Parameter estimation with reluctant quantum walks: a maximum likelihood approach

The parametric maximum likelihood estimation problem is addressed in the context of quantum walk theory for quantum walks on the lattice of integers. A coin action is presented, with the real parameter θ to be estimated identified with the angular argument of an orthogonal reshuffling matrix. We provide analytic results for the probability distribution for a quantum walker to be displaced by d units from its initial position after k steps. For k large, we show that the likelihood is sharply peaked at a displacement determined by the ratio d/k which is correlated with the reshuffling parameter θ. We suggest that this ‘reluctant walker’ behaviour provides the framework for maximum likelihood estimation analysis, allowing for robust parameter estimation of θ via return probabilities of closed evolution loops and quantum measurements of the position of quantum walker with ‘reluctance index’ r = d/k.


Introduction
Quantum estimation theory [1], [2], [3], [4], is developing as an important component of quantum information and computation theory, which extends the concepts and techniques of the classical theory into the quantum framework; see early [5], [6] and recent reviews: [7], [8], [9]; The classical Fisher information techniques and the classical Cramer-Rao inequality have found various versions of quantum counterparts (see e.g. the last three reviews above).The quantum estimation theory plays a central role in the general framework of quantum technology, see e.g.[10].The maximum likelihood (ML) principle concerns especially this work as a technique that would for example provide parameter estimation in the form of operational valued functions, the measurement of which would yield estimates for interesting parameters of various quantum probability mass functions (pmf).Within this general framework we address the problem of the estimation of an angular variable that determines the coin matrix of a one dimensional quantum walk QW ( [11] - [26]), on the lattice of integers by the ML estimation method (MLE).
Previous works have also addressed similar parameter estimation problems within the context of QWs see e.g.[27,28,29]), wherein the quantum Fisher information and the quantum Cramer-Rao inequality have been used as tools in the various estimation task.In some cases also problems of multi-parameter estimation problems have been addressed.In those works the generic behaviour of the walker itself and the ensuing estimation questions are typically captured using numerical simulations.We note however that there is a need for obtaining analytic and exact solutions for quantum estimation problems in general models and in QWs in particular.In addition to exact solutions it is also desirable to have estimation problems formulated in the language of quantum information and computation theory i.e. in such ways that concepts such as quantum channels, unitary dilations, quantum measurements as well as measures of resources e.g.entanglement or other general resources are used and utilized extensively.With such broad aims in mind we address in this work the estimation problem mention previously by using a QW framework and the technique of maximum likelihood estimation (MLE) in the quantum context.The QW-MLE problem formulated is solved by a novel exact closed form analytic solution for any finite number of steps for the pmf of the problem of QW on the integers with an arbitrary orthogonal coin matrix.The analytic likelihood function constructed from that pmf is shown to be sharply peaked and so it allows for an in-principle exact parameter estimation.These results are further elaborated to provide an operational implementation of the estimation procedure in terms of quantum observables, quantum measurements and CP maps and their unitarily dilated equivalent forms.Finally a analysis of the computational aspects of computing the quantum likelihood function from data identified by quantum measurements of positional probabilities of a diffusing QW is presented.
An outline of the paper is as follows: The quantization scheme of the SO(2) QW on integers is presented and its solution is derived.(Chapt.2).The QW-MLE is developed and the analytic expression of the likelihood function is derived and graphed.Various relevant concepts (evolution loop, return probabilities, reluctant walk, translational symmetry etc) are introduced (Chapt.3).Complexity issues for the likelihood function estimation are presented in Chapt.4, and a general discussion is given in Chapt. 5. Closing a two part appendix summarizes some mathematical properties needed. .Also the coin Hilbert space is taken to be H c ∼ = Z {0,1} and the projection operators . The so called "U quantization" rule of the CRW modifies the classical evolution operator V cl by introducing the transformation where U is the unitary (reshuffling) matrix in coin space.The k'th step map reads , and the related pmf is p The quantum effects in the resulting quantum random walk (QW) are attributed to the non-diagonal reshuffling matrix U , which renders V q as a form of entanglement creating operator (comparable to the Bell states generating operator) [35], [31].
The final result, in terms of U n (x), the Chebyshev polynomial of the second kind of degree n , is summarized in Proposition 1: The Z-SO(2) QW's kth step evolution unitary operator reads for the initial coin state e.g.ρ c = |0⟩ ⟨0| , reads
where U k (ξ) is a polynomial of degree k in ξ, to be specified shortly.Multiplying eq. ( 9) by M (ϕ) yields and from eq.( 2) by substitution k → k + 1, we obtain and by equating eqs.(3)(4) and elaborating, we obtain Then k = 1 in eq.( 2), leads to 2ξM (ϕ 1 ) are the second kind Chebyshev polynomials, and by means of eq. ( 2) would determine the density matrix of QW at any step k via equation (1).(Note that eq. ( 1) for k = 1, is a form of half canonical decomposition of operator V q , and that the walker's step operators are diagonal in the continuous basis with delta-function orthogonality between kets (see Appendix C).This results in the kth power of V q involving the kth power in the kernel matrix M (ϕ)).
By means of this result the power kth step evolution kernel matrix Referring to eq.( 1), and to the fact that matrix M k (ϕ; θ) depends only on variable ϕ we obtain via the spectral decomposition of phase and step operators that the evolution operator V k q depends only on operator Φ.More explicitly V k q depends only on the step operators E ± .Indeed in terms of the Euclidean algebra generators, the unitary evolution operator reads or in terms of in terms of the step operators More concisely The evolution map ρ w → E k q (ρ w ) for the initial coin state e.g.
and is provided by means of the positive trace preserving map where the generators (A k , B k ) of the map are normal operators (commute with their Hermitian conjugate), and satisfy the trace preserving relation The occupation probabilities, expressed in terms of the parameter λ(θ) = cos θ, as an argument of the 2 F 1 hypergeometric function, which for the parameter ranges applicable is truncated to a polynomial in λ , Y (2k) d (λ) , are exactly determined in Proposition 2: The pmf p (k) (d; λ) assigns zero probability to odd steps, and even steps k have equal occupation probabilities for sites distanced ±|d| units away from the initial 0 site: where the polynomial □ Proof : Explicit evaluation of pmf: we proceed with the evaluation of p (k) (x|θ) which reads, To proceed we need to evaluate the three traces of the square bracketed expressions above.For the infinite case, recalling the matrix M from eq. ( 6) and choosing the initial coin state to be |c⟩ = |0⟩ , we get This results into the next expression for the distribution function, in terms of some factorized integrals, where λ = cos θ.Because of independence of the sign of d in the exponential, the indicated integrals above are real e.g.: leading to the analytic form of the pmf The probability distribution of the walk is solved for analytically and is evaluated in terms of the functions This integral can be evaluated by writing the kth order Chebyshev polynomial of the second kind in a series form, expanding cos(dϕ) in powers of cos ϕ and then performing the integral on the remaining terms involving ϕ.Doing this gives us a new definition for This double sum series is not very useful.To simplify it recall that the kth order Chebyshev polynomial of the second kind of an arbitrary argument x can be written as a terminating series with the following form Therefore we can rewrite the integrand of eq.8 as Since this is a terminating sum we can freely exchange the order of the integration and summation yielding We will convert the above real integral into a complex contour integral and use the calculus of residues to evaluate it.To this end let the integral and make the substitutions z = e iϕ , dz dϕ = iz and cos ϕ = z+z −1

2
, cos(dϕ) = transforming the integral into where p = k − 2n (note p ≥ 0).The integrand only contains polynomial terms and so we can clearly see that it is regular everywhere except for at the origin where there is a pole.To find the residue of this pole we start by using a binomial series to expand the term raised to the power p, i.e.
This then splits the integrand into two series It should be noted that since the only change in the exponent of each series is the sign of d, the actual sign of d does not matter.Therefore, without loss of generality, we can take d ≥ 0. We now look for the residue by examining the coefficient of the z −1 term.First we must consider when this term actually exists in the series (a non-zero residue).Since −2j − 1 is odd, it is necessary that p ≡ d (mod 2).Since p = k − 2n and −2n ≡ 0 (mod 2) this means that k ≡ d (mod 2) (i.e.d and k have to be both even or both odd for a non-zero residue, a fact we already knew).Now, taking p ≡ d (mod 2), over the range of the summation, p + d − 2j − 1 and p − d − 2j − 1 form monotone decreasing series taking the value of every odd integer in the interval between when j = 0 and j = p.We want −1 to lie in this interval so we get four conditions on p and which are reduced to only one p ≥ d which is satisfied and implies that n > k−d 2 .Now we look at what the actual value of the residue at the origin is, c.f. eq.( 10), and find Res(z = 0) = p 1 2 (p+d) .So we can now evaluate the integral to be Returning to eq.( 9) we obtain As a final remark note that in eq. ( 17) the term with the lowest power of λ occurs for the greatest value of n which is λ k−2 k−d in the single sum form as above is expressed in terms of the Gaussian hypergeometric series Note first that this series terminates if one of its first two arguments, a or b, is a negative integer.If e.g. a = −m then the infinite series collapses to In the process of putting eq.( 11) into the form of the polynomial above we will make use of the following identities for Pocchammer symbols This leads to the following form of which is identified with the form as required.□

Remark 1
By means of the notations (Y , the pmf in eq.( 7), reads This is identified with the law of cosines for a triangle of sides α, β, γ of respective lengths d and of angle θ between sides α and β.The values 0, π 2 , π, 3π 2 , θ of the reshuffling angle θ and their cosines λ(θ) = 1, 0, −1, 0, cos θ, correspond to QWs with diagonal, offdiagonal coin matrices I, Y = iσ y , −I, −Y, R(θ); (for the effect of non-diagonality of reshuffling matrix to the quantization of a CRW in relation to quantization rules see, [31]).The respective probabilities distributions p , are associated, as indicated, to CRW (γ : exact squares) or QW (γ : not exact squares).The conditional appearance of law of cosines in the final expression of pmf is a manifestation of the quantum character of the walk.

Remark 2
By means of the last final relation and the contiguity relations of the hypergeometric function we can cast the final expression for the pmf of the reluctant QW as follows Propositions 1 and 2 provide an exact analytic solution to the pmf of the Z-SO(2) QW for arbitrary parameter θ and any displacement d away from the origin.The novel feature is the maximum of the pmf achieved at the center of the coordinates i.e. λ = r = 0 , and in general, for given λ, also at a specific d determined by the ratio r := d/k which we term the reluctance.This "reluctant walker" behaviour will allow for the parameter estimation of θ (see below).The complicated form of the pmf issued in equation ( 7), enforces numerical plotting of the maximization of the likelihood function (see figures), instead of an analytic calculation of the maximum.This inconvenience stems from the fact that the pmf does not belong to the exponential family of probability density functions, for which the MLE usually admits an analytic treatment [30].

Operational approach to the MLE problem
Turning to the MLE of θ parameter for the group theoretical models of QW above leads to the following considerations.

MLE: Z-SO(2) case.
Let the position projection operator |d⟩ ⟨d| , which would provide the probability that after k steps the walker reluctantly moved d steps away from its initial cite.
The CP evolution map E k q is generated by step operators E ± satisfying the translation invariant condition Ad(E †s + )•E k q = E k q •(Ad(E †s + )), so the occupation probability for initial state |Ψ⟩ = |0⟩ , reads The likelihood function defined for a set of position points denoted by vector − → x = (x 1 , ..., x n ) , reads where we denote p (k) (x j ; θ) by p (k) (x j |θ) and designate the likelihood function as n (θ|x).Next let positions − → x be occupied by walker after k steps.
Maximization of the logarithm of likelihood l n (θ) = 0, for each j and the determination of the θ roots of function p (k) (x j |θ).
Let us choose an evolution loop for the QW i.e.L (k) =ρ (0) w = |x⟩⟨x|, where |x⟩, x ∈ Z, a basis vector.This choice implies that initial and final state for QWer be the same, say some |x⟩, x ∈ Z.This choice of loop implies a drastic simplification for the likelihood function since now the lattice position vector becomes − → x = (x, ..., x) and imposes the simplification Since p (k) ∈ [0, 1] , we obtain l (k) n = log p (k) < 0, thus the likelihood must be negative and its maximization requires l The evolution loop idea implies the involvement of return probabilities in the evaluation of likelihood function of the reluctant QW resulting into a factorization of l (k) n into factors depending separately on n and k, as in equation (12).Likelihood maximization amounts to maximization of the distribution p (k) (x|θ), for given k ∈ N for those θ ∈ [−π, π) for which the distribution satisfies the positivity inequality p The complicated form of the pmf issued in equation ( 7), enforces numerical plotting of the maximization of the likelihood function (see figures), instead of an analytic calculation of the maximum.This inconvenience stems from the fact that the pmf does not belong to the exponential family of probability density functions, for which the MLE usually admits an analytic treatment [30].
Finally we point out an alternative experimental scenario for likelihood estimation, which does not require the use of translational invariance (to be explained below) in constructing probabilities for multiple QW trials sampled such that there is net zero displacement at various positions.Rather, in each trial the QW is allowed to run from the origin x = 0, and then a measurement is taken after k steps to establish whether or not it has returned.In this scenario, with p := p (k) (0, λ) and n 0 recordings of measurements at x = 0 out of n trials, the log likelihood is so that, at the optimum ( assuming now p ′ ̸ = 0) , the odds p/(1 − p) of return are the ratio n 0 : (n − n 0 ); that is, we simply have in terms of the relative frequency of return f := n 0 /n, The possible solutions in λ = cos θ are therefore the level set at height f of the polynomial function p (Proposition 2 and equation ( 7), and provided −l ′′ > 0).Such a "high reluctance" protocol might be appropriate for small parameter values, where the first intersection is provided by the dominant peak of the pmf around the origin λ = 0 (see figure 2(a)).

QW-MLE with return probabilities
The treatment of this subsection is presented in order to show transparently the ideas of return probabilities, of closed loops and of the symmetry of translation invariance of the QW and how all these are used for constructing likelihood function of MLE task.
Start with the QW density matrix at k-th step ρ w → E k q (ρ w ) that reads, q ) a,b .Then we proceed by noting that the position states are generated from the zero cite (vacuum) state as As shown in Prop. 1, the evolution channel is Note that the generators A k ( Φ; θ), B k ( Φ; θ), are commuting with step operators , a property that give rise to translation invariance.
Then we have where the second equation above follows again from mentioned commutativity.Elaborating last equation is cast in the form: q (P a )).The likelihood function employing transition probabilities as above for paths with initial site x j in and final site x j f = x j in + d j , i.e. a site displaced by d j integers , reads Assuming that the statistical model x j f | that perform quantum measurements on the initial and the k-th step walker density matrix.In such a case the likelihood function is cast in the form For the special case of the return probability with x j in = x j f = x the likelihood function should be computed at the point d = 0.
Before closing this section we show how to compute the distribution of return probabilities formally in terms of the Kraus generators of the evolution channel.The return probability of zero displacement reads for some integer x as follows: Applying the Hadamard (or element-wise or entry-wise) product (def.: or equivalently if x = x = 0, Recall the k-th step Kraus generators A k and B k and their kernels (matrix elements in the continuous basis), A k and B k respectively, we obtain their expression in the discrete basis as The likelihood function now reads .
The above integrals have been carried out analytically in Proposition 2, and the likelihood function and its maximum has been studied and analyzed.

On the computational complexity of likelihood function
In this chapter we address the question of employing a QW for the task of estimation of a unknown parameter.In the estimation methodology of likelihood function the need of a large among of data is indispensable for the success of the estimation task.This requirement implies that a trading is set up between the size of gathered data and the number of operations needed for generating and collecting those data.In the concrete context of the QW-MLE framework the trading pair of actions corresponds to the number of CP maps on the walker systems density matrix generating the walker spread and a measure quantifying that spreading in terms of number of sites and their occupation probabilities.More specifically such a diffusion measure is the standard deviation of the position of a QWer for which the process of QW is know to exhibits a computation advantage in comparison to a classical random walk.In the biological processes of phylogenetic evolution, where to the problem of estimation is a central one and in which random processes are also employed, offers a paradigmatic case for possible use any quantum advantages, (see the related discussion in [43]).
Next we provide an analytic discussion of the computation complexity of the likelihood function in the context of estimation and the advantages offered by QW-MLE developed so far.
Recall that in a fixed number of steps k, a CRWer diffuses over a range of order O( √ k), while a QWer is quadratically faster and diffuses in a range of order O(k).This quadratic speed up implies that a MLE of a parameter via QW utilizes a more extended set of points for the same number of steps, and in this way renders the QW based estimation algorithm more effective in comparison with its classical counterpart.This well known feature of the QW makes it an attractive process in applications ( [11] - [26]).Indeed we see that it also becomes an important feature from the point of view of the estimation problem.Actually the question of right size of the sample needed for the MLE to be successful for the particular case of canonical densities and others is a problem of extensive discussion in various fields (see for example [42], [44]).
The MLE based algorithm uses a set of n QWs to build its likelihood function.Therefore, a data box of size k × n, representing the number of steps × the numbers of QWs, constitutes the data resources for the estimation problem in hand.Given that bigger data box is expected to produce better estimations, the standard dilemma: larger number of steps or larger number of QW?, should be decided efficiently so that the parameter θ is nearly optimal.Both in classical estimation theory and in the present quantum likelihood estimation approach, k and n are considered as quantities constituting two competing scarce resources.That is, the data box dilemma transcribes in our context to then question: "fewer QWs running for a longer time, or many QWs running for a shorter time?".Notice then the significance of the quadratic speed up of QW (faster diffusion rate of a QW), in connection with this data box dilemma.
A QRW produces O(k) × n data points, in comparison to a CRW that could be used instead, which produces O( √ k) × n data points, so it is expected to provide a more efficient (lower cost) parameter estimation, (see [43] for the data box dilemma in the context of quantum phylogenetics and [44], and references therein, for its original form in classical phylogenetics).

Discussion
Quantum walks can be employed as devices for estimating unknown parameters.The step operator of a QW on the integer lattice containing a local coin operator (chosen to be a single angle θ parametrized SO(2) matrix) and a non local conditional step operator, provide is a suitable framework for applying a maximum likelihood estimation technique for determining θ.The QW dynamics is solved analytically for the case of arbitrary finite number of step.This is accomplished by determining the unitary evolution channel of the walk and its Kraus generators.The underlying algebraic structure manifested by the Euclidean algebra is employed to show the translational invariance of the walk.That symmetry suggest the return probabilities of the walk describing closed loop events in the course of the evolution as a convenient probability mass function for building the likelihood function.The likelihood maximization of such reluctant walk is shown by combining analytic exacts results and some numerical investigation.Two important aspects of the those findings are presented: first, the operational aspect that shows how quantum projection measurements on the time evolved density matrix of the walker system can be used for computing the return probabilities and their associated likelihood function; second, complexity of evaluating the likelihood and the gathering of data.The data box dilemma is presented and its importance for effective MLE is analyzed.It is argued that the known feature of a QW to manifest a quadratic speed up in the diffusion rate on the integer lattice in comparison with the corresponding rate of a classical random walk, offers a computational complexity advantage in employing a QW for the task of MLE of an unknown parameter.Some prospects for future extensions of this work would include the following items: the extension of MLE via QW schemes to the case of multi-parameter coin matrices, further, the possibility of using QW schemes for estimating parameter of other gates or (CPTP) completely positive trace preserving maps of interest, since it is known that a QW can be regarded as a universal computational primitive.Also QW-MLE for walks on other lattices and typologies would be a challenge (e.g.unpublished work by the authors that addresses the parameter estimation task in circular lattices for a unitary coin matrix provides interesting exact results).

2 =
λ d , i.e. the polynomial Y (k) d has a monomial factor d that factorizes out of the entire series.Next we show how Y (k) d

Fig. 2
Fig. 2 Upper panel: plots of the pmf with respect to λ for d = 0.The curves have k values of k = 2 ℓ ,ℓ = 3, . . ., 9 .Lower panel: same as above, for fixed r = d/k = 0.25 where the derivatives with respect to θ are denoted by primes.
the generic variables a, b ∈ Z, and define the projectors P a = | a⟩ ⟨a | and P b = | b⟩ ⟨b |.If initially ρ w = | a⟩ ⟨a | then the transition probability from site a to site b after k steps is obtained by means of a projection measurement as for QW-MLE contains the data generation statistical assumption that d j = d ∈ Z, implies that the likelihood function is computed in terms of transition probabilities of paths of equal length d.These probabilities are computed by standard projection operators P x j in = | x j in x j in | and P x j f = | x j f