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Noise correlations behind superdiffusive quantum walks

We study how discrete-time quatum walks behave under short-range correlated noise. By considering noise as a source of inhomogeneity of quantum gates, we introduce a primitive relaxation in the uncorrelated stochastic noise assuption: binary pair correlations manifesting in the random distribution. The transport properties for both spatial and temporal noise regimes were examined, taking into account different quantum gates. For spatial inhomogeneities, we show noise correlations driving quantum walks from the well-known exponentially localized regime (stochastic and uncorrelated noise) to superdiffusive spreading. This scenario display an interesting performance, in which the superdiffusive exponent is almost invariant to the inhomogeneity degree. Both the time-asymptotic regime and the finite-size scaling unveil an emergent superdiffusive behavior also for quantum walks undergoing temporal noise correlation, replacing the diffusive regime exhibited when noise is random and uncorrelated. However, results report some quantum gates insensitive to correlations, contrasting with the spatial noise scenario. Results and following discussions help us understand the underlying mechanism behind superdiffusive quantum walks, including those with deterministic aperiodic inhomogeneities.

Designing and controlling such quantum processes for a long-time dynamics is an essential matter, which have the noise among the main obstacles [22][23][24]. Quantum error-correcting methods [25,26] and fault-tolerant protocols [27,28] have pointed to the need for better understanding of the noise nature on the system. Thus, ingredients aiming to symbolize some sort of interaction between system and environment have been studied. In general lines, noise drives the discrete-time quantum walk at a slower spreading rate in the long-time limit. White noise fluctuations on the time evolution operator usually leads to a diffusive wave-function spreading [19,[29][30][31], while an arbitrary spatial inhomogeneity is responsible for a localized behavior [19,20,[32][33][34]. There are also studies considering the simultaneous existence of both, in which the diffusive behavior in the long-time limit has been reported [35,36]. The consequences of an instantaneous stochastic noise over the quantum walk stability have been recently reported, where a maximally coherent initial state achieves a breathing dynamics or even a standing self-focusing in a long-time regime [37].
Noise correlations have attracted significant attention by quantum characterization, verification and validation techniques [38][39][40][41]. This aspect has also been considered by quantum walks, with reports of significant changes in the walk profile. As example, we observe a quantum walker exhibiting a superdiffusive spreading in one-dimensional systems in which quantum gates follow aperiodic time-dependent sequences, such as Fibonacci [42,43] and Thue-Morse [43]. On the other hand, systems in which quantum gates are temporally ruled according to the Rudin-Shapiro distribution display a nearly diffusive behavior [43]. The behavior exhibited for the system with Fibonacci temporal sequencing has been connected with the power-law decay of the timecorrelation function of the trace map [44]. Dynamically localized quantum walks have been reported by considering an occurrence of possible phases during the timeevolution, as a consequence of additional interactions between the system and the environment [45].
Nonstochastic scenario has also been explored in the spatial framework, both for different quantum gates distributed along the lattice sites and for systems with position-dependent phase defects. Aperiodic Fibonacci and Thue-Morse sequencing show a superdiffusive spreading, either embedded into the quantum gate distribution [43] or the step operator, where the allowed jumps symbolize connections between non-neighboring quantum gates [46]. Spatial localization of quantum walker has been reported for systems in which quantum gates are distributed analogous to the Aubry-André model [47] and the Rudin-Shapiro ordering [43]. Localization-delocalization transitions have been found for systems in which the statistical regime of randomness [48] or the absence of spatial periodicity [49] are controllable.
Reports above can be sorted in two distinct scenarios: uncorrelated stochastic noise or correlated inhomogeneities following deterministic sequences. However, less extreme, and perhaps more realistic, scenarios need further understanding. How do discrete-time quantum walks behave under short-range correlated noises? In this paper, we examine the impact of a primitive relaxation in uncorrelated stochastic noise assumption: the emergence of binary pair-correlated in the random distribution. Let us consider a homogeneous lattice with quantum gateŝ C n,t (θ 1 ), in which a general noise process deviates some quantum gates from their ideal operation, leading them to effectively behave likeĈ n,t (θ 2 ). We explored both the spatial and the temporal inhomogeneous scenarios, in which we assume discrete-time quantum walks effectively ruled by two distinct quantum gates, just like in Refs. [36,42,43]. We show noise correlations driving quantum walks with spatial inhomogeneities from the well-known exponentially localized (stochastic and uncorrelated noise) to the superdiffusive spreading. This scenario display an interesting performance, in which the superdiffusive exponent is almost unvarying to the inhomogeneity degree ∆θ. A superdiffusive asymptotic behavior is also reported for quantum walks undergoing temporal noise correlation, contrasting with the diffusive regime exhibited when noise is random and uncorrelated. However, results show the superdiffusive regime unreachable for specific quantum gates settings.

II. MODEL
We consider quantum walks in one-dimensional lattices of interconnected sites. The walker consists of a qubit (two-state quantum system) with the internal degree of freedom. Thus, the quantum walker state |Ψ belongs to a Hilbert space H = H P ⊗ H C , where H C is a complex vector space of dimension 2 associated with the internal degree of freedom, here spanned in the basis {| ↑ , | ↓ }. The position Hilbert space H P is spanned by the basis {|n } with the lattice nodes n ∈ Z.
Each step of evolution consists in quantum gatesĈ n,t located in the lattice sites, which act on the quantum walker and shuffles its internal state. Such state establishes the spatial redistribution performed by the shift op-eratorŜ. Thus, starting from an initial state |Ψ t=0 , the dynamical evolution is accomplished by recursively applying the unitary transformation |Ψ t+1 =Û |Ψ t , witĥ U =Ŝ · (Ĉ n,t ⊗ I P ).
An arbitrary quantum walker state in the t-th time step can be given as in which ψ ↑ n,t and ψ ↓ n,t are complex amplitudes that satisfy n |ψ ↑ n,t | 2 + |ψ ↓ n,t | 2 = 1. Quantum gatesĈ n,t , the ones responsible for mixing the qubit internal degree of freedom, are arbitrary SU (2) unitary operator given bŷ C n,t = cos[θ n,t ]Ẑ + sin[θ n,t ]X, q 1 q 1 q 2 q 2 q 1 q 2 q 2 q 1 q 2 q 1 q 1 q 2 q 1 q 1 q 2 q 1 q 2 q 1 q 1 q 1 a Exponent q 2 q 2 q 2 q 1 q 1 q 2 q 2 q 2 q 1 q 1 q 1 1. Schematic illustration of spatial and temporal noise scenarios, in which a general noise process D deviates some quantum gates from their ideal operation, leading them to effectively behave likeĈn,t(θ2) = DĈn,t(θ1). Main features of the quantum walker spreading are described by considering the characteristic power-law behavior of standard deviation achieved by using the shift operatorŜ, given aŝ It is well-known that such quantum walk protocol, with single and steady quantum gates, provides the asymptotical behavior of ballistic spreading (except for θ = π/2 for which the particle remains confined). This scenario is modified by disturbances on the quantum gates [21]. Consider a homogeneous lattice with quantum gateŝ C n,t (θ 1 ), in which a general noise process D deviates some quantum gates from their ideal operation, leading them to effectively behave likeĈ n,t (θ 2 ) = DĈ n,t (θ 1 ). Such error processes can be unitary, arising from over-or underrotation in qubit control pulses [50]. By assuming a local noise, where interferences act randomly and independently on individual quantum gates located along the lattice positions, we effectively deal with a spatial inhomogeneity, since distinct quantum gates are distributed at each position n. By considering such spatial arrangement as temporally persistent, the resulting quantum walk is localized for anyĈ n (θ 2 ) other thanĈ n (θ 1 ) [19,20]. On the other hand, when all quantum gates simultaneously feel the same disturbances, but in randomly specified time steps along the time evolution, results report a diffusive spreading for anyĈ t (θ 2 ) other thanĈ t (θ 1 ) [36]. Fig. 1 shows a schematic illustration of both scenarios, as well as their corresponding features of quantum walker spreading. Localized, diffusive, subballistic and ballistic regimes are identified by considering the characteristic power-law behavior σ(t) ∼ t α of the standard deviation where n 2 (t) = n n 2 |Ψn(t)| 2 and n(t) = n n|Ψn(t)| 2 .
Here, we introduce a minimally biased noise model, which exhibits a short-range correlation nature: a binary pair-correlation. Just as in Ref. [36,42,43], we assume only two quantum coins are involved. We set C(θ 1 ) according to the percentage p = 0.5 andĈ(θ 2 ) to (1 − p), withĈ(θ 2 ) appearing in pairs. By considering a spatial-dependent noise scenario, quantum gates are fixed during the time evolution, but their random drawn at each site has the constraint thatĈ(θ 2 ) always appear in pairs (adjacent sites). For time-dependent noise, in which quantum gates are the same at all lattice sites but randomly change at each time step, we assume the constraint ofĈ(θ 2 ) appearing in temporal dimers (two consecutive time steps).
The initial state of the quantum walker can be written as in which the initial position n 0 of the quantum walker is defined in the central site of the lattice. We consider open chains as the boundary condition throughout the analysis, with large enough lattice sizes so that the wavefunction does not reach the edges over the time course described. Due to its stochastic nature, we stablish an ensemble of fifty subsequent and independent quantum walks to evaluate its average behavior.

III. RESULTS AND DISCUSSION
A. Spatial noise Using the numerical method described above, we start by examining the weight of proposed correlation over the asymptotic behavior of the quantum walker. In Fig. 2 we show a snapshot of average wave-function profiles, taking as reference the effective scenario of quantum gates disposed ramdomly and independently. We set the binary pair (BP) correlated θ 2 as Hadamard quantum gates and θ 1 = π/3. In the absence of correlations, we observe the probability distribution of the quantum walker strictly around the initial position. Wave-function profile exhibiting an exponential decay exposes a signature of Anderson localization, which corroborates the Refs. [19,20,51]. A distinct scenario is described by systems with binary pair correlations, since the wavefunction are no longer concentrated around the starting position. Now, exponentially decaying tails give way to wave-packet fronts exhibiting sharp cutoff, which suggests a delocalized regime of spreading.
To better understand the previous results, we follow the time evolution of the wave-function spreading by using the standard deviation σ(t), defined in Eq. 4. Its characteristic power-law behavior σ(t) ∼ t α quantifies the spreading properties of wave-function, as for instance the ballistic (α = 1.0), the diffusive (α = 0.5) and the localized (α = 0.0) behavior. In Fig. 3a, we explore the same θ settings used in Fig. 2, with addition of a noiseless quantum walk. Hadamard quantum gates are taken as reference for binary pair correlated noise. As expected, noiseless quantum walks display a ballistic spreading, characterized by σ(t) growing linearly as time evolves. On the other hand, the localized behavior is verified when θ 1 and θ 2 are ruled by a random and independent spatial distribution. Such aspect is characterized by σ(t) saturating after an initial transient, i.e. σ(t) ∼ t 0 . The main point here is the superdiffusive spreading for quantum walks subjected to binary pair correlations, caracterized by exponent α ≈ 0.74. This asymptotic behavior persists even with the replacement of the quantum gate θ 1 , from π/3 to 4π/15 (see Fig. 3b). The initial transient is longer as the error ∆θ = |θ 1 − θ 2 | decreases. A supplementary analysis is shown in Fig. 3c, in which we report a size-dependence of the wave-packet width in long-time regime. In function of initial transient, we consider lattice sizes ranging from N = 10.000 to N = 1.000.000 sites and the same quantum gates used before. As expected, systems with a stochastic and uncorrelated spatially ordering of quantum gates exhibit a size-independent scenario [σ(t ∞ ) ∼ N 0 ], which is a signature of localized regime. On the other hand, the finite-size scaling for systems with binary pair-correlated spatial noise shows σ(t ∞ ) ∼ N 0.72 , in full agreement with superdiffusive spreading reported before.
Results suggest the superdiffusive behavior an intrin- sic component from noise with binary pair correlations, regardless of how far θ 2 is from θ 1 . However, the question now is whether such behavior really prevails over other quantum gates. Could other quantum gates provide a diffusive or subdiffusive spreading? This question is answered in Fig. 4, where we explore the asymptotic exponent α as a function of θ 1 , taking into account the binary pair-correlated quantum gates (a) θ 2 = π/4, (b) θ 2 = 4π/15 and (c) θ 2 = π/3. Fig. 4a extends the previous analysis to further θ 1 quantum gates and confirms a dominant superdiffusive regime for quantum walks with binary pair-correlated noise. Figs. 4b-c show the same behavior when other quantum gates play the role of binary pair-correlated. The asymptotic exponent α stands approximately unvarying (∼ 0.73) when noise is correlated, even for a considerably small ∆θ. On the other hand, uncorrelated inhomogeneity leads to a stagnation of the spread after an initial transient and hence a σ(t) ∼ t 0 in the asymptotic regime, in full agreement with Ref. [19,20,51]. We observe exceptions for quantum gates θ 1 equivalent to the pair-correlated quantum gates θ 2 , in which the fully extended (ballistic) regime is achieved. On the other hand, an employment of Pauli X quantum gates results in a localized regime, coming from swapping the amplitudes of states | ↑ and | ↓ that corresponds to a negation operation. These results can be understood when we look at the quantum gates playing the role of altering the probability of the quantum walker (qubit) moving to the right or to the left, therefore, they establishes the transmission probability. With all quantum gates identical, the system is translation invariant and the generalized eigenfunctions are described by Bloch waves, infinitely extended over the whole lattice. When quantum gates vary randomly in space, the system is no longer translational invariant. The spreading is inhibited by interferences effects between multiple scatterings of qubit wave-function, altering the eigenfunctions to exponentially localized. A noise with random binary pair-correlation seems to develop transparent (resonant) states, that emerge as if they were extended for a small finite sample and contribute to the spreading of quantum walker. This scenario helps us understand the superdiffusive quantum walks reported for Fibonacci and Thue-Morse spatial inhomogeneities [43], since binary pairs spontaneously appear in such aperiodic sequencing. Although a similar phenomenology has been reported in electronic transport [52], we observe a absence of superdiffusive, diffusive or localized regimes, with thresholds between them depending on inhomogeneity degree. Our results suggest a prevailing superdiffusive scenario, asymptotically independent of ∆θ.

B. Temporal noise
This section is devoted on studying walks with timedependent varying quantum gates. We start by analysing the standard deviation (see eq. 4), establishing a comparative analysis between systems with binary pair correlations and their uncorrelated or noiseless counterparty. In Fig. 5 we evaluate the time-evolution by considering θ 2 = π/4 with (a) θ 1 = π/3 and (b) θ 1 = 4π/15. We observe noiseless quantum walks displaying a ballistic behavior, characterized by power law σ(t) ∼ t. Systems in which θ 1 and θ 2 appear randomly and independently exhibit a diffusive character, corroborating previous studies [43]. Our analysis unveils superdiffusive quantum walks again emerging from binary pair correlations. As well as the scenario in which noise affects the spatial distribution of quantum gates, such character seems to be independent of how far θ 2 is from θ 1 , since the superdiffusive behavior is the same for θ 1 − θ 2 = π/12 and θ 1 − θ 2 = π/60. This behavior is reinforced by analysis of finite-size scaling of the wave-packet width in asymptotic regime (see Fig. 5c). By exploring lattice sizes ranging from N = 10.000 to N = 1.000.000 and the same quantum gates, we observe agreeing results: a diffusive character for quantum gates undergoing an uncorrelated alternation, identified by σ(t ∞ ) ∼ N 0.5 ; a superdiffusive behavior (∼ N 0.73 ) for systems with binary pair correlations.
In order to know whether such superdiffusive behavior also extends when other quantum gates are involved, we examine the power-law exponent α [σ(t) ∼ t α ] as a function of θ 1 , taking into account the different dimerized quantum gates (see Fig. 6). As before, we also present data for the regimen without correlations as reference, named random. We observed a predominant superdiffusive behavior, whether the dimerized quantum gates are (a) Hadamard, (b) θ 2 = 4π/15 or (c) θ 2 = π/3. This scenario is present even for the Pauli-X quantum gates, whose bit-flip character between | ↑ and | ↓ has proved to be dominant when the noise interfered only in the spatial arrangement of the quantum gates. The characteristic asymptotic exponent α is similar to that presented in the spatial noise regime (∼ 0.73). However, we observe an anomalous behavior for θ 1 = 2π/3 and θ 1 = 5π/3, which suggests these quantum gates as insensitive to binary pair-correlation. The slow convergence to the diffusive regime as θ 1 approaches 2π/3 or 5π/3 is a consequence of a very long initial transient to asymptotic behavior. This scenario points to the possibility of specific quantum gates performing as a filter of correlations, which can contribute to the progress of tools and algoritms for quantum processes under noise influence [53].

IV. SUMMARY AND CONCLUDING REMARKS
In summary, we have studied the transport properties in discrete-time quantum walks undergoing a noise correlated. By considering a relaxing in the uncorrelated stochastic noise premiss, we considered the emergence of binary pair-correlated in the random distribution. We have explored both the spatial and temporal noise scenarios, always drawing a comparative analysis with systems under uncorrelated noise. The dynamics of qubit was computed from a sample mean of independent noises. In systems with spatial inhomogeneity, we observe the binary pair correlation driving the quantum walks from exponentially localized regime (stochastic and uncorrelated noise) to superdiffusive spreading. Such regime holds maintained regardless of the difference between quantum gates θ 1 and θ 2 , either by analyzing the time-asymptotic regime, as well as the finite-size scaling, which have unveiled a superdiffusive exponent almost unvarying to the degree of inhomogeneity. The binary pair correlation seems to develop resonant states that favors delocalized quantum walks, even when existing in the temporal scenario. In such systems, the superdiffusive spreading also emerges from the binary pair correlation, taking the place of the diffusive quantum walks observed for independent and random temporal inhomogeneity. However, some quantum gates exhibit a remarkable effect of insensitivity to correlations, which seems attractive for studying correlation filter for quantum processes [53]. Our results bring new aspects about superdiffusive quantum walks reported in aperiodic inhomogeneities, since binary pairs can spontaneously appear in traditional sequences such as Fibonacci. Studying the entanglement is one of future prospects, which can brings new features to recent fidings reported in Ref. [54]. To conclude, recent experimental achievement in a time-multiplexing system based on an unbalanced Mach-Zehnder interferometer with a feedback loop [55] makes us believe that the proposed scheme here is feasible for prompt implementation. Such setup has been proven capable of controlling quantum gates both over space and time, designing inhomogeneities.