Two-walker discrete-time quantum walks on the line with percolation

One goal in the quantum-walk research is the exploitation of the intrinsic quantum nature of multiple walkers, in order to achieve the full computational power of the model. Here we study the behaviour of two non-interacting particles performing a quantum walk on the line when the possibility of lattice imperfections, in the form of missing links, is considered. We investigate two regimes, statical and dynamical percolation, that correspond to different time scales for the imperfections evolution with respect to the quantum-walk one. By studying the qualitative behaviour of three two-particle quantities for different probabilities of having missing bonds, we argue that the chosen symmetry under particle-exchange of the input state strongly affects the output of the walk, even in noisy and highly non-ideal regimes. We provide evidence against the possibility of gathering information about the walkers indistinguishability from the observation of bunching phenomena in the output distribution, in all those situations that require a comparison between averaged quantities. Although the spread of the walk is not substantially changed by the addition of a second particle, we show that the presence of multiple walkers can be beneficial for a procedure to estimate the probability of having a broken link.


I. INTRODUCTION
In the field of quantum walks, many results have been recently obtained, both from the theoretical [1,2] and from the experimental [3][4][5][6][7][8] point of view.Being the quantum analogue of the classical random walk, the quantum walk [9] is a simple yet interesting model that can be exploited in several ways.It has been proved that quantum walks can be used to implement quantum search algorithms [10], and they can be considered as a universal computational primitive [11,12].They have applications in the simulation of biological processes [13] and an equivalence with quantum lattice gas models [14] has been proposed.It is worth to keep in mind that the scenario with only a single walker is however classically simulable, for example using coherent light instead of a single-photon state in an optical implementation [15,16].In order to have a true quantum behavior, more than one walker has to be considered.In this case, the total evolution clearly depends on the quantum nature of the particles involved.This more complex scenario has been already partially studied in the presence of a fixed unitary evolution [16][17][18][19][20][21][22].
It is also interesting to take into account the possibility of having missing links (percolation graphs), modelling for example physical situations in which one does not have perfect control on the structure.Several authors have investigated this scenario for a single-particle quantum walk from different perspectives, e.g.studying decoherence [23], modelling transport phenomena [24] or describing asymptotic behaviors [25].Moreover, one can analyze quantum walks on geometries more complex than the simple line, for instance higher-dimensional lattices [26] or graphs [27].Relating the behavior of the walk to some general characteristics of the considered structure would surely be of great interest.
With this in mind, the purpose of this paper is to investigate the behavior of a quantum walk when two of these aspects are jointly considered.We study a walk with two quantum particles, using as underlying lattice a simple line with bonds randomly missing with time.This is interesting in order to evaluate how the possible absence of links (due, for instance, to a non-perfect experimental setup) could influence the evolution of many walkers, a fundamental step toward a complete exploitation of the quantum walk model.The simulation of the same scenario in more complex geometries, being more computational demanding, is left as a possibility for future investigations.
The overall structure of the paper is as follows.In Section II we will briefly review the concept of discretetime quantum walk on the line, with and without percolation.Section III will be devoted to fix our notation when a second indistinguishable particle is added to the system.The analysis of a two-particle quantum walk on the line with time-dependent missing bonds will be the topic of Section IV.Depending on the input state, we can simulate the average behavior of two proper quantum particles (bosons or fermions) as well as that of two "classically indistinguishable" particles.In Section IV A we will explain the appearance of a raised (diminished) probability of finding both bosons (fermions) at the same position, with respect to the distribution for classically indistinguishable particles and to the result deducible from the single-particle output probability.Eventually, in the remainder of Section IV, we will characterize particular aspects of the output distribution by means of single numbers: the meeting probability (i.e. the probability of finding both particles at the same position), the spread of the walk, and the final average distance between the walkers.We will study how these quantities depend on the percolation regime considered and on the probability of missing bonds.Conclusions are left for Section V.

II. SINGLE-PARTICLE DISCRETE-TIME QUANTUM WALK
In this Section we will briefly review the definition and the evolution of a discrete-time quantum walk on the line, with and without percolation.

A. Without percolation
A discrete-time quantum walk can be thought as the quantum counterpart of a classical random walk, where at each step the walker can move left or right with a certain fixed probability.In this scenario, each step can be considered as composed by two parts: a (possibly biased) coin toss and a shift in the direction associated with the random output.
Quantum mechanically the position of the walker is described by orthonormal quantum states {|i } i , that span the Hilbert space H P , while the coin can be implemented adding a two-dimensional Hilbert space H C , so that the global system can be described in H P ⊗ H C .Indeed, the coin toss can be realized by means of a fixed unitary operation applied to the C component of the system.Throughout this manuscript, we will use for this purpose the Hadamard matrix: expressed in the standard basis of the coin {|↑ , |↓ }, whose states correspond to the classical idea of "head" and "tail".This matrix represents the simplest choice for an unbiased walk, with no direction privileged by the unitary evolution.Although it has been shown that the Hadamard coin leads to all possible quantum walks if different starting states are considered [28], such equivalence breaks down when percolation is introduced.Our choice has hence to be considered as motivated by the sake of simplicity.For the same reason, we will consider most of the time the initial state which leads to an output probability distribution symmetric around the initial position.In the following we will drop the labels "P" and "C", using the convention of writing the position always in front of the coin state.
The second part of a single step corresponds to the shift of the walker in the lattice, toward a direction depending on the coin output.Formally, this can be written as the following operator acting non-trivially on the whole space H P ⊗ H C : Hence, a single step of the evolution can be obtained applying the operator to the vector |ψ in , representing the initial state of the system.In order to obtain a different result with respect to a classical random walk, we must perform a measurement in the position degree of freedom only after a certain number of steps N .The probability distribution characterizing the result will be where the subscript "1" reminds us that it represents the probability associated with a single walker.After such measurement, that instance of quantum walk will be considered concluded with output given by (5).Notice that all the coefficients appearing in the evolution operators are real.This implies that the real and imaginary part of the initial state (expanded in the basis |i |↑ , |i |↓ ) evolve independently.A conjugation of the state amplitudes, hence, does not change the final probability distribution.It is useful to keep such property in mind, since it will be needed in the following.
It is well known that the average distance travelled by a classical random walker scales with the number N of steps as √ N , while for a quantum walker it scales linearly with it [2].A typical plot of a probability distribution for a quantum walk can be observed in Figure 1a.

B. Percolation in 1D quantum walks
The situation previously described changes if we consider a lattice whose bonds can be randomly missing with time.Typically, in a percolation graph each bond is actually present only with a fixed probability 0 ≤ p ≤ 1, and the structure is periodically changed, respecting such probabilistic constraint.Depending on the frequency of the changes, there can be two extreme regimes of percolation: statical (different lattices for different walks) or dynamical (different lattice at each step).The first scenario can be associated with a time-scale for the latticeimperfections evolution which is long with respect to the typical time required to finish a single walk, but small with respect to the average time needed between different runs of the apparatus.On the other hand, the regime of dynamical percolation appears when the imperfections in the lattice evolve so quickly that the missing bonds can change position even between consecutive steps of the same walk.In the following both these extreme cases will be considered.
There are two possibilities to keep track of missing bonds in the evolution of the particle.
1. Change the shift operator (3), making it position and time dependent, leaving the coin operator H C unmodified.If the walker cannot cross a missing bond, in this approach it does not move and its coin state is flipped, so that the unitarity of the evolution is preserved.Formally, the operator applied on the i th site will be one of the following: respectively when all near bonds are present, when the following or the previous one is missing, or when they are all missing.
2. Change the coin operator, making it position and time dependent, guiding the walker along an alternative direction if a bond is missing; the shift operator is left unaltered.
This second approach presents a problem when a regime of dynamical percolation is considered, since a shift over a missing edge could be allowed, independently of the choice of the unitary operation H C used as coin (e.g. when all the bonds near the walker will be missing at the following step).This problem does not appear when static percolation is considered, if the initial state is opportunely chosen (depending on the lattice configuration).Indeed, if the walker moves to a particular site, we are guaranteed on the existence of a way out that can be used.In order to compare the two percolation regimes, in the following we will use the first approach, based on the redefinition of the shift operator.Such choice also closely represents the intuitive idea of a walk with percolation: a standard coin is tossed and the walker tries to move based on the output.If this should be impossible, it turns around and waits for the next step.As already stated, every time that the linear lattice changes, the missing-bonds positions are chosen randomly: each link is actually independently present only with probability 0 ≤ p ≤ 1.In order to get rid of this randomness, the output probability has to be averaged over many different lattice sequences.Typically, this kind of quantum walk behaves qualitatively similarly to a classical random walk, and it leads to a final Gaussian-like probability distribution.An example can be seen in Figure 1, where the outcomes of a 300-step walk is shown in different percolation regimes.While in the standard quantum walk only odd (or even) positions can be occupied, there is no such constraint when our strategy of time evolution with missing bonds is adopted (modified shift operator and fixed coin).Notice also how in the example shown in Figure 1, the walker spread is much larger for dynamical percolation than for the statical case.This is a typical feature emerging after a certain number of steps.The reason can be understood comparing the two percolation regimes.
The evolution of a single-walker quantum walk on the line in the presence of dynamical percolation has been first studied by Romanelli et al. in [23].They showed how for a small number of steps the evolution is coherent, with a variance depending quadratically on the number of steps.This is because at the beginning the walker does not meet any broken link.This consideration allowed them to estimate the typical number of steps in which percolation effects start to become relevant as After this point the spreading behavior is diffusive, with a variance that scales linearly with the number of steps and a p-dependent pre-factor.The spread in the statical scenario, instead, is limited by the average length of consecutive links actually present.Indeed, for a given lattice after the first ∼ 1 1−p steps, the walker will typically meet a broken link.From there on, independently of the total number of steps, its spreading will be limited by the length of the connected segment in which it is located on that run of the quantum walk.Hence no matter how many steps N 1 1−p , the averaged spread of the walker in a static percolation regime will be localized in a region In order to have a symmetric distribution we started from the initial state (2).In the last two plots, we considered lattices characterized by a percolation parameter p = 0.75 and we averaged over 500 different outputs.
determined by the average length of a connected segment L:

III. TWO INDISTINGUISHABLE WALKERS WITHOUT PERCOLATION
In this Section we want to review the problem of a quantum walk with two walkers.The global Hilbert space of the system will be where each H k can be expanded as H k = H P k ⊗ H Ck .The singlestep operator characterizing the evolution is chosen to be the single-particle operator U of Equation ( 4), applied independently to both systems: where the labels represents the Hilbert space of application.This choice guarantees that both particles evolve independently, without any kind of interaction.Particularly, we want to focus on indistinguishable particles, for which the final output is obtained considering the set of projectors: leading to a probability distribution With such definition, the probability of finding one particle in position i and the other one in j = i is given by 2P 2 [|ψ in ] (i, j), while the probability of finding them at the same location i is just P 2 [|ψ in ] (i, i).Such description is chosen to have a properly normalized probability distribution on the whole plane (i, j).
In considering indistinguishable walkers, we can think at three characteristic behaviors: bosons, fermions and classically indistinguishable particles.In the first two cases, the initial state has to be opportunely symmetrized (or anti-symmetrized for fermions): where the indeterminate form Sym − |ψ ⊗2 is set to 0 by definition.The case of classically indistinguishable particles, instead, simply corresponds to consider initial separable states |ψ 1 1 |ψ 2 2 , since their indistinguishability is already taken into account in the choice of Π ij in (8).Given two single-particle states |ψ k , the output probability distributions obtained from (9) in the three cases can be labelled respectively by P (±) 2 and P (cl) 2 .The dependence upon the states |ψ k , k = 1, 2, will always be dropped to simplify the notation.While the fist two distributions cannot in general be simply expressed in terms of single-particle outputs (5), for two classically indistinguishable particles the following equivalence holds: as can be easily proved using the factorized structure of both evolution (7) and projector (8).Sufficient conditions for having the equality It is worth noticing that not every entangled state in H 12 can be expressed in a symmetrized form as in (10).An example of bipartite state left out by that particular structure is when i = j.In the following, however, we will restrict to either states of a symmetrized form as in (10) 2 .Such analysis aims at detecting the features of the output distribution due to the quantum bosonic or fermionic statistics of the particles.
Since we are considering an evolution operator (7) without inter-particle interaction, in order to maximize the multi-particle effects, in the following we will always consider states where the two walkers start from the same spatial position (the origin).In addition to this, to avoid asymmetrical spreading, we will focus mainly on combinations of the two single-particle coin states: that lead to the symmetric spreading around the origin shown in Figure 1.Considering their symmetrization and anti-symmetrization, as given by (10), we obtain the two maximally entangled coin states corresponding to purely bosonic and fermionic statistics.
On the other hand, we can consider also their tensor product a separable states that describes two classically indistinguishable walkers.
Notice that with the choice (15) the general relation (11) further simplifies: being |ϕ ± exchanged via a complex conjugation, their single-particle output distributions P 1 [|ϕ ± ] are the same.For this reason from now on we will refer to it simply by P 1 , without further indication about the state.Hence P (cl) 2 (i, j) can be written as The probability distributions obtained with the three symmetrical-spreading initial states can be observed in Figure 2, where typical bunching and anti-bunching behaviors can be noticed.

IV. TWO INDISTINGUISHABLE WALKERS IN THE PRESENCE OF PERCOLATION
When the possibility of having missing bonds is considered, a random component is introduced in the evolution.As in Section II B, we can get rid of this by averaging over several different output probability distributions: where a runs over the different realizations of the quantum walk.Each plot is obtained averaging over a number of different outputs between A = 500 and A = 5000, depending on the precision required.The qualitative result does not change significantly in that range, nor upon the number of steps considered (in our case typically N = 15).These facts allow us to draw conclusions about the general qualitative behavior of the walk in the presence of percolation.Depending on the frequency with which the lattice is changed, as we have already stated, we can have two regimes of percolation: dynamical, if the lattice is modified at each step of every run of the walk, or statical, if the present bonds are left unmodified during each run, but can be in different positions in different walks.The two results are qualitatively dissimilar, as can be appreciated in Figure 3 where the output of a quantum walk with initial state |ψ S (15) is plotted for a value of percolation parameter p = 0.75 (representing the probability of having each bond).In particular, we can notice how the distribution is more spread out over the allowed region with dynamical percolation, while in the statical regime it is concentrated in a few particular points.In both cases, however, some high values of probability seem to appear along the diagonal, representing the output situation in which the two walkers exit at the same position.A more detailed analysis of this phenomenon can be found in Section IV A.
After that, in the remainder of this Section, we will address the problem of characterizing the averaged output of the quantum walk by means of a single number, describing one particular aspect of the result.This approach would be extremely useful in higher dimensional lattices, where the output of a multi-particle quantum walk can no longer be graphically plotted.A comparison between statical and dynamical percolation is also performed wherever necessary.Particularly, in Section IV B we will focus on the meeting probability [19], that in certain conditions could be useful to infer the unknown value of p, whereas in Section IV C and IV D we will respectively consider the spread of the walk and the average distance between the positions at which the particles are detected.We will show that while the latter is highly dependent upon the quantum statistics of the particles, the spread of the walk is not.Furthermore, we will show that for the considered symmetrical-spreading states, as well as for many other cases of interest, such spread is exactly the same as in the single-walker case, showing that this feature of the quantum walk is not affected by the presence of a second particle.

A. Analysis of peaks (valleys) for bosons (fermions)
Here we want to consider the appearance of diagonal peaks (for bosons) or valleys (for fermions) in the averaged output probability distribution (17) obtained when percolation is involved, dynamical as well as statical.
In order to properly define a peak/valley, we need a reference probability distribution to use as a comparison.To do this, if we are considering the symmetrization (10) of two different single-particle states |ψ 1 = |ψ 2 , we can compare the averaged probability for two quantum particles P (±) 2 with the averaged result P given by two classically indistinguishable particles, starting from the same couple of states |ψ k , k = 1, 2. On the other hand, also P (cl) 2 could be compared with the symmetrized product of the two single-particle averaged distributions P 1 [|ψ i ].We will see that in general both of these comparisons could lead to an enhanced probability along the diagonal (diminished in the case of fermions): the first is due to the quantum statistics of the particles, while the second is a mere effect of the averages.
1. Peaks/valleys of P (±) with respect to P (cl)   Given two states |ψ 1 , |ψ 2 ∈ H P ⊗ H C , we can evaluate the expression for P (±) 2 (i, j) along the diagonal i = j, using an expansion similar to Equation (A5) in Appendix A. For a single run of the quantum walk U N (a) we get: From this, it follows that at least when so that ∀ lattice site j one finds: Usually when |ψ 1 and |ψ 2 lead to different probability distributions, or when off-diagonal terms i = j are considered, the second line in (21) has not a fixed sign, so for A 1 it does not contribute significantly.This is no longer true when we consider diagonal terms i = j for states with the same final distribution (e.g.|0 |ϕ + , |0 |ϕ − ).In such a case, being sum of squared terms, the second line in ( 21) is always positive.These contributions hence add up to create the peaks over the first term, which in this case is exactly the product of averaged single-walker probabilities.Notice that here, differently from before, it is the averaging process itself that lead to the peaks, not the quantum statistics of the particles.
From this analysis, it follows that the first phenomenon arises for sure when we are dealing with single-particle states satisfying ψ 1 | ψ 2 = 0, while the second effect appears when they also lead to the same probability distribution.The choice of symmetrical-spreading states |0 |ϕ ± satisfies both these condition at the same time.Indeed, this observation allows us to interpret Figure 5, where and the product of P 1 (i)P 1 (j), along with their differences, are plotted for such states in a dynamical percolation regime.We can see how P (±) 2 presents higher (smaller) terms on the diagonal with re-spect to P (cl) 2 , which in turn has small peaks on the diagonal if compared with the product of single-particle distributions.

B. Meeting probability
A quantity which can be quite easily obtained is the overall probability for the two walkers to be on the same site when the measurement is performed.Such quantity, also called meeting probability [19], corresponds to the sum of the diagonal terms in the output probability matrix P 2 (i, j) (9).
In this Section we want to see how such quantity depends upon the percolation parameter p used, for different input states, in the regimes of dynamical and statical percolation.In particular, we will focus on the three initial coin states |φ + , |ψ − and |ψ S , defined in ( 14), (15), for two walkers starting at the same position.For each one of them, fixed a parameter p, we numerically obtain the meeting probability M (a) (p) for |{a} a | = A = 5000 different lattice sequences.In Figure 6 we plot the mean values obtained, together with their standard errors describing the uncertainty on the estimation of the real mean value.In formulas: The obtained results for a 15-step quantum walk are plotted in Figure 6 for both dynamical and statical percolation.We can notice how in the first scenario the meeting probability decreases for p 0.92 and slightly increases after this point, while in the statical case it is monotonically decreasing ∀p.The minimum position in the dynamical case seems to depend on the number of steps, moving slowly toward p = 1 when this is increased.Numerically we did not go over 25 steps, for which the minimum seems to be around 0.94 (only the bosonic state |φ + has been considered, since the effect is more evident in it).
The monotonicity in p obtained in the statical case could also be used to infer the value of p given a measured meeting probability M (p).From this perspective, it is worth trying to further reduce the number of measurements experimentally required.We can for example just consider the averaged probability C(p) of finding both walkers at the origin to deduce the value of p. Intuitively, this value will be close to 1 for small p, since both walkers are constrained by the missing bonds to stay close to the starting point, and it decreases with increasing p while the walk spreads further away.This behavior is numerically confirmed, as it is shown in Figure 7 for both dynamical and statical percolation.Having a plot with an almost constant slope over the whole region p ∈ [0, 1], P 2 (0, 0) is more suitable to infer the value of p in a statical percolation scenario, with respect to the dynamical case.

C. Spread of the walk
In this Section we want to discuss the spread of the two-particle quantum walk, in order to see if there are any differences with respect to the single-walker case briefly reviewed in Section II.The intuitive idea of spreading for a single walker, starting from a localized position i 0 , can be well described by the quantity where the initial coin state is not explicitly written.Notice that if the walker evolves symmetrically, such quantity corresponds to the variance of the output distribution.How can we generalize this expression to the twowalker case?As we have done since the beginning, we are mostly interested in let both particles start from the same lattice position.Indeed, since in our model there are no long-range interactions, this situation is the one that could maximize the multi-particle effects.For this reason we will consider here only the case where both walkers start from the origin (with arbitrary coin states), but the interested reader can find in Appendix B a short discussion about spread measures useful when different starting points are involved.Considering the origin as initial position, the straightforward generalization of ( 24) is: where the label 2 stays for "two walkers".Once two single-particle coin states have been fixed, we can label with the spread corresponding with the different statistics of the particles, obtained using respectively in (25).In the same way, with an over bar we will label the value of the spread averaged over many different lattice sequences in a percolation regime.The overall factor 1/2 in our definition is introduced only to ease the comparison with the single-walker definition of V 1 (24), as the following Proposition enlightens.Proposition 1.Let us consider two classically indistinguishable walkers, starting from the origin with a separable coin state |c 1 |c 2 .Their spread can be written as average of the two single-particle spreads: and the respective single-particle distributions.This Proposition shows how, if the two walkers are described by any separable state, each particle spreads independently of the other: the fact that we cannot distinguish them in the measurement leads only to the average that appears in Equation (26).One can now wonder whether the presence of a proper quantum statistics, introduced by the symmetrization of the vector describing the state, can somehow alter this result.The following Proposition addresses this question.For the proof we refer to Appendix B, where a more general result involving orthogonality between states in H P ⊗ H C is provided.Proposition 2. If the two single-particle coin states |c 1 , As already noted, the two symmetrical-spreading states |ϕ ± are orthogonal, hence from this result we can conclude that for the three coin states |φ + , |ψ − and |ψ S , mainly considered throughout the paper, the quantum statistics has no effect on the spread.Furthermore, |ϕ ± lead to the same probability distribution, so that also the two spreads V 1 (ϕ ± ) coincide.Equation (25) therefore implies so that for these symmetric states there is no advantage in adding a second walker, when the spread of the walk is concerned.The only way the bosonic (or fermionic) statistics could influence the spread is by choosing two non-orthogonal initial states for the walkers, which is not the most natural configuration that one may encounter.

D. Average distance between the walker
A final property that can be analyzed is the average distance between the positions where the two walkers are detected.Formally such quantity can be expressed as where as usual the over line represents the average over different runs of the walk.Its numerical evaluation, for a 15-step quantum walk and for different percolation parameters p, is plotted in Figure 8. Two particles starting from the origin with coins |φ + , |ψ − and |ψ S (14), (15) are considered.As can be expected, independently of the percolation parameter, the value of this quantifier for fermions is larger than the one for bosons, with an intermediate behavior for the separable coin |ψ S .
If we consider the dependence upon p in the dynamical percolation regime, we can see that the average distance between bosons increases approximately in a linear way, with a small reduction just before p = 1.This can be interpreted as a trade-off between two trends: on the one hand with less missing bonds the two walkers can move on a larger segment of the line, increasing the likelihood of finding them further away.On the other hand, if the two particles are less disturbed, their distribution approaches the one obtained without percolation shown in Figure 2a, characterized by an high degree of bunching.For fermions, instead, the same considerations add up increasing the averaged distance of separation, since they tend to show an anti-bunched distribution in the limit p → 1 (Figure 2b).
If a comparison is performed between the plots obtained with dynamical and statical percolation for a 15step walk, one can notice that in the second regime the averaged distance between the particle is much smaller.This can be understood by thinking that with statical percolation in each run the particles are constrained to stay close together in a fixed region of space, whereas in the dynamical case they have always the chance of getting further away.The observed behavior is hence typical once the number of steps is larger than the averaged length of the connected segment L (6) available to the walkers in a regime of statical percolation.

V. CONCLUSIONS
We studied the behavior of a one-dimensional quantum walk when the possibility of having missing bonds is considered together with the presence of a second walker.We focused on two percolation regimes, statical and dynamical, associated respectively with slow and fast varying imperfections.To get rid of the introduced randomness, our numerical simulations have been averaged over many (∼ 500 − 5000) different lattice sequences, realized associating to each bond a probability p ≤ 1 of being present.Starting from two symmetrical-spreading singlewalker states, we combined them in three different ways, in order to observe the behavior of two bosons, fermions or simply two classically indistinguishable particles, i.e. without symmetrizing their quantum states.
We described how the bosonic (fermionic) quantum statistics of the walkers affects the final probability distribution, inducing the presence of characteristics peaks (valleys) in the diagonal values of the detection probability.This feature, easily observed through a comparison with the output of an unsymmetrized state, is not due to percolation but emerges as a consequence of the imposed exchange symmetry between two orthogonal single-particle states.On the contrary, the leftover peaks that can still be observable in the distribution of two classically indistinguishable particles are just a statistical effect of the averages over different unitary evolutions, associated with different realizations of the un- derlying lattice structure.
The advantage of studying low-dimensional lattices, with no more than two walkers, is the possibility of representing the output detection probability through a 3D plot.This visual approach, however, cannot be extended to more complicated scenarios, involving multiple walkers and/or more complex structures.The research for quantities that could meaningfully describe some aspects of the result, being at the same time easily accessible from both a theoretical and experimental point of view, will therefore be of extreme importance in the future.Here we considered three of them, describing their qualitative behavior for a symmetrical-spreading two-particle quantum walk with percolation.In particular we saw how the meeting probability (or the simpler probability of finding both walkers at the origin) can be used to infer the amount of lattice imperfections, described by the percolation parameter p. Due to the monotonicity observed in Figures 6b and 7b, such strategy works better in a regime of slow varying imperfections, i.e. of statical percolation.When we considered the average distance among the detected particles, we found different behaviors for bosons and fermions, since the latter tend to anti-bunch when the lattice structure allow them to do so.Eventually we considered the spread of the walk, focusing on the dependence upon the second walker more than on the decoherence introduced by the percolation.Our analysis shows that for many cases of interest, i.e. when the two initial single-particle states are orthogonal, the spread is not affected by the quantum statistics of the walkers, being always equal to its single-particle value.
This investigation considered for the first time the effects of possible missing links (that could happen, for instance, in an experimental setting) on the features of walks where the quantum nature of the involved particles is relevant.Due to the importance of quantum walks for the purposes of quantum computation and simulation, and in particular for the role that many walkers will have in their future exploitation, these results pave the way for a more complete understanding of these models.
starting positions of the walkers, they will never be at the same place at the same time.Since there is no distance interaction, the two particles will evolve independently and their statistics will have no effect.
More generally Equations (A4) and (A5) do not exclude the possibility of having particular relations among the coefficients λ j , γ j and δ j for which the equality holds.However, such conditions would not be simply expressed in terms of the initial |ψ k , k = 1, 2 as required, since such initial states are not trivially obtained from the expansions (A2), (A3) of their evolutions.

FIG. 1 .
FIG.1.Probability distribution for a quantum walk on the line after N = 300 steps, without missing bonds (a) and with dynamical (b) or statical (c) percolation (different ranges for the position values were chosen in order to increase the readibility of the plots).In order to have a symmetric distribution we started from the initial state (2).In the last two plots, we considered lattices characterized by a percolation parameter p = 0.75 and we averaged over 500 different outputs.

FIG. 3 .
FIG. 3. P2(i, j) for a quantum walk of two walkers starting from the same position with coin state |ψS (15), measured after 15 steps.Panels (a) and (b) correspond respectively to a dynamical and statical percolation regime, with p = 0.9 and a number of averages A = 1000.

FIG. 6 .
FIG. 5. Averaged output probability distributions for a two-particle 15-step quantum walk in the presence of dynamical percolation (p = 0.75, A = 2000).We considered boson-like (a), fermion-like (b) and classically indistinguishable (c) combinations of two single-particle states |ϕ± that spread symmetrically from the origin.In Panel (d) we plotted their averaged single-particle behavior, while the differences among previous scenarios are shown in Panels (e), (f ), (g).

FIG. 7 .
FIG. 7. (Color online, legend with the same vertical ordering of the data) Average probability of finding both walker in the origin, as a function of the percolation parameter p, for a 15-step quantum walk starting from the origin with three different coin states.(a) and (b) correspond respectively to dynamical and statical percolation regimes.The average is calculated over 5000 different lattice realizations and the errors are evaluated as the standard errors on the mean values obtained.

FIG. 8 .
FIG. 8. (Color online, legend with the same vertical ordering of the data) Average output distance between two particles starting from the same point, after 15 steps, for both regimes of percolation as a function of p. Three initial coin states (|φ+ , |ψ− and |ψS ) are considered and the result is averaged over 500 different lattice sequences.