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quantum mechanics gives the most accurate description of many physical systems of interest . in turn
, the most accurate characterization of a quantum device is given by its quantum mechanical model .
thus , efficient methods for the honest estimation of the distribution of parameters in a quantum mechanical model are of utmost importance , not only for building robust quantum technologies , but to reach new regimes of physics .
bayesian experimental design ( see , e.g. @xcite ) is a methodology to ascertain the utility of a proposed experiment .
bayesian experimental design has been successfully applied to problems in experimental physics , such as in the recent examples of @xcite and @xcite . in classical theories of physics and statistics , the measurement simply reveals the state of the system at that instant .
by contrast , quantum theory presents with the following physical ( and conceptual ) barrier : no single measurement can reveal the state .
rather , each potential kind of experiment admits a probability distribution from which we draw our data .
thus , the methodology of experimental design seems tailor - made for quantum theory .
the structure of the paper is as follows .
we begin by reviewing the general outline of bayesian experimental design .
we then apply the technique to devise an algorithm for the estimation of quantum hamiltonian parameters .
we show that in a particular case , this strategy is nearly globally optimal and demonstrate its improvement over standard algorithms numerically . finally we conclude with a discussion on the applicability of this technique to real experiments on more complex quantum systems .
we assume some initial experiment @xmath0 has been performed and data @xmath1 has been obtained .
the goal is to determine @xmath2 , the probability distribution of the model parameters @xmath3 given the experimental data . to achieve this we use bayes rule @xmath4 where @xmath5 is the _ likelihood function _ ,
which is determined through the process of modeling the experiment , and @xmath6 is the _ prior _ , which encodes any _ a priori _ knowledge of the model parameters .
the final term @xmath7 can simply be thought as a normalization factor .    at this stage
we can stop or obtain further data .
experimental design is well suited to quantum theory since an arbitrary fixed measurement procedure does not give maximal knowledge as is often assumed in the statistical modeling of classical system .
we conceive , then , of possible future data @xmath8 obtained from a , possibly different , experiment @xmath9 .
the probability of obtaining this data can be computed from the distributions at hand via marginalizing over model parameters @xmath10 we can use this distribution to calculate the expected _ utility _ of an experiment @xmath11 where @xmath12 is the utility we would derive if experiment @xmath9 gave result @xmath8 .
this could in principle be any function tailored to the specific problem .
however , for scientific inference , a generally well motivated measure of utility is _ information gain _ @xcite . in information theory ,
information is measured by the entropy @xmath13 thus , we search for the experiment which maximizes the expected information in the final distribution . that is , an optimal experiment @xmath14 is one which satisfies @xmath15
as an example of how to apply the bayesian experimental design formalism to problems in quantum information , we consider a simple situation with a single qubit .
in particular , we suppose that the qubit evolves under an internal hamiltonian @xmath16 here @xmath17 is an unknown parameter whose value we want to estimate .
an experiment consists of preparing a single known input state @xmath18 , the @xmath19 eigenstate of @xmath20 , evolving under the hamiltonian @xmath21 for a controllable time @xmath22 and performing a measurement in the @xmath20 basis .
this is the simplest problem where adaptive hamiltonian estimation can be used and is the problem studied in reference @xcite .    in the language of bayesian inference , the data @xmath23 is the outcome of the measurement .
an experiment @xmath0 consists of a specification of time the @xmath22 that the hamiltonian is on , while the model parameter @xmath3 is simply @xmath17 .
the likelihood function is given by the born rule @xmath24 experimental design is a decision theoretic problem based on the utility function @xmath25 the optimal design is any value of @xmath22 which maximizes this quantity .
we proceed by performing the optimal experiment and obtaining data @xmath8 . using bayesian inference we update our prior @xmath26 via bayes rule : @xmath27
if we are not satisfied , we can repeat the process where this distribution becomes the prior for the new experimental design step .
this algorithm is depicted in figure [ fig : overview ] .
overview of a step in the online adaptive algorithm for finding locally optimal experiments .
top : method for calculating the utility function @xmath28 , given a simulator and a prior distribution @xmath29 over model parameters @xmath3 .
bottom : method for updating prior distribution with results @xmath1 from chosen actual experiment . ]
the preceding problem had a single unknown variable .
if we desire an estimate @xmath30 of the true value @xmath3 , the most often used figure of merit is the _ squared error loss _ :
@xmath31    the _ risk _ of an estimator @xmath32 is its expected performance with respect to the loss function : @xmath33 for squared error loss , the risk is also called the _
mean squared error_. the average of this quantity with respect to some prior @xmath34 is the _ bayes risk _ of @xmath35 , @xmath36 and the estimator which minimizes this quantity is called a _ bayes estimator_. in this case the bayes estimator is the mean of the posterior distribution . let us assume then that the estimators we choose are bayes .
let us also choose a uniform prior for @xmath3 .
then , the final figure of merit is the average mean squared error ( amse ) : @xmath37 we would like a strategy which minimizes this quantity .
non - adaptive fourier and bayesian strategies were investigated and compared to an adaptive strategy in reference @xcite .
their adaptive strategy fits into the bayesian experimental design framework when the utility is measured by the _ variance _ of the posterior distribution : @xmath38 where @xmath39 is the mean of the posterior .
recall that the mean is a bayes estimator of amse , so @xmath40 . for a single measurement
this utility function satisfies @xmath41 . that is
, maximizing the utility _ locally _ at each step of the algorithm is equivalent to minimizing the amse at each step .
hence , when using the negative variance as our utility function , the adaptive strategy summarized in figure [ fig : overview ] is an example not only of a local optimization , but also a _ greedy algorithm _ with respect to the amse risk . in the future , we shall refer to this choice of utility function together with the local optimization algorithm as the greedy algorithm for this problem .
we can write the risk of this strategy recursively as follows .
suppose at the @xmath42th , and final , measurement we have the updated distribution @xmath43 .
then , the risk of the local strategy is @xmath44 where @xmath45 is the locally optimal design satisfying @xmath46 the expected risk at any other stage is @xmath47 where @xmath48 is , again , the locally optimal design satisfying @xmath49 then , the bayes risk of the greedy strategy is @xmath50    again , it is clear that the greedy algorithm is globally optimal on the final decision , as there is no further hypothetical data to consider .
that is , the optimal solution at the @xmath42th measurement is @xmath51 where @xmath45 is the locally optimal design satisfying @xmath46 however , the globally optimal risk at any other stage @xmath52 where now @xmath53 is the globally optimal design satisfying @xmath54 then , the bayes risk of the greedy strategy is @xmath55    in general , @xmath56 . nor is it the case that @xmath57 for an arbitrary prior .
however , for the special case of the uniform prior , we have found numerically that the bayes risk of the greedy strategy and the bayes risk of the global strategy are similar enough that the greedy strategy is useful .      in reference
@xcite , it was shown via simulation that the posterior variance of the greedy strategy is best fit by an exponentially decreasing function of @xmath42 , the total number of measurements .
in contrast , all off - line strategies decrease at best as a linear function of @xmath42 .    in figure
[ fig : results ] , we show that the local information gain optimizing algorithm also enjoys an exponential improvement in accuracy over naive off - line methods
. moreover , we show nyquist rate sampling is unnecessary and , indeed , sub - optimal .
all results stated are obtained using a uniform prior on @xmath58 $ ] and are computed numerically by exploring every branch of the the decision tree , in contrast to simulation .
in order to be `` fair '' to the off - line methods , we restricted the adaptive methods to explore the same experimental design specifications .
that is , for this particular problem , the adaptive algorithm was allowed to select measurement times from @xmath59 $ ] , where @xmath60 is the total number of measurements . in principle , these methods could only do better with a larger design specification .
performance of the estimation strategies .
the bayesian sequential and the strategy labeled `` nyquist '' sample at the nyquist rate .
the `` optimized '' strategies find the global maximum utility ( using matlab s `` fmincon '' starting with the optimal nyquist time ) . in each case , @xmath61 measurements are considered .
left : the ideal model discussed in the text .
right : a more realistic model with @xmath62 noise and an addition relaxation process ( known as @xmath63 ) which exponentially decays the signal ( to half its value at @xmath64 ) .
, title="fig : " ]   performance of the estimation strategies . the bayesian sequential and
the strategy labeled `` nyquist '' sample at the nyquist rate .
the `` optimized '' strategies find the global maximum utility ( using matlab s `` fmincon '' starting with the optimal nyquist time ) . in each case , @xmath61 measurements are considered .
left : the ideal model discussed in the text .
right : a more realistic model with @xmath62 noise and an addition relaxation process ( known as @xmath63 ) which exponentially decays the signal ( to half its value at @xmath64 ) .
, title="fig : " ]
summarizing , we have shown for the problem of estimating the parameter in a simple hamiltonian model of qubit dynamics an adaptive measurement strategy can exponentially improve the accuracy over offline estimation strategies .
moreover , we have shown that sampling at the nyquist rate is not optimal in the case of strong measurement .
we have derived a recursive solution to the risks for both the local and global optimal strategies . using this solution
, we numerically found that the local strategy is nearly optimal in the special case of a uniform prior . that the greedy algorithm is nearly optimal in a case relevant to experiment demonstrates that an adaptive bayesian method may be computationally feasible , in that an implementation need not consider all possible future data when choosing each experiment .
together , these results demonstrate the usefulness of an adaptive bayesian algorithm for parameter estimation in quantum mechanical systems , especially in comparison with other algorithms in common use . in the presence of noise ,
this improvement becomes still more stark , as demonstrated by the results shown in figure [ fig : results ] .
why is it the case that the nyquist times are not optimal ?
first , why should we expect them to be optimal ?
the nyquist theorem states that a signal which contains no frequencies higher than @xmath65 is completely and unambiguously characterized by a discrete set of samples taken at a rate greater than or equal to @xmath66 .
however , the classical notion of sampling fails for the strong - measurement case that we consider here .
what we have is a periodic _ probability _ distribution which can be sampled , not a periodic function whose _ values _ can be ascertained .
that is there is no _ signal _ , in the classical sense of the word , which can be reconstructed .
the failure of the nyquist rate sampling is exemplified in figure [ fig : utils ] .     the information gain ( left ) and variance ( right ) utilities for the prior followed by three simulated measurements .
the vertical grid lines indicate the nyquist times .
note that the times at which the utilities are maximized do not necessarily increase with the number of measurements.,title="fig : " ]   the information gain ( left ) and variance ( right ) utilities for the prior followed by three simulated measurements .
the vertical grid lines indicate the nyquist times .
note that the times at which the utilities are maximized do not necessarily increase with the number of measurements.,title="fig : " ]    in this paper , we have chosen to measure success via the squared error loss .
although this is a standard metric , note that it is not practically useful in the context of estimating the parameters of a quantum mechanical system .
we motivate this claim as follows .
a typical application of our algorithm is to inform control theory algorithms , which can achieve significantly higher fidelities if given a distribution over hamiltonians rather than a single best estimate . indeed , in the case of nuclear magnetic resonance , the physical ensemble of qubits produces a real distribution of hamiltonians to which control theory algorithm must be robust against @xcite .
any single estimate of the hamiltonian parameters will thus artificially exclude dynamics which will appear as decoherence in the resultant pulses .
thus , we must measure the success of our algorithm via a loss function of the true _ distribution _ and estimated posterior . noting that relative entropy is broadly considered the correct loss function for probability estimators , our algorithm , which maximizes expected information gain , becomes the optimal solution .
we expect that in more complicated systems , the bayesian adaptive method will remain useful , especially in applications such as optimal control theory , where having a distribution over hamiltonians is significantly more useful than a single best estimate .
u.  von toussaint , t.  schwarz - selinger , m.  mayer , and s.  gori , _ nuclear instruments and methods in physics research section b : beam interactions with materials and atoms _ * 268 * , 21152118 ( 2010 ) , issn 0168 - 583x , http://dx.doi.org/10.1016/j.nimb.2010.02.062 .
using bayesian experimental design techniques , we have shown that for a single two - level quantum mechanical system under strong ( projective ) measurement , the dynamical parameters of a model hamiltonian can be estimated with exponentially improved accuracy over offline estimation strategies . to achieve this 
 , we derive an adaptive protocol which finds the optimal experiments based on previous observations . 
 we show that the risk associated with this algorithm is close to the global optimum , given a uniform prior . 
 additionally , we show that sampling at the nyquist rate is not optimal .     
 address = institute for quantum computing , university of waterloo , waterloo , ontario , canada , altaddress = department of applied mathematics , university of waterloo , ontario , canada , email=csferrie@uwaterloo.ca     address = institute for quantum computing , university of waterloo , waterloo , ontario , canada , altaddress = department of physics , university of waterloo , ontario , canada , email=cgranade@cgranade.com     address = institute for quantum computing , university of waterloo , waterloo , ontario , canada , altaddress = perimeter institute for theoretical physics , waterloo , ontario , canada , altaddress = department of chemistry , university of waterloo , ontario , canada