Quantum walks on a ring as a versatile single-parameter sensor

1 Introduction

The concept of quantum walks (QWs) was originally introduced as the quantum counterpart of classical random walks (Aharonov et al., 1993; Venegas-Andraca, 2012). Since then, QWs have emerged as a versatile tool across several quantum technologies, playing a significant role in quantum computing (Qiang et al., 2024), quantum algorithms (Wong, 2015; Wong et al., 2016; Shenvi et al., 2003), and quantum key distribution (Vlachou et al., 2018). Their use extends beyond quantum information science, finding concrete applications in fields such as protein biology (Mohseni et al., 2008; Hoyer et al., 2010) and medical diagnostics (Dubovitskii et al., 2025).

In continuous-time quantum walks (CTQWs), a particle is confined to a set of discrete locations, identified with the vertices of a graph, and may tunnel from one site to another at any time with a finite amplitude. In the ideal case, dynamics is unitary and governed by the system Hamiltonian, often taken as H = − ζ A, where ζ is the tunneling rate and A is the adjacency matrix of the graph. For a given underlying graph structure, an entire class of Hamiltonians can be introduced by varying the tunneling amplitude or the chiral phases, with the only constraint being Hermiticity (Zimborás et al., 2013; Frigerio et al., 2021; Frigerio et al., 2022). In this article, we consider CTQWs on cycle graphs. These structures have been extensively studied for their simplicity and symmetry, and they are also amenable to analytical treatment, revealing peculiar properties in contexts such as transport (Finocchiaro et al., 2025). By exploiting additional degrees of freedom, nontrivial effects are introduced into the dynamics, typically increasing the nonclassicality of the dynamics (Gualtieri et al., 2020) and enhancing performance in different tasks (Wang et al., 2022; Candeloro et al., 2020a; Candeloro et al., 2020b; Caruso, 2014; Apollaro et al., 2016; Cattaneo et al., 2018; Candeloro et al., 2020b).

In this framework, precise characterization of the Hamiltonian parameters is crucial for achieving robust quantum protocols. The present work aims to analyze the bounds to precision in estimating those parameters and, more generally, to investigate the performance of quantum walks on ring graphs as sensors of external perturbations. The sensing performance of the system is assessed by evaluating the quantum Fisher information (QFI), which, through the quantum Cramér–Rao bound, provides the ultimate precision of any unbiased estimator of the quantity of interest. Unlike previous studies focusing on transport or algorithmic applications, here we systematically characterize the metrological use of CTQWs on rings, including the interplay between electric- and magnetic-type perturbations. While, in general, there is no guarantee that the ultimate bound to precision is achievable by realistic measurements, we prove that this is the case at least for energy measurement in thermometry and position measurement in the estimation of electric-like perturbations.

In Sections 2 and 3, we analyze the estimation of a single parameter, consisting of a localized perturbation caused by a longitudinal field (e.g., an electric field) or a phase factor due to the flux of an axial field (hereafter referred to as a magnetic perturbation). We analyze both static estimation strategies, in which the parameters are encoded onto the ground state of the perturbed system, and dynamical ones, in which encoding is achieved through Hamiltonian evolution. In Section 4, we use these results as a benchmark to determine whether the presence of a finite value of one field can modify, and possibly enhance, the estimation of the other. We identify the features that make the system a versatile tool for metrology, and determine the optimal estimation regimes. Section 5 focuses on the multiparameter case, assessing the feasibility of joint estimation of both fields. Here, we show that the resulting statistical model is nearly sloppy, indicating dependence only on specific parameter combinations and an overall poor performance as a multiparameter sensor. Finally, Section 6 analyzes quantum walks on ring graphs as thermometers, focusing on whether and how a magnetic-like perturbation can improve the sensitivity of the system in estimating low temperatures. Section 7 closes the article with some concluding remarks, and Supplementary Appendices A,B provide some details on the performance of position measurement and on the effect of noise and decoherence, respectively.

2 Static strategies for single-parameter estimation

A graph is defined as a pair G = ( V , E ), of which V is a set of sites, namely, the possible spatial positions of the walker, and E is the set of edges, indicating the possible transitions between sites (Venegas-Andraca, 2012). The systems may be described in a N-dimensional Hilbert space H N with a canonical basis | j 〉, j = 0 , … , N − 1 denoting the vertices of the graph, that is, the possible spatial positions of the walker. Let us now consider a QW where the unperturbed Hamiltonian is the adjacency matrix H = − A. We also set ℏ and the hopping rate ζ to unit. H is a circulant matrix for any value of N, and thus it is diagonalized by a discrete Fourier transform, with eigenvalues and eigenvectors given by Equations 1, 2: where n = 0 … N − 1. The ground state is always unique, while the multiplicity of the highest energy state depends on the parity of N. The ground state is

e n = − 2 ⁡ cos 2 π n N . (2)

2.1 Electric-like perturbation

We now introduce a perturbation, consisting of a site-localized projector of amplitude λ (also referred to as the electric parameter), chosen, without loss of generality, to be on site | 0 〉. The perturbed Hamiltonian is now:where the perturbation may be interpreted as the action of a localized electric field, changing the energy landscape (Anderson, 1958) or the oracle in a Grover-like search protocol (Grover, 1996; Farhi and Gutmann, 1998; Childs and Goldstone, 2004). In this setting, negative and positive values of lambda correspond to the nature of the local perturbation, which may either increase or decrease the local energy, acting as a well or a barrier. One possible realization of such a system is presented by Benedetti et al. (2021), where the same Hamiltonian in Equation 4 has been realized experimentally in a two-dimensional waveguide array. Here we assume that after switching on the perturbation, the system is left to relax to the new ground state | ϵ 0 〉, on which measurements are performed, where

H λ = − A + λ | 0 〉 〈 0 | , (4)

H λ | ϵ n 〉 = ϵ n | ϵ n 〉 . (5)

The QFI for a parameter α encoded in a pure state | ψ α 〉 is given byand determines the ultimate precision achievable in the estimation of α according to the Cramér–Rao bound shown as Equation 7:where M is the number of repeated measurements, α ̂ is any unbiased estimator of α, and the inequality can be saturated by measuring a suitable observable.

F α = 4 〈 ∂ α ψ α | ∂ α ψ α 〉 − 〈 ∂ α ψ α | ψ α 〉 2 . (6)

Var α ̂ ≥ 1 M F α , (7)

For N = 3 and N = 4, the ground state of the perturbed Hamiltonian in Equation 4 may be analytically obtained. For larger N, we calculate the QFI by numerically solving Equation 5 or using the alternative expressionFor N = 3, we have Equation 9:

F α = 4 ∑ n ≠ 0 | ⟨ ϵ n | ∂ α H α | ϵ 0 ⟩ | 2 ϵ 0 − ϵ n 2 . (8)

F λ = 8 λ 2 + 2 λ + 9 2 . (9)

In Figure 1, we show the QFI for rings up to N = 10. The results show a sharp dependence of the QFI on λ in the regime λ ≃ 0, while the information contained in the ground state rapidly drops when | λ | is away from 0. For each system size N, there is an optimal value λ opt of the perturbation where the QFI reaches its maximum. The peak value increases with N, whereas λ opt approaches 0 as the dimension grows. Notice, however, that the value of λ opt may be tuned by a different choice of the tunneling rate in the unperturbed Hamiltonian. As a matter of fact, by setting H = − ζ A + λ | 0 〉 〈 0 |, the QFI becomes a function of the ratio λ / ζ, making it possible to tune the value of λ opt to an arbitrary non-zero value.

Concerning the scaling of the peak value of the QFI F ( λ opt ) with the system size, we notice that for increasing N, the value of λ opt is vanishing, making it convenient to exploit standard perturbation theory, expanding the ground state as Equations 10, 11.

The derivative | ∂ λ ϵ 0 〉 therefore contains no linear terms in the vicinity of λ = 0. Using the QFI formula for pure states, it is easy to notice that the QFI itself contains no linear terms: Equation 12 is an approximation that holds for sufficiently large N. To find an analytical expression, one can expand the QFI in terms of the eigenvalues and eigenvectors of the unperturbed system, arriving at Equation 13:

F λ opt = N ≫ 1 F 0 + O λ 2 . (12)

F 0 = 4 ⟨ ∂ λ ϵ 0 | ∂ λ ϵ 0 ⟩ + O λ 2 = ∑ k = 1 N − 1 1 sin 4 π k N ≃ 4 45 N 2 + 8 9 + O 1 N . (13)

The emerging optimal strategy is therefore the following: if the true value of λ is large, one must set the tunneling rate ζ such to tune λ / ζ toward λ opt. If the true value is small, then the optimal choice is to use a large ring. We also notice that in this particular situation, a position measurement saturates the QFI, as further discussed in Supplementary Appendix C.

2.2 Magnetic-like perturbation

Let us now consider a perturbation which acts by adding a phase φ / N to every link in the graph, thus making the tunneling amplitudes complex. This mimics the effects of an axial field on the dynamics of the walker (Delgado et al., 2007) and will be referred to as a magnetic perturbation, while φ will be called the magnetic parameter accordingly. The Hamiltonian may be written as Equations 14, 15:

H φ = − ∑ j = 0 N − 1 e − i φ / N | j + 1 〉 〈 j | + h . c , (14)

H φ | ε n 〉 = ε n | ε n 〉 . (15)

The above Hamiltonian can be realized with cold atoms (Aidelsburger et al., 2013) in the single-excitation regime, while with waveguides, this would be considerably more challenging [notice, however, that this task would be straightforward in the case of discrete-time quantum walks (Crespi et al., 2013)].

The Hamiltonian in Equation 14 is still circulant, and therefore, the eigenvectors are unchanged, while the eigenvalues are modified to

ε n = − 2 ⁡ cos φ N − 2 π n N . (16)

This result carries a direct implication: because the eigenvectors are independent of the parameter φ, the related QFI when measuring the ground state is identically zero. No information about the phase can be extracted from measurements performed on stationary states.

3 Dynamical strategies for single-parameter estimation

Here we consider strategies in which the parameter of interest α is encoded onto an initial state ρ 0 (which does not depend on α) by the action of a unitary operator, yielding ρ ( α ) = U ( α ) ρ 0 U † ( α ). This setting could be implemented by cooling down the unperturbed system to its ground state, which is known for all N, as in Equation 3. Having reached the ground state, the perturbation is then turned on, letting the system evolve under the modified Hamiltonian, U ( α ) = exp − i H α t. In this situation, the quantum Fisher information can be obtained by (Liu et al., 2015; Pizio et al., 2019):where Var G α ( t ) = ⟨ G α ( t ) 2 ⟩ − ⟨ G α ( t ) ⟩ 2 denotes the variance of the effective encoding generator, given by Equation 18:

F α = 4 Var G α t , (17)

G α t = − ∫ 0 t U † s ∂ α H U s d s . (18)

3.1 Electric-like perturbation

Let us now evaluate the performance of the dynamical procedure in the estimation of the electric parameter λ, as imposed by the Hamiltonian in Equation 4. To warm up, let us start from the case N = 3, where the dynamics is confined to a two-dimensional subspace. The time evolution can be obtained analytically, and upon exploiting Equation 17 with the Hamiltonian H λ in Equation 4, the QFI is given bywhere the energy gap E g is given by Equation 19 (see Supplementary Appendix A for full derivation)

F λ , t = 4 E g 6 2 λ + 1 2 E g t − sin E g t 2 + 2 E g 2 1 − cos E g t 2 + 1 4 λ + 1 2 E g t + 8 ⁡ sin E g t 2 − 1 36 λ + 1 λ + 9 E g t − 8 λ ⁡ sin E g t 2 ,

E g = λ + 1 2 + 8 . (19)

The behavior of F ( λ ) in the weak-field limit ( λ → 0 ) is oscillating:whereas for higher values of the field λ, the oscillatory contribution is smoothed out. In the strong-field limit ( λ ≫ 1 ), a quadratic behavior may be observed:

F λ = 32 81 sin 2 3 t 2 , (20)

F λ = 8 9 t 2 . (21)

This result is optimal as it yields a QFI of the order F ( λ ) ∝ t 2, which saturates the Heisenberg limit for unitary parameter encoding (Zhou et al., 2018).

For larger values of N, Equation 17 may still be used to evaluate the QFI upon numerical diagonalization of the perturbed Hamiltonian. The results are shown in Figure 2, where we plot the QFI F ( λ ) as a function of the interaction time for different sizes of the ring graph. As shown in the plot, the QFI grows quadratically with multiperiodic oscillations. The best performance is obtained for N = 3.

3.2 Magnetic-like perturbation

As a matter of fact, the dynamical estimation of magnetic perturbations also faces the same issue as the static case: the perturbation leaves the eigenstates unchanged. Consequently, the parameter φ appears in the time evolution of the unperturbed ground state only as a global phase factor, resulting in a vanishing QFI.

4 Assisted estimation

To further enhance the estimation of λ, and to enable an estimation of φ, we analyze the regime in which the system is simultaneously perturbed by both fields. This configuration will be referred to as assisted estimation, as one of the parameters is not introduced for its own metrological study but only as a tool to improve the estimation of the other. Hence, we assume that one of the parameters is known and tunable, that is, may be set to a desired value. More specifically, the idea behind assisted estimation is that one of the parameters is first estimated assuming no or little knowledge about the other, and then the information obtained in the first step is used to assist the estimation of the other. More generally, the following analysis provides tools to optimize estimation of one of the parameters when, for some reason, the other is known and tunable. This could be the case in which the local electric perturbation is due to the structure of the system, while the magnetic one is due to some external, controllable field, or vice versa.

We analyze both static and dynamical assisted estimation strategies, with the doubly-perturbed Hamiltonian given by Equations 22, 23:

H λ φ = H φ + λ | 0 〉 〈 0 | , (22)

H λ φ | ψ n 〉 = E n | ψ n 〉 . (23)

4.1 Assisted static estimation of an electric perturbation

The first analysis focuses on the static measurement strategy, and therefore some knowledge about the perturbed ground state is needed. We refer to the quantum Fisher information as in Equation 8, which specifically reduces to Equation 24:where the index m runs over all the excited states and their corresponding energies, E m .

F λ = 4 | 〈 ψ 0 | 0 〉 | 2 ∑ m ≠ 0 | 〈 ψ m | 0 〉 | 2 E 0 − E m 2 , (24)

For a system with N = 3, the Hamiltonian of Equation 22 can be explicitly diagonalized, and the overlaps can be written in closed form as Equation 25:where the eigenvalues are given by Equation 26:with the auxiliary function given by Equation 27:

E k = λ 3 + 2 λ 2 + 9 3 cos Φ + 2 k π 3 , k = 0,1,2 , (26)

Φ = arccos λ 3 − 27 ⁡ cos ⁡ φ λ 2 + 9 3 / 2 . (27)

The quantum Fisher in this setting can be reduced to Equation 28:

F λ = | ⟨ ψ 0 | 0 ⟩ | 2 | ⟨ ψ 1 | 0 ⟩ | 2 E 1 − E 0 2 + | ⟨ ψ 2 | 0 ⟩ | 2 E 2 − E 0 2 . (28)

One can notice that F λ can be increased by closing the gap between the ground state and the first excited state. This gap has a minimum for φ = π and goes arbitrarily close to zero in the limit λ → 0. For this reason, for N = 3, the possibility to tune φ can effectively improve the performance, but only in the low-field region ( λ ∼ 0 ).

More generally, when N ≠ 3, perturbed eigenvalues cannot always be expressed in closed form. However, by employing perturbative theory for λ → 0, one can observe from Equation 29 that the eigenvalues in first-order approximation are:

E k 1 = − 2 ⁡ cos 2 π k + φ N + λ N + O λ 2 . (29)

This reveals that even for N > 3, the ground and the first excited states have the same energy (except for contributions O ( λ 2 )) whenas is shown in Figure 3, where we compare the curves associated with N = 3 and N = 10.

φ opt = π , (30)

4.2 Assisted static estimation of magnetic-like perturbation

Given that F φ is identically zero in the solely magnetically perturbed case, here we investigate whether the introduction of a localized electric-like perturbation enables the estimation of φ. The QFI can be written as follows:

F φ = 4 ∑ n ≠ 0 | ⟨ ψ m | ∂ φ H | ψ 0 ⟩ | 2 E 0 − E m 2 . (31)

In general, we lack a simple closed form of the above expression. However, from the shape of Equation 31, and because the eigenvalues do not change with respect to the previous case, we expect behavior similar to that encountered in Section 4A. The degeneracy between the ground state and the first excited state, obtained for λ → 0 and φ → π, makes the system highly sensitive to tiny changes in the parameter, as shown in Figure 4. This stays true independently of the size of the system, making ring graphs highly sensitive probes, although only in a restricted range of values of φ, and showing that assisted estimation is a convenient strategy.

4.3 Assisted dynamical estimation of an electric perturbation

In this section, we study whether the assisted estimation scheme may also be conveniently exploited in the dynamical case.

To better understand the increasing behavior of the QFI without focusing on the oscillations, we notice that the derivative of the state can be expressed with Leibniz’s product rule and spectral decomposition over the eigenstate basis:where c k = ⟨ ψ 0 | ψ k ⟩ represents the overlap between the initial state and an eigenstate of the perturbed Hamiltonian. We notice that the three terms account for different phenomena happening in the temporal evolution. The first term in Equation 32 describes the spectral mixing of the initial state between the new eigenstates triggered by a variation of λ. Hence, it does not depend explicitly on time and remains O ( 1 ) in time throughout the evolution. The second term represents the sensitivity of the energy spectrum to a variation in λ and is the only term growing linearly in time. The third term instead accounts for the known phenomenon of Berry phase, which describes the curvature of the Hilbert space caused by a parametric variation of the Hamiltonian. This is purely geometric and remains O ( 1 ) in time as well. The QFI can be correspondingly expressed as the sum of three distinct terms, scaling as O ( t 2 ), O ( t ), and O ( 1 ), respectively. Plugging only the leading term into Equation 6 yields Equation 33:where V = ∑ k | ψ k 〉 ( ∂ λ E k ) 〈 ψ k |. Or equivalently, by substituting the variance explicitly, Equation 34 (See Figure 5):

F λ = 4 t 2 Var ψ 0 V ≡ A λ t 2 , (33)

F λ = 4 t 2 ∂ λ E k 2 ∑ k | c k | 2 − ∑ k | c k | 2 . (34)

Examining the N = 3 ring, it is possible to obtain an analytical form of the derivative ∂ λ E k by seeing the characteristic polynomial as an implicit equation for eigenvalues. We arrive at Equation 35:

∂ E k ∂ λ = − ∂ P / ∂ λ ∂ P / ∂ E k = E k 2 − 1 3 E k 2 − 2 λ E k − 3 . (35)

To maximize F λ in this measurement setting, the optimal strategy requires driving φ toward the critical values 2 π , 4 π, as observed from the numerical analysis in Figure 6.

The behavior strongly depends on the perturbation strength. In the weak regime λ ≪ 1, the coefficient A λ presents very sharp peaks, requiring very precise tuning of the parameter φ. In the strong perturbation regime λ ≫ 1, an improvement in QFI is observed across the entire period of φ, as the peaks enlarge. However, when λ → ∞, the coefficient reaches the constant value observed in the unassisted case. This is consistent with the interpretation that a strong electric-like perturbation acts as a potential barrier on a site, turning the ring into a chain. As the phase loses its meaning in a graph with no loops, the phase effect is lost, and the system is back to the electric-only situation.

The general trend holds regardless of the system dimension N, as shown in the lower panels of Figure 6. When the perturbation is small, the optimal values of φ are ( N ± 1 ) π. However, we always reach a constant value of A λ when λ becomes high. We also notice that the limit value for λ → ∞ decreases with N.

4.4 Assisted dynamical estimation of magnetic-like perturbation

Let us now consider the case in which λ is tuned to assist the estimation of φ. We recall that unassisted estimation fails regardless of the value of φ. Hence, any non-zero value of F φ would constitute an improvement.

The behavior of F φ as a function of time reveals a leading quadratic increase ∝ t 2 paired with oscillations (see Figure 7), similar to the behavior of F λ in the previous section. Following the same procedure used for F λ, we split the derivative of the state into three terms and isolate the resulting quadratic term in F φ, which is expressed as:where K = ∑ k | ψ k 〉 ( ∂ φ E k ) 〈 ψ k |. The optimal prefactor may be obtained by looking at the derivative of the eigenvalues, as we have done in the previous section. For N = 3, we have P ( E ( φ ) , φ ):while for larger N, the characteristic polynomial should be analyzed numerically.

F φ ∼ 4 t 2 Var e 0 K = A φ t 2 , (36)

∂ E k ∂ φ = − ∂ P / ∂ φ ∂ P / ∂ E k = 2 ⁡ sin ⁡ φ 3 E k 2 − 2 λ E k − 3 , (37)

The results are illustrated in Figure 8, where a few interesting features are revealed. At first, we notice that the minimal probe, N = 3, is again the most sensitive system, even if the performances remain lower than those of the electric case. Then, regardless of the dimension, three distinct features can be identified: the dominant one is an overall decreasing trend as we move from the center of the periodic domain, φ = N π / 2, toward the edges, φ = 0 (or N π). Superimposed on this is an oscillatory behavior, whose maxima are located at φ = n π ( n = 0 , ± 1 , ± 2 , … ). If only these two effects were present, the optimal precision would simply be achieved at φ = N π / 2 for even N, and at φ = ⌊ N / 2 ⌋ π and φ = ⌈ N / 2 ⌉ π for odd N. Notice, however, that for φ = k π , the QFI vanishes and that the local maxima are reached in the neighborhoods of those values. The behavior changes when approaching φ = 0 , N π, where A φ remains very small over a much broader region. This problem can be circumvented for odd-N rings by opting for a static measurement scheme, whereas for even-N rings, where this value corresponds to a vanishing QFI, one must resort to a larger interaction time.

5 Joint estimation

Multiparameter quantum estimation theory extends the single-parameter formalism to scenarios in which a group of parameters is estimated simultaneously through a joint measurement (Albarelli et al., 2020). The multiparameter analog of the Cramér–Rao bound is expressed in terms of the covariance matrix V ( θ ̂ ) of the estimators of a vector of unknown parameters θ = { θ 1 , … , θ d } and is expressed as Equation 38:where the QFI matrix may be written (for pure states) as Equation 39:

V ( θ ̂ ) ≥ 1 M Tr F − 1 = C S M , (38)

F μ ν = 4 Re ⟨ ∂ μ ψ θ | ∂ ν ψ θ ⟩ − ⟨ ∂ μ ψ θ | ψ θ ⟩ ⟨ ψ θ | ∂ ν ψ θ ⟩ . (39)

The precision limit expressed by this bound is not always achievable because the optimal observable for a given parameter may not commute with those of the others. On the other hand, it is often the case that quantum noise due to incompatibility is not the major source of uncertainty. Rather, it is the fact that the quantum state of the sensor is only weakly dependent on some of the parameters, usually referred to as the sloppy parameters of the model, such that the quantum Fisher information matrix (QFIM) is singular or nearly singular. To quantify this effect, we introduce the overall sloppiness (He and Paris, 2025) of the sensor as Equation 40:

s = 1 det F . (40)

A large value of s signals that joint estimation is inherently inefficient because the sensors actually depend on a reduced number of parameters. In our case, large values of s indicate that the quantum state of the walker does not depend independently on the two perturbations, but rather only on a combination of them.

Our results indicate that ring graphs are not the best choice for joint estimation. For the static scheme, the bound C S and the sloppiness s are large in a large range of values of the parameter. These limitations are even enhanced in the dynamical scheme, where both C S and s exhibit rapid, high-frequency fluctuations as a function of φ. This instability is not lifted by any value of λ, suggesting high variability in the system. Some results are shown in Figure 9; the fluctuations rapidly become more irregular and of higher frequency for higher interaction times.

6 Temperature estimation

Quantum thermometry aims to exploit the spectral structure of quantum systems to optimize estimation of temperature (Correa et al., 2015; Kiilerich et al., 2018; Kucsko et al., 2013; Neumann et al., 2013). In this section, we analyze quantum walks on a ring as quantum thermometers. In particular, we exploit the fact that magnetic perturbations modify the spectrum of the Hamiltonian, providing a means to improve the sensitivity of the system to temperature. The Hamiltonian of this system is H φ, given in Equation 14, and temperature is encoded in the thermal equilibrium state of the system as Equation 41, that is,

ρ j T = e − E j / T ∑ j n − 1 e − E j / T . (41)

The QFI for the temperature in this system can be easily calculated using Equation 42 (Paris, 2015) and reads:which simplifies to Equation 43: where:

F T = ∑ j n − 1 1 ρ j ∂ ρ j ∂ T 2 , (42)

F T = Var ρ E T 4 = 〈 E 2 〉 − 〈 E 〉 2 T 4 , (43)

Z = ∑ j n − 1 e − E j / T , (44)

〈 E 〉 = 1 Z ∑ j n − 1 E j e − E j / T , (45)

〈 E 2 〉 = 1 Z ∑ j n − 1 E j 2 e − E j / T . (46)

Let us now investigate how to optimize temperature estimation by tuning φ. To this aim, we take the derivative of F T with respect to φ. The extramality condition corresponds to Equation 47, the following implicit relation:where Cov ( A , B ) represents the covariance between operators A and B over the thermal state, that is Equation 48,

T = Cov E 2 , ∂ φ E − 2 ⟨ E ⟩ Cov E , ∂ φ E 2 Cov E , ∂ φ E , (47)

Cov A , B = ⟨ A B ⟩ − ⟨ A ⟩ ⟨ B ⟩ . (48)

The full derivation is shown in Supplementary Appendix B.

In the low-temperature limit, the thermal occupation of higher energy levels becomes negligible, allowing the system to be effectively described as a two-level quantum system defined by the ground and first excited state using Equation 49:where Δ is the difference in energy between the first excited state and the ground state, whose labels are, respectively, k 1 and k 0. In this case, the QFI is given by Equation 50 (Paris, 2015),

ρ ≃ 1 Z | ϵ k 0 〉 〈 ϵ k 0 | + e − Δ T | ϵ k 1 〉 〈 ϵ k 1 | , (49)

F T = Δ 2 4 T ⁡ cosh Δ 2 T . (50)

The variance of any temperature estimator is thus bounded by Equation 51:where

δ T 2 ≥ T 2 ⁡ g Δ / T , (51)

g x = 2 x 2 1 + cosh ⁡ x . (52)

The maximum precision is achieved at the peak of g ( Δ T ), which occurs at Δ T ∼ 2.4. Consequently, to attain the Landau bound ( δ T 2 ∝ T 2 ) at arbitrarily low temperatures, we require a system whose gap can be tuned to arbitrarily small values, a condition that is fulfilled by the magnetically perturbed ring. By recalling the spectrum in Equation 16, as long as φ < π, ϵ 0 is the ground state, and ϵ 1 is the first excited state, such that Δ ( φ ) = ϵ 1 ( φ ) − ϵ 0 ( φ ). Hence, one can readily invert this relation to obtainwhich shows how to tune the magnetic phase φ to optimize the estimation of low temperatures. The validity of the two-level approximation can be checked over different temperatures and ring sizes in Figure 10.

φ N = π N − arcsin − Δ 4 ⁡ sin π N , (53)

7 Conclusion

In this article, we have investigated quantum sensing of electric and magnetic-like perturbations, using quantum walks on an n-dimensional cycle graph as a probe. We have considered two distinct measurement protocols: a static strategy, which involves performing measurements on the ground state of the perturbed system, and a dynamical strategy, which consists of a unitary encoding process governed by the perturbed Hamiltonian and starting from the ground state of the unperturbed system. The overall Hamiltonian of the system is given by the chiral adjacency matrix of the graph (with phase φ, referred to as the magnetic-like perturbation), and a pointwise defect located at a site with strength λ (electric-like perturbation). The goal is to optimize the estimation of the two parameters λ and φ, either individually or jointly.

First, we analyzed single-parameter estimation, where the Hamiltonian contains only one of the two perturbations. For the estimation of λ, we derived analytical expressions for the QFI for both static and dynamical protocols. While the static strategy provides high precision only for small λ, the dynamical protocol makes F λ grow quadratically, thus approaching the Heisenberg limit, F λ = 8 / 9 t 2 in the limit of large λ. We have also found that the limitations of the static protocol can be partially mitigated by introducing tunability of the energy scale ζ in the unperturbed Hamiltonian, so that H ̃ λ = − ζ A + λ | 0 〉 〈 0 |. In this formulation, the effective parameter entering the QFI becomes λ / ζ, which can be tuned into the desired precision regime even when λ itself is far from 0. Nonetheless, for large λ, the energy cost of the estimation strategy increases. In contrast to the electric case, the estimation of φ in the magnetic-only perturbation case is not possible because the Hamiltonian is a circulant matrix, making the eigenvectors independent of the phase φ itself, and the QFI vanishes.

We therefore considered an alternative strategy, termed assisted estimation, in which one parameter is assumed known and tunable, and this information is exploited to enhance the precision of the estimation of the other. Concerning the estimation of the electric perturbation, tuning φ can improve the precision. In the static protocol, setting φ → π significantly increases the value of F λ, even though the estimation remains efficient only for small values of the electric parameter λ, as summarized in Figure 3. In the dynamical protocol, quadratic growth of the QFI is maintained, and tuning the magnetic parameter φ can increase the value of the prefactor; see Figure 6. We should note, however, that the advantage holds only for small λ, while the prefactor becomes independent of φ for increasing λ. Regarding the dependence on the dimension of the ring, the two approaches exhibit opposite behaviors, with the static protocol favoring high-N values and the dynamic protocol being optimal for N = 3, as shown in Figures 3, 6. Similar results are obtained when we switch the roles of the perturbations, setting λ to a convenient value to estimate φ. We recall that in the unassisted case, the estimation of the magnetic parameter φ is impossible. In the assisted static scheme, however, having a small electric perturbation allows the estimation of the magnetic one, although only in the region φ ∼ π, as shown in Figure 4. On the other hand, in the assisted dynamical procedure, we find that F φ grows quadratically in time; see Figure 7. The optimization of the prefactor over λ is illustrated in Figure 8. In both the static and dynamical procedures, the best results are obtained for small ring sizes (optimal when N = 3).

We have also analyzed joint estimation of the two parameters. In this case, although the incompatibility vanishes, the system performs poorly due to the high degree of sloppiness, which makes the global bound on the overall precision very large over large parameter ranges.

Finally, we investigated the use of the system as a quantum thermometer. More specifically, we considered the system in equilibrium with a thermal bath at a temperature T and analyzed the precision in estimating T by tuning the magnetic parameter φ to maximize the QFI F T. We obtained an analytic expression for F T and the optimal phase, showing that it is possible to achieve near-optimal low-temperature sensitivity that follows the Landau bound.

In addition, given the interesting results mentioned above, we analyzed the effect of noise and decoherence on the performance of the dynamical estimation schemes. We found that an optimal time emerges, maximizing the QFI before the degrading effect of noise causes loss of information (see Supplementary Appendix D for more details).

Our results demonstrate that quantum walks on cycle graphs provide a versatile platform for sensing both electric and magnetic-like perturbations, with dynamical protocols offering Heisenberg-limited precision. Assisted estimation proves particularly useful when one parameter can be controlled, enabling sensing in otherwise inaccessible regimes. However, joint estimation remains challenging due to sloppiness of the encoding. Finally, the system’s performance as a quantum thermometer highlights its potential for thermometry at low temperatures, where optimal sensitivity can be achieved by tuning the magnetic parameter.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/frqst.2026.1894991/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Berihu Teklu Gebrehiwot, Khalifa University, United Arab Emirates

Reviewed by: Abdallah Slaoui, Mohammed V University, Morocco

Hichem Eleuch, Université Polytechnique Hauts-de-France, France