights, but not at exactly the same point. In this pa-per, we use a defensive strategy [16, 41], where 90% ofresampled points use a covariance which is 10% of thecurrent covariance, Σ = 0.1Σk, and 10% of the resam-pled points use the full covariance matrix Σ = Σk. Thisallows for small perturbations in parameter space aroundthe high weight sample points, including the correlationsbetween different parameters seen in the covariance ma-trix, and a small number of large excursions, to exploremore of the parameter space than is currently being cov-ered by the sample points with large weight. There aretwo specific design considerations relevant to the choiceof the distribution, q. The first is to ensure that havingmore samples will give rise to more accurate estimatesof quantities of interest, which is manifest empirically asrobustness. Put simply, this demands that samples areproposed in a way that explores possible but potentiallylow probability states. The second is to ensure that theSMC method is computationally efficient, i.e. that it getsas accurate an estimate as is possible with a given num-ber of samples. This demands that samples are placedin high probability areas. A defensive proposal is an ad-vanced, but relatively standard technique, used in parti-cle filters and SMC samplers, that combines robustnesswith efficiency by having two elements to the proposal,one that is designed to ensure the sampler is robust andthe other that is designed to ensure that it is efficient.

When the new sample points have been selected, they

are initially assigned the weight w(i)0 = 1/N and then

the unnormalized weight for the new candidate pointsis calculated reusing the entire record of measurementincrements,

w(i)k =

∏r,k p(dyr(tk)|λ

(i))∏

r,k p(dyr(tk)|λ(i))w

(i)0 (8)

Once all of the new weights have been recalculated theycan be renormalized, and the integration of the HybridSME can continue as normal. Where the new sampleweights are still degenerate and the effective number ofparticles is still below the threshold value, the resampling(and recalculation) needs to be performed again.

To summarize, the SMC method used here can be de-scribed in the following five steps:

(a) Initialize the individual density matrices ρc,λ(i) usingthermal mixed states, and select a set of classical pa-

rameter values (sample points or particles) λ(i) usinga uniform distribution covering the full range of pos-sible parameter values, and assign uniform weights to

each of these w(i)0 = 1/N .

(b) Evolve the quantum state using the individual SMEs(2), using the corresponding classical parameters and

6

FIG. 1: (Color online) Schematic process, showing the mainsteps in the SMC Sampler for a one parameter example.

updating their weights using (7). Continue this evo-lution until Neff drops below the threshold value.

(c) If Neff is below the threshold value, the classicalparameter values are resampled using a cumulativeprobability distribution calculated from the particleweights. This resampling creates a new set of par-ticles/sample points, where the classical parametersare selected around the ‘parent’ values. The defen-sive strategy introduces small perturbations aroundthe parent values and the occasional large perturba-tion to explore a wider parameter space – the newweights associated with each of the new parame-ter values/sample points are uniformly distributed atthis point.

(d) Once the new values have been selected, the com-plete evolution of each quantum state is recalculatedusing new initial thermal states and the individualSMEs (using the same measurement record), and theuniform weights from step (c) are recalculated using(8).

(e) Return to step (b) with evolution of the quantumstate and weights determined by the individual SMEs

and the weight update (7), until Neff drops belowthe threshold value again, at which point the resam-pling step (c) and the re-weighting step (d) are againrequired.

A schematic example of the estimation process for a onedimensional parameter example is shown in Figure 1. Inthis example, it is possible to see that the initial uniformweighting of the particles evolves so that the relativelylarge number of particles below a parameter value of 1.0carry very little weight, and the distribution of particlesimmediately after resampling is concentrated more to-wards the values above 1.0. The re-weighted parametervalues shown in (d) represent a better approximation tothe underlying probability distribution than those shownin (b), which contains significant gaps towards the peakof the distribution. For a more detailed description of theimplementation, a full description of the SMC method isgiven as pseudo-code in [16].

The recalculation over the entire measurement historyis an unfortunate, but necessary, computational cost inthe SMC sampler. Recalculating the weights for the en-tire history of measurement increments will often take asignificant amount of time. However, the need to regu-larly resample the entire set of particles reduces as thedistribution of the particles improves to reflect the infor-mation contained in the measurements [16]. This meansthat the computational load introduced is biased towardsthe start of the calculation of a quantum trajectory. Inaddition, for resampled points very close to the parentparticles, some approximations are possible based on thefact that the ratio between the products in (8) is veryclose to one. It is not possible to remove the recalcula-tion entirely however without constraining the resampledparameter values and therefore not exploring the full pa-rameter space.

V. EXAMPLE SYSTEM – DUFFINGOSCILLATOR

The properties of the quantum trajectories generated bythe Duffing oscillator have been studied extensively interms of the appearance of chaotic behavior from quan-tum systems in the classical limit [42–52], but it is alsoa model used for a number of other practical systemswhere quantum effects in classical nonlinear systems areof interest. For example, it has been used to describe themotion of a levitated particle in an electromagnetic trap[53, 54], and is the basis for the analysis of the propertiesof vibrating beam accelerometers [55–57]. The Hamilto-nian for the Duffing oscillator can be written in the gen-eral form, using dimensionless position and momentumoperators q and p,

H(λ) =1

2p2+

1

2ω2q2+

1

4µq4+g cos(t)q+

Γ

2(qp+ pq) (9)

where the vector λ = (ω, µ, g) contains the three Hamil-tonian parameters of interest: the natural (linear) oscil-

7

lation frequency ω, the nonlinear coefficient µ, and thestrength of the external driving term g. The measure-ment is applied via a Linblad operator L =

√2Γa, and

a is the harmonic oscillator lowering operator so thatq = (a† + a)/

√2 and p = i(a† − a)/

√2 with [a, a†] = 1

and ~ = 1. We fix the measurement strength so thatΓ = 0.125 for all of the results presented here. Thefinal term in the Hamiltonian is included because, incombination with the dissipative measurement process,it generates linear damping in momentum. This is a use-ful numerical addition because it keeps the phase spacecontained, thereby restricting the numbers of states re-quired in the simulation, without affecting the underlyingphysics.

The numerical integration of the individual SMEs usesa method developed by Rouchon and colleagues [58, 59]specifically for stochastic master equations. This methodhas been demonstrated to provide significant benefitsin terms of accuracy versus computational resourceswhen compared to standard methods, such as Milstein’smethod [60], for both systems involving small numbersof basis states [59] and large numbers of basis states [61].We also employ a moving basis method used by Schack,Brun and Percival [42, 43] to shift basis states to be cen-tred on the current expectation value of the state. Al-though not strictly necessary [42, 43], we shift the basisafter each time step. This comes at a computational costbut it also ensures that the number of basis states em-ployed is minimized. Once the evolution of the individualSMEs has been calculated, using the appropriate set ofparameters, the combined density operator is calculatedby averaging over all of the individual states, weightedappropriately by the particle weights.

The increment to the state ρ(n)c,λ for the time step from

tn = n∆t to tn+1 = (n+ 1)∆t is calculated using

ρ(n+1)c,λ =

Mn,λρ(n)c,λM

†n,λ + (1− η)Lρ

(n)c,λL

†∆t

Tr[Mn,λρ

(n)c,λM

†n,λ + (1− η)Lρ

(n)c,λL

†∆t] (10)

where ∆ρ(n)c,λ = ρ

(n+1)c,λ − ρ(n)c,λ and Mn,λ is given by

Mn,λ = I −(iH +

1

2L†L

)∆t+

η

2L2(∆W (n)2 −∆t)

+√ηL(√

ηTr[Lρ(n)c,λ + ρ

(n)c,λL

†]∆t+ ∆W (n))

where the ∆W ’s are independent Gaussian variables withzero mean and a variance equal to ∆t. Once the incre-ment has been calcluated, center of the basis is movedto the new location of the state in phase space, as givenby the expectation values of the phase space operators,

(q(n+1),λ, p(n+1),λ) = (Tr[qρ(n+1)c,λ ],Tr[pρ

(n+1)c,λ ]), using the

displacement operator [42, 43],

D(p(n+1),λ, q(n+1),λ) = exp(i(p(n+1),λq − q(n+1),λp)

)(11)

and the conditioned state in the shifted basis is given by

ρ(n+1)c,λ → D(p(n+1),λ, q(n+1),λ)ρ

(n+1)c,λ D(p(n+1),λ, q(n+1),λ)†

(12)

VI. RESULTS

The Duffing Hamiltonian (9) has four classical param-eters but we will fix the measurement strength so thatΓ = 0.125 and we will concentrate on the estimationof the other three parameters: the linear oscillator fre-quency ω, the coefficient of the nonlinear term µ, andthe magnitude of the drive term g. The estimated val-ues for these three parameters are denoted by ω, µ, andg respectively. For all of the examples shown below, theactual values for parameter values were set to be ω = 1.2,µ = 0.15, and g = 3.0. The numerical integration of theSMEs was performed using time steps ∆t = 2π/500 sothat there were 500 steps per period of the drive term.The individual SMEs for each particle/sample point useda moving basis with 15 harmonic oscillator states, and thecomposite state was calculated by combining the densitymatrices from the individual SMEs using (5), using amoving basis with 60 harmonic oscillator states.

Figure 2 shows two examples for the estimation of thelinear oscillator frequency ω. The examples correspondto the same stochastic record (i.e. the same realization)but with different measurement efficiencies. The bluelines correspond to the case where the measurement is100% efficient (with η = 1). This shows a rapid con-vergence to the actual value, ω = 1.2, within about 50-100 periods/cycles of the drive term. The 3 sigma er-rors predicted for the estimate are also shown, togetherwith the resampling events as blue circles. The conver-gence is fairly rapid and the estimate is relatively stableonce converged. The red line on the same figure showsan example where the measurement is inefficient, corre-sponding to a measurement efficiency of 40% or η = 0.4(chosen to match the estimated efficiency reported in [5]).In this case, the convergence is much slower, indicatingthat the measurement record contains less informationupon which a parameter estimate can be constructed.In this case, the estimated parameter value only stabi-lizes after around 150-200 cycles of the drive term, andthe larger estimated errors indicate this increased uncer-tainty. In both cases shown, there are slight variationsin the estimated values (seen around 200-250 cycles) butthese are relatively small and are well within the esti-mated errors. In addition, where the estimation processtakes longer, the number of resampling events (red cir-cles) tends to increase and they often occur later in theprocess than the corresponding resampling events for effi-cient measurements, leading to increased computationaldemands to recalculate the weights after resampling. Inaddition to the estimates, Figure 2 also shows the pu-rity of the full estimated quantum state for both casesas an inset. For efficient measurements, the conditioned

8

quantum state purifies very rapidly (1-2 periods of thedrive term) and remains pure throughout the estimationprocess. For inefficient measurements, the conditionedquantum state purifies somewhat but then the purityfluctuates between 0.8 and 0.9. The state remains mixedbecause information about the quantum state is beingcorrupted by extraneous noise. This is a characteristicof inefficient measurements in quantum systems, and itis not affected by, and does not itself affect, the classicalparameter estimation process.

FIG. 2: (Color online) Examples of estimated values for thelinear parameter (ω) using SMC sampler with efficient mea-surements (η = 1.0, solid blue line) and inefficient measure-ments (η = 0.4, solid red line) with an actual linear parametervalue ω = 1.2 (solid black line) and 101 sample points (otherparameters are given in the text). Three standard deviationerrors are indicated in each case with dotted lines, and the re-sampling points are indicated by circles along the solid blackline. The inset figure shows the purity values for the estimatedstate in each case.

Figure 3 shows the evolution of the effective numberof particles Neff as a function of time for the examplesshown in Figure 2. The resampling events are marked onFigure 2 as large dots, but they are also seen in Figure 3as large jumps in Neff after the resampling. The data inthis figure is useful when optimizing the resampling pa-rameters. It provides information regarding the averagenumber of particles being used. An efficient SMC pro-cess would expect to have rapid fluctuations in Neff inthe initial phases of the estimation process, with frequentresampling, which would become more gradual drops inNeff as the estimates improve. As time increases, andmore measurements are added, the resampling events be-come less frequent, as is shown in Figures 2 and 3.

The estimation of the frequency of the linear oscilla-tor term is relatively straightforward, and this is alsofound to be the case for the magnitude of the drive termg. Estimating the coefficient of the nonlinear term µ ismore challenging however. When the external drive isvery small, the Duffing oscillator will appear to be ap-

FIG. 3: (Color online) Examples of the effective number ofparticles Neff for the estimates shown in Fig.2 for efficientmeasurements (η = 1.0, solid blue line) and inefficient mea-surements (η = 0.4, solid red line).

proximately linear and estimating the degree of nonlin-earity is problematic. As the amplitude of the drive isincreased, the system will explore more of the nonlin-ear potential and µ will become easier to estimate. Thisfact is reflected in the results obtained. For the param-eter values selected, the drive term is sufficiently strongto explore the nonlinearity of the potential, but not suf-ficiently strong so as to require very large numbers ofbasis states or to make the estimation process easy com-pared to the other two parameters. An example of theestimation of the nonlinear coefficient is shown in Figure4, where the convergence to a stable value takes muchlonger than either example shown in Figure 2, requiringover 500 periods of the drive term to stabilize the esti-mated value (note the different x-axis compared to Figure2).

Each of the examples shown in Figures 2 and 4 showthe estimation of one parameter, the other parametersare assumed to be known. The estimation of one pa-rameter is relatively straightforward and a value can befound using a grid-based method (as was the case in [15]and [27]). The number of particles required for the esti-mation of ω and µ is around 101 sample points in each ofthe SMC examples shown above. The number of resam-pling events is around 4-6 in the cases shown in Figures 2and 4, and the maximum number of quantum trajectoriesthat would need to be calculated is approximately equiv-alent to 200-300 trajectories on a fixed grid Hybrid SME.The expected errors for a fixed grid approach are relatedto the grid spacing, which is related to the initial rangeover which these grid points are initially distributed. Forthe cases considered here, with an initial distribution ofpoints for a parameter value λ between 0.5λ ≤ λ(i) ≤1.5λ. Assuming that the actual values of λ are uniformlydistributed across each interval, the expected error fora fixed grid approach with Ngrid points would be lim-

ited by σλ,grid > (λ/Ngrid)/√

12 ' 0.1%− 0.15%λ. This

9

FIG. 4: (Color online) An example of estimated values forthe nonlinear parameter (µ) using SMC sampler with effi-cient measurements (η = 1.0, solid blue line) with an actualnonlinear parameter value µ = 0.15 (solid black line) and 101sample points (other parameters are given in the text). Threestandard deviation errors are indicated in each case with dot-ted lines, and the resampling points are indicated by circlesalong the solid black line.

FIG. 5: (Color online) An example of values for all threeparameters (ω, µ, and g) estimated simultaneously using SMCsampler with efficient measurements (η = 1.0, solid blue line)and 1001 sample points (other parameters are given in thetext). Three standard deviation errors are indicated in eachcase with dotted lines, and the resampling points are indicatedby circles along the solid black line.

value is achievable only in the long time limit and theactual error is likely to be significantly larger than this.In the examples given above, the SMC sampler producesparameter estimates with errors approaching this limitwithin a few hundred cycles. There is therefore a smallbut potentially significant benefit in using the SMC sam-pler method for one parameter estimation.

Moving from single to multiple parameter estimationpresents a serious problem for grid-based methods. Thenumber of points required scales exponentially in thenumber of dimensions to achieve the same accuracy. Theerror from a grid-based approach results from approxi-mating an integral of functions in D dimensions, wherethe error is O((Ngrid)

(−1/D)). The error for an SMC sam-pler comes from approximating the integral directly (us-ing Monte-Carlo integration) and therefore is O(1/Ngrid)whatever value D takes [39]. (See reference [40] forproofs for the convergence of SMC and particle filterbased methods). So, for D = 1, the two approachesoffer similar scaling of error with Ngrid, in higher dimen-sions, an SMC sampler will asymptotically outperform agrid-based method as Ngrid tends to infinity. Of course,differences in constants of proportionality mean that acomputational benefit from using the SMC sampler in asmall number of dimensions (number of parameters) isnot guaranteed. Estimating all three parameters in ourexample, at a level of accuracy equivalent to the one pa-rameter examples above, would require around ten mil-lion grid points, (300)3 = 9 × 106. With SMC methods,this number is dramatically reduced.

Figure 5 shows an example of the simultaneous esti-mation of all three parameters using 1001 sample points.The values for ω and g still converge rapidly whilst µtakes longer to establish a stable estimate. When com-paring this with a grid based method, we note that thenumber of trajectories is larger than for the single pa-rameter case and the number of resampling events is alsoincreased, approximately 20 in the case shown in Figure5. This is equivalent to a run-time for approximately10,000 trajectories on a fixed grid. Using the same as-sumptions as before, this would give errors limited byσλ,grid > (λ/ 3

√Ngrid)/

√12 ' 1.5%λ. The errors found

using the SMC sampler described above are nearly anorder of magnitude smaller than this limit for one of thethree parameters (ω) and comparable for the remainingtwo parameters (µ and g). There is an additional bene-fit, in that the sample points not only provide estimatesof the parameter values, they also provide informationregarding the correlations between the different parame-ters. For the example shown in Figure 5, the mean vectorand the estimated covariance matrix (S) are given by ω

µg

=

1.19810.15573.0874

S =

(4.0308 × 10−3 −1.6449 × 10−6 9.2270 × 10−6

−1.6449 × 10−6 3.9795 × 10−4 −2.8254 × 10−6

9.2270 × 10−6 −2.8254 × 10−6 1.1586 × 10−2

)

10

Note also that in Figure 5, the standard deviation ofthe linear parameter (ω) is larger than in Figure 2 andthe convergence is slower than for the single parametercase. This is partly due to the larger uncertainty gener-ally in the three unknown parameters, and in part dueto the slower convergence of the nonlinear parameter (µ).The coupling between the parameters, shown by the non-negligible correlations shown in the covariance matrix,means that uncertainty in the nonlinear parameter in-creases the standard deviation of the other two parame-ters.

The use of an SMC sampler to estimate the Hamil-tonian parameter values directly from the quantum tra-jectories is more efficient than an equivalent grid-basedmethod but it still presents a computational challenge.Solving a single SME can be simplified using a stochasticintegration method designed specifically for SMEs, likeRouchon’s method [58, 59], and using efficient numericaltools, like moving basis states [42, 43]. However, solv-ing many simultaneous SMEs to determine the evolutionof the particle weights still requires significant compu-tational resources. The number of combinations of pa-rameter values explored using the SMC sampler is sig-nificantly less than that required by a conventional grid-based method, but each sample point explored requiresthe full trajectory to calculated, or recalculated after re-sampling. The number of SMEs required to be calculatedcan be said to be relatively small but it is still not a triv-ial exercise. In their favor, SMC methods are amenableto parallelization [16], since the evolution of SME andthe recalculation of each trajectory after resampling arelargely independent processes and can be distributed sim-ply across a number of processors. However, at present,it is more likely that this type of technique is more likelyto be used for post-processing experimental data ratherthan as part of an on-line closed-loop control system.

VII. CONCLUSIONS

Continuous quantum measurements, and their asso-

ciated stochastic master equations (SMEs), provide ameans to monitor the dynamical evolution of a quan-tum system and to provide an estimate of the underly-ing quantum state. In addition, the quantum trajecto-ries resulting from the integration of stochastic masterequations contain useful information about the param-eters that govern the evolution of the system. Hybridstochastic master equations provide a means to extractthe information regarding these classical parameters. Hy-brid SMEs involve running many parallel SMEs, each onehaving a different value for the parameter (or parame-ters). The classical probabilities attached to the individ-ual SMEs and the associated parameter values can thenbe found by integrating a Kushner-Stratonovich equa-tion. This classical estimation process is numericallycostly, and is even more so when estimates are requiredfor multiple parameters. This paper has demonstratedhow such estimates can be found using a technique takenfrom classical state estimation and nonlinear filtering, aSequential Monte Carlo (SMC) sampler. The SMC sam-pler used in this paper has been demonstrated to allowthe simultaneous estimation of three Hamiltonian param-eters, together with their statistical correlation and theassociated quantum trajectories, in a computationallytractable form, with a relatively small number of can-didate parameter values and parallel SMEs.

Even with such methods, the computational task insolving the Hybrid SME is formidable, and is currentlybeyond the point where it could be used as part of aclosed-loop quantum control system. At present, thestrength of such techniques is in the ability to post-process experimental measurement data to verify thequantum states used in an experiment but also to providean independent, in-situ means to check the parametersthat govern their evolution.

Acknowledgments: JFR would like to thank the USArmy Research Laboratories (contract no. W911NF-16-2-0067). JFR would also like to thank Hendrik Ulbrichtand Peter Barker for helpful and informative discussions.

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