1. INTRODUCTION

Quantum walk (QW) is the quantum equivalent of a random walk, which benefits from quantum features such as quantum superposition, interference, and entanglement [1–4]. In contrast with a classical walker, the quantum walker spreads quadratically faster in position space [1–5]. This notion brings up a motivation to introduce algorithms for quantum computers that solve the problem exponentially faster than the best classical algorithm [6,7]. On the other hand, QW provides a powerful model to describe energy transport phenomena in heterogeneous systems, either biological, in the case of photosynthesis [8,9], or solid-state ones, in the case of Luttinger liquids [10]. In addition, nonclassical features play a substantial role in the dynamics of a quantum walker as a sign for quantum coherence effects in biological systems [11,12]. Finally, it is worth noting that the QW model is applicable to simulate a wide range of quantum phenomena such as topological phases [13,14], neutrino oscillations [15,16], and relativistic quantum dynamics [17–19].

A relevant quantity, useful to characterize the walk, is the mean square displacement (MSD) of the walker in the absence of bias. The linear growth of the MSD as a function of the evolution time has become the universal identifier of what is known as normal transport. Any stochastic process that does not follow a linear growth trend with time is called anomalous. In particular, processes characterized by a superlinear growth of the MSD are usually addressed as superdiffusive [20–22]. There are several cases in which superdiffusion settles in transport or propagation processes, such as complex biological environments [8,9,23], chaotic Hamiltonian systems [24,25], transport in disordered systems [26–28], quantum optical systems [29], and single-molecule spectroscopy [30,31]. On the other hand, numerical evidence has been noted for the existence of subdiffusive transport [32–36]. Also, an extremely slow process of matter-wave spreading, subdiffusion transport, has been experimentally implemented in Bose–Einstein condensates [37].

In recent years, there has been substantial interest in formulating QW models that can exhibit different transport behaviors with respect to the typical one, consisting in a MSD growing quadratically with time. The spreading behavior of quantum walkers can be modified by suitably tuning the evolution of a quantum walker through various types of disorder [38–40] and decoherence effects [41]. Due to the latter ones, it has been shown that the ballistic growth of the variance changes to a superdiffusive one, reaching the diffusive spread [40]. Moreover, by means of the same techniques, the subdiffusive region, between diffusive and Anderson localization regime [42], can be exploited [43]. On the contrary, there are few reports in which a QW in presence of evolution nonlinearities avoids complete trapping in a finite region of the lattice. In these cases, the spreading has to slow down to a subdiffusive case, but it does not converge, generating a phenomenon known as delocalization of the wave packet [44].

In the context of quantum probing strategies [45–49], one may characterize properties of a complex environment, such as fragile biological samples, by a small and controllable quantum probe that interacts with the bigger complex system. These features may be conveyed through error analyses of the quantum probe, having a natural connection with the theory of quantum parameter estimation [50–55], where unknown parameters are inferred via repeated measurements on the system of interest. The quantum Fisher information (QFI), which depends on the kind of probe–environment interaction, is used to measure this error through the quantum Cramér–Rao bound.

Despite the high number of studies about transport features in a QW, as far as we know, investigations in a quantum probing fashion have remained elusive. The idea behind this work is to infer how much information about the features of the QW network can be extracted using quantum estimation merits from the characteristics of the quantum walker evolution. In quantum metrology, the extractable information about an unknown parameter, such as a phase $\phi$, is usually given by the QFI, which is also linked to the measurement accuracy of the estimation strategy. We take QFI for granted to investigate the different transport regimes of information due to the properties of the quantum network. From another perspective, the growth pattern of QFI is a faithful indicator of describing defects and perturbations occurring in complex networks, both classical and quantum. We find that the disorder pattern plays a significant role in the spreading pattern of quantum information. The growth trend of QFI allows one to characterize the anomalous and normal spreading pattern and also Anderson localization of the walker.

2. DISCRETE-TIME QW-BASED QUANTUM PROBE

A. State-of-Art Quantum Probing Scheme

Due to the sensitivity of the probe to the perturbations induced by the environment, quantum probes may effectively infer characteristics of the environment. One of the possible strategies is to send a quantum probe through a Mach–Zehnder-like interferometer (MZI). As illustrated in Fig. 1, an MZI consists of a lower arm, the so-called reference path where the probe travels freely, and an upper arm, where the probe acquires a phase shift $\phi$ due to environmental action. Finally, at the probe, a suitable procedure to estimate environmental properties of interest is measured. Typically, the parameter $\phi$ is encoded in a unitary operator ${U_L}(\phi)$. Here, the phase shifter is length-dependent so that the probe is capable of capturing the coding process with respect to the phase shift length. We aim to analyze the phase sensitivity in terms of controllable phase shift length.

 

Fig. 1. Typical representation of a quantum probing protocol using a MZI. A phase difference $\phi$ is applied between two paths, which is acquired by the probe thanks to an adjustable path length $L$.

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In parameter estimation schemes [50–54], a parameter of interest, like $\phi$, is estimated by processing the data by means of estimator observable outcomes $\hat O = \{{o_1},{o_2}, \ldots ,{o_n}\}$. The measurement sensitivity of the phase parameter is given by the Cramér–Rao inequality $\delta \phi \ge 1/\sqrt {M{{\cal F}_\phi}}$, where $\delta \phi$ is the mean square error in the measure of parameter $\phi$, $M$ is an asymptotically large number of independent measurements, and ${{\cal F}_\phi}$ is the QFI given by

In this equation, $p(\phi |o) = {\rm Tr}[{\hat M_o}\rho]$ is the conditional probability distribution that may give information about parameter $\phi$ a result of the measurement outcome, where $\hat M$ is a set of positive operator-valued measurements (POVMs) [54] and $\rho$ is the density operator. The optimal estimator is the one that saturates the Cramér–Rao inequality and maximizes the QFI. According to the symmetric logarithmic derivative (SLD) approach, the QFI is ${{\cal F}_\phi} = {\rm Tr}[L_\phi ^2\rho]$, where ${L_\phi}$ is the SLD operator satisfying the equation ${d}\rho /{d}\phi = \{L,\rho \} /2$, with $\{\cdot , \cdot \}$ indicating the anticommutator. For a pure state ${\rho ^2} = \rho$, the SLD operator reduces to ${L_\phi} = 2{d}\rho /{d}\phi$. It is worth mentioning that QFI provides an upper bound to the sensitivity, which is independent of the choice of measurement. Also, the set of optimal POVMs is given by the eigenstates of the SLD operator, ${L_\phi}$. However, in operational conditions, ${L_\phi}$ may not be the optimal observable that one tends to perform in the measurement setup, so optimization strategies, such as maximum likelihood method, have been suggested to supply an optimal estimator that saturates [54]. Within the scope of this paper, we focus on the probing strategy in a discrete QW network to address maximum extractable information spreading pattern using QFI.

B. Discrete QW as a Quantum Probe

Here, we design a quantum probing protocol to address the spreading behavior of a QW and the extractable information that the walker can assess, examining the problem by quantum estimation strategy. The discrete-time QW process is a multipath interferometer setup, equivalent to a chain of MZIs. A typical representation of a QW network is displayed in Fig. 2 in four steps. Each line linking two different vertical bars corresponds to a mode along which the walker can travel. As can be seen in Fig. 2, the rightmost path is the reference path along which the quantum probe (walker) travels freely. The length-dependent phase shifter is implemented by the sequence of (discrete) sites that the quantum walkers pass through along the multipath dynamics. A measurement strategy is finally employed after each step to perform the optimal estimator and extract the information spreading about the phase. It is important to note that the probing continues until the relative error converges to the lowest value for the 99.9% confidence interval. Here, we numerically address the achievable bound QFI that one may tend to measure by a suitable and practical measurement strategy.

 

Fig. 2. Typical representation of a quantum probing protocol using disordered QW dynamics. A phase difference $\phi$ between coin (internal) states of the walker is applied after each step, including possible fluctuations $\Delta {\phi ^\prime _{i,j}}$ depending on step $i$ (time) and position $j$ within the circuit. This way, static and dynamic disorder can affect the QW process.

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The dynamics of a quantum walker on a one-dimensional lattice is defined on the joint Hilbert space ${\cal H} = {{\cal H}_{\textbf{p}}} \otimes {{\cal H}_{\textbf{c}}}$ of position (${{\cal H}_{\textbf{p}}}$) and coin (${{\cal H}_{\textbf{c}}}$) subspaces of the walker [2,3]. The coin basis states set is ${{\cal B}_{\textbf{c}}} = \{\uparrow , \downarrow \}$, which can be seen as an internal degree of freedom of the walker, while the position space is spanned by the discrete set ${\{|x\rangle _{\textbf{p}}}\}$, which represents the sites of the lattice. The evolution of the quantum walker is determined by the coin operator $\hat {\cal C} = \frac{1}{{\sqrt 2}}(| \uparrow {\rangle _{\textbf{c}}}{{\langle \uparrow {|_{\textbf{c}}} + | \uparrow \rangle}_{\textbf{c}}}{{\langle \downarrow {|_{\textbf{c}}} + | \downarrow \rangle}_{\textbf{c}}}{{\langle \uparrow {|_{\textbf{c}}} - | \downarrow \rangle}_{\textbf{c}}}\langle \downarrow {|_{\textbf{c}}})$ and the shift operator

The main aim of quantum estimation strategy is to evaluate the maximum extractable knowledge about an unknown parameter, which we call $\phi$, from repeated measurements on the probe. One can numerically obtain the QFI ${{\cal F}_\phi} = 4{\rm Tr} [L_\phi ^2\rho]$ using the walker state at step $t$, $|{\Psi _t}\rangle = \hat U(\phi)|{\Psi _{t - 1}}\rangle$, and its derivative with respect to parameter $\phi$, which is

Similar to earlier suggestions [57,58], the QW circuit is a multipath interferometric arrangement, where an input state is split into many beams by a chain of beam splitters. In the measurement stage of a multipath interferometer [57,58], output modes are recombined and then are detected by $M$ identical photodetectors to construct the probability distribution of detector $i$. This allows one to obtain the joint-probability distribution $p(\phi |x)$ using a POVM set given by

3. APPLICATION: SINGLE PROBE AND PAIR OF PROBES

In the following, we study the case of both a single probe and a pair of probes sent into the QW network under the effect of static and dynamical disorder. We limit our analysis to the case in which the random fluctuations $\Delta \phi ^\prime (t,x)$ can be only 0 or $\pi$. Afterwards, the degree of disorder, $p$, is defined as the percentage of random phases that the walker experiences during the evolution. This simply means that a $p$ percentage of the random fluctuations $\Delta \phi ^\prime (t,x)$ is selected to have $\pi$ value and to be randomly distributed in time and position. By iterating over enough random phase samples, it is possible to numerically calculate the QFI in the presence of a given percentage of randomness $p$. A key point in the particle evolution is the type of disorder: static disorder, where the phase fluctuations are frozen in time, or dynamic disorder, where the imposed phase can change in both time and space, as happens in the $p$-diluted model [40]. The degree of disorder $p$ is directly connected to the time evolution of the probability distribution of the quantum walker (see Appendix A for the probability distribution of a quantum walker corresponding to both cases of static and dynamic disorder).

A. Single Probe

Let us first consider the simplest instance, where a particle starts the QW in the position $|0{\rangle _{\textbf{p}}}$ with the $| \uparrow {\rangle _{\textbf{c}}}$ coin state, that is, $|{\Psi _0}\rangle = |0{\rangle _{\textbf{p}}} \otimes | \uparrow {\rangle _{\textbf{c}}}$. For both static and dynamic disorder, we simulate the behavior of the average QFI for different degrees of disorder $p$. For each value of $p$, the simulation is performed by averaging over ${10^4}$ different phase maps. The average QFI is plotted in Fig. 3 as a function of the step number $t$. As a result of the power-law fitting data, we find that the average QFI is ${\cal F} \propto {t^\alpha}$, where the range of $\alpha$ is $0 \le \alpha \le 2$ and $1 \le \alpha \le 2$ for the static disorder and the dynamic disorder, respectively. Without any disorder ($p = 0$), the QFI of the QW grows quadratically in time ${\cal F} \propto {t^2}$, analogously with a ballistic transport pattern, as displayed in Fig. 3(a). Also, this means that the upper-bound phase variance is proportional to the inverse of the number of steps $\delta \phi \propto {t^{- 1}}$. Similar to earlier studies, the phase sensitivity decreases with the number of available paths [57]. For static disorder [Fig. 3(I)], the superdiffusive ($\alpha \gt 1$) to subdiffusive ($\alpha \lt 1$) transition is reachable by increasing the value of disorder $p$ out of 50 steps. Therefore, the way information spreads in the quantum network can be determined as a result of the disorder strength. For dynamic disorder, we plot the average QFI in Fig. 3(II). Here, we use the QFI to probe the transition from the ballistic regime with $p = 0$ [Fig. 3(a)], superdiffusive with $p = 0.1$ [Fig. 3(b)] and diffusive one with maximum degree of disorder $p = 1$ [Fig. 3(c)], analogous to the case of a classical probe. This indicates that the output information, which is inferred trough measurements performed exclusively on the probe, shows a superdiffusive to classical transition in transport pattern. The observed fluctuation in the QFI value is due to the limited number of iterations that can be realized for each step number. Similar to the variance of the position operator of the quantum walker [40,41,43] (see also Appendix A), QFI provides a simple measure to quantify the transport pattern of the walker.

 

Fig. 3. QFI for a quantum walker. Average QFI ${\cal F}$ of a quantum walker versus the number of steps $t$ for static (top panels) and dynamic (bottom panels) disorder, at different degrees of disorder. (a) $p = 0$, (b) $p = 0.1$, (c) $p = 1$. In all plots, the initial input is $|{\Psi _0}\rangle = |0{\rangle _{\textbf{p}}} \otimes | \uparrow {\rangle _{\textbf{c}}}$. The simulated data (blue dots) are plotted along with a power-law fitting function (red line).

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It is worth mentioning that the probing pattern depends on the number of steps that a walker takes. In the static disorder case, we find out that the QFI eventually tends to a given value. This is a clear signature of Anderson localization given by the average upper bound of QFI, as clearly seen in Fig. 4(a), where the growth trend declines until average QFI reaches a maximum value. As an additional clarification, the QFI is fitted with the power function ${\cal F} \propto {t^{\alpha (t)}}$, where $\alpha (t)$ is the step-dependent coefficient in a nonlinear fitting process. In Fig. 4(b), $\alpha (t)$ is plotted by increasing the step number of the QW process. We observe how the growth trend depends on the step number $t$, and how it significantly declines for higher $t$. This property shows that information stops spreading within the quantum network as an indicator of particle localization.

 

Fig. 4. (a) Logarithmic scale of average QFI ${\cal F}$ of a quantum walker and its fitted curve ${\cal F} \propto {t^\alpha}$ as a function of the number of steps $t$ for complete static disordered $p = 1$; (b) step-dependent coefficient $\alpha (t)$ and its fitted curve as a function of the number of steps $t$. The simulated data (blue dots) are plotted along with a power-law fitting function (red line).

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Fig. 5. QFI for two quantum walkers. Average QFI ${\cal F}$ of a quantum walker versus the number of steps $t$ for (a) ordered case ($p = 0$), (b) complete static disorder ($p = 1$), and (c) complete dynamic disorder ($p = 1$). In all plots, the blue circles and red squares represent, respectively, indistinguishable and distinguishable two-particle inputs. The simulated data (blue dots) are plotted along with a power-law fitting function (red line).

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B. Pair of Probes

To enrich the physics of the phenomenon, we now consider two input quantum walkers, either distinguishable or indistinguishable, in position $|0{\rangle _{\textbf{p}}}$ with initial opposite coin states. When two identical particles arrive at a beam splitter with temporal delay, one in each input port, they are distinguishable in the eyes of the beam splitter. For distinguishable particles (walkers) named 1 and 2, the input state is separable, that is,

4. CONCLUSION

In this work, we have proposed a quantum probing protocol using the QW process to infer information about defects and perturbations occurring in both quantum and classical networks. This goal has been achieved by applying quantum metrology techniques to the QW process. We have exploited QFI to describe extractable information concerning an unknown phase $\phi$ that the quantum walker acquires at each step, plus random fluctuations, through the QW. Even though the framework is general, we have studied coherent static and dynamic disorder in the QW to describe the transport pattern of information about the unknown parameter $\phi$. We have found that different disorder regimes, corresponding to a disorder percentage $p$ in the QW process, lead to different spreading patterns, including ballistic, superdiffusive, classical, subdiffusive regimes, and Anderson localization. QFI, as a spreading pattern indicator, is independent of the number of quantum walkers, in contrast with the position variance dimension, which grows according to the particle number [40]. Ultimately, our results show that QW can play the role of a readout device of information about the internal characteristics of complex networks.

Our results provide a general characterization of the spreading pattern using the quantum probing scheme of QFI bound. One may put the complex system in an interferometric scheme (like an MZI) to examine how a quantum probe interacts with the complex system. Also, we have applied the disordered QW dynamics to explain the system–probe interaction. In Refs. [40,66], the control over the parameter $p$ by performing a selective random disorder for photons in a bulk optics scheme has been experimentally demonstrated. Indeed, it is challenging to implement an optimal estimation strategy with accuracy associated with QFI bound [54]. The quantum Cramér–Rao bound needs available prior knowledge to adopt a local approach and the large number of experimental observations. Also, the SLD operator itself may not represent the optimal observable to be measured. In fact, projection over eigenstates of the SLD operator determines the POVM, and not the estimator. If it is practically possible, one may try to perform the optimal measurement, which are projections over the eigenstates of the SLD operator. Moreover, one can compute the classical Fisher information (FI) by exploring all possible POVMs for a fixed input state. If the maximized FI reaches the QFI obtainable, we can conclude that the POVMs are optimal. This strategy is largely determined by how successful the POVMs can be implemented in each scheme. To saturate the Cramér–Rao bound, one may pursue the classical postprocessing of data using maximum likelihood, which is known to provide an asymptotically efficient estimator [50]. In addition, the adaptive measurement strategy is employed to decrease the number of measurements [67]. Furthermore, machine learning might be a powerful tool to obtain the goal of interest [68,69].

This protocol is on track to be experimentally implemented, and several platforms are likely candidates. An example of a potential candidate is a cascade of balanced beam splitters arranged in a network of MZIs [38]. In this scheme, phase shifters are involved in implementing the randomized disorder to a certain degree. This experiment faces many challenges, but the most challenging one is probably utilizing phase-stable POVMs (possibly an optimal set or performing an optimization protocol, such as the maximum-likelihood method, to reach the quantum Cramér–Rao bound) along multiple paths. Indeed, the experiment also depends on measurement components that have low noise and phase stability. Our work is expected to motivate further studies, such as studying different types of disorder like the spatiotemporal correlated disorder [70], probing the QW on a high-dimensional graph network [71], detecting memory effects of the environment on quantum walkers [72], and assessing material characteristics of complex random media [73].

APPENDIX A: FURTHER ANALYSIS

In this Appendix, we address the probability distribution and the variance displacement of a single walker under the effect of different degrees of static and dynamic disorders. Let us consider a given initial state,

(A1)$$|{\Psi _0}\rangle = \frac{1}{{\sqrt 2}}|0{\rangle _{\textbf{p}}} \otimes \left({| \uparrow {\rangle _{\textbf{c}}} + | \downarrow {\rangle _{\textbf{c}}}} \right),$$

to explore the probability distribution of a quantum walker in the presence of both static and dynamic disorder. We limit our analysis in the case that a $p$ percentage of phase fluctuations are selected out of 0 and $\pi$. We show the density plots of the evolution of single particles in Fig. 6, in terms of step number and walker position. Here, both static disorder [Fig. 6 (left panels)] and dynamic disorder [Fig. 6 (right panels)] is realized with different degrees of disorder $p$. (a) $p = 0$, which corresponds to a standard ordered QW; (b) $p = 0.1$; (c) $p = 1$, a completely disordered QW. The horizontal axis denotes different positions that the walker can reach while the vertical one represents the step number $t$, increasing from top to bottom. We averaged the probability distributions over 10,000 random phase-map realizations. As can be seen, the particle follows different transport patterns affected by static [Fig. 6 (left panels)] or dynamic disorder [Fig. 6 (right panels)]. Also, we show how the degree of disorder would affect the probability distribution of the quantum walker. It seems that the probability of finding a particle in the center rises by increasing the degree of disorder $p$.

 

Fig. 6. Probability distribution. Average probability distribution of an unbounded discrete QW on the line as a function of step number and position, for different values of disorder. (a) $p = 0$, (b) $p = 0.1$, (c) $p = 1$, by averaging over ${10^4}$ phase maps for both static (left panels) and dynamic (right panels) disorder. In all cases, the initial input state is $|{\Psi _0}\rangle = |0{\rangle _p} \otimes ({| \uparrow {\rangle _c} + | \downarrow {\rangle _c}})/\sqrt 2$.

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Fig. 7. Position variance. Position variance ${\sigma ^2}$ of a quantum walker on the line in terms of the step number $t$ for different values of disorder. (a) $p = 0$, (b) $p = 0.1$, and (c) $p = 1$, by averaging over 10,000 phase maps for both static disorder (top panels) and dynamic disorder (bottom panels). In all cases the initial input is $|{\Psi _0}\rangle = |0{\rangle _p} \otimes ({| \uparrow {\rangle _c} + | \downarrow {\rangle _c}})/\sqrt 2$. The simulated data (blue dots) are plotted along with a power-law fitting function (red line).

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In the QW problem, the operator $\hat X$ that measures the walker position is the operator of interest, since the variance of the position operator provides a simple measure to quantify the spread of the walker [41],

(A2)$${\sigma ^2}(\hat X) = \langle {\hat X^2}\rangle - {\langle \hat X\rangle ^2}.$$

This measure has been proven to be particularly useful to compare the effects of different kinds of disorder on the spreading pattern of the walker(s). Generally, the variance of the position operator is given by ${\sigma ^2}(\hat X) \propto {t^\alpha}$, where range $\alpha \gt 1$ is called superdiffusive and $\alpha \lt 1$ subdiffusive. For instance, an ordered QW presents a ballistic spread with ${\sigma ^2}(\hat X) \propto {t^2}$, while the classical random walk is diffusive with ${\sigma ^2}(\hat X) \propto t$.

As an example, we study the spreading behavior of a walker for the given input state $|{\Psi _0}\rangle$ of Eq. (A1) with three different values of disorder. The position variance of a quantum walker generally depends on the type of the disorder, the degrees of disorder ($p$), and the number of steps ($t$). The variance of a single walker is plotted versus the number of steps for both static disorder [Fig. 6 (top panels)] and dynamic disorder [Fig. 6 (bottom panels)]. For an ordered QW, the walker shows a ballistic pattern, displayed in Fig. 7(a). In static disorder, a superdiffusive pattern appears because of the increasing value of disorder degree to $p = 0.1$ [see Fig. 7 (top panel (b))]. Interestingly, in Fig. 7 (top panel (c)), the subdiffusive pattern also can be simulated by increasing the value of the disorder degree to the maximum. In the dynamical disorder case, the quantum walker exhibits a transition from ballistic to superdiffusive behavior, and then a classical spreading pattern by increasing the degree of disorder to $p = 1$.

Disclosures

The authors declare no conflicts of interest.

Data Availability

All data generated or analyzed for this study are included in the present study.

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