Stochastic Quantum Walks

• Stochastic quantum walks are quantum-walk models combining coherent propagation with classical randomness through stochastic measurements, dissipative transitions, or random control fields.
• They employ continuous-time Lindblad and discrete-time CPTP formulations to analyze key transport regimes, scaling exponents, and recurrence behaviors on graphs.
• Applications range from quantum simulations and PageRank algorithms to photonic implementations, demonstrating practical insights into complex quantum systems.

Searching arXiv for recent and foundational papers on stochastic quantum walks. Stochastic quantum walks are quantum-walk models in which coherent propagation is combined with classical randomness, dissipative transitions, stochastic measurements, or random control fields. In the literature represented here, the term covers several closely related constructions: continuous-time open-system walks written in GKSL/Lindblad form on graphs, discrete-time walks whose step or coin is randomized and then ensemble-averaged through a CPTP map, and discrete-time unitary walks driven by time-random gauge fields or phases (Govia et al., 2016, Ellinas et al., 2012, Wójcik, 2024). Across these formulations, the central technical question is how interference, irreversibility, and graph structure jointly determine transport, asymptotics, and observables such as recurrence, entropy growth, and stationary distributions.

1. Formal model classes

In the continuous-time formulation most commonly called a quantum stochastic walk, the state is a density operator $\rho$ on the vertex Hilbert space of a graph and evolves according to a Lindblad master equation,

$$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$

A widely used interpolation introduces a parameter $\omega\in[0,1]$,

$$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$

so that $\omega=0$ gives a purely coherent quantum walk and $\omega=1$ gives a purely dissipative walk. In these constructions, $H$ is taken as the adjacency matrix or another graph-derived Hamiltonian, while the $L_k$ encode environment-induced transitions; asymmetric rates $\gamma_{nm}\neq\gamma_{mn}$ make directed transport possible (Govia et al., 2016, Domino et al., 2016).

A distinct discrete-time line of work introduces randomness directly into the step rule and studies the ensemble-averaged evolution. In one representative model on $\mathbb Z$, the coin reshuffling operator depends on a dichotomic random variable $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$0, $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$1, chosen independently at each step. The averaged density matrix then evolves under a CPTP map

$$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$2

which permits exact recurrences for probabilities and moments (Ellinas et al., 2012).

Another discrete-time construction keeps each realization unitary but randomizes a gauge field. On a 1D cycle, the basic walk has

$$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$3

and stochasticity enters because a phase operator is applied only with probability $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$4 at each step. The two cases emphasized are the position- and spin-dependent field

$$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$5

and the electric field

$$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$6

Although both are weak, their dynamical consequences differ sharply (Wójcik, 2024).

2. Transport regimes and dynamical behavior

A recurrent theme is the crossover between quantum-ballistic and classical-diffusive transport. In the discrete-time stochastic-averaging model with dichotomic coin noise, the second moment yields two asymptotic regimes. If $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$7, then $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$8, so the root-mean-square displacement scales as $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$9. If $\omega\in[0,1]$0, then $\omega\in[0,1]$1, so the root-mean-square displacement scales as $\omega\in[0,1]$2. The crossover criterion is

$\omega\in[0,1]$3

which the paper states as $\omega\in[0,1]$4 (Ellinas et al., 2012).

In the continuous-time GKSL framework, the second moment is also used to classify transport. The asymptotic scaling law

$\omega\in[0,1]$5

defines an exponent $\omega\in[0,1]$6 with $\omega\in[0,1]$7 for ballistic transport and $\omega\in[0,1]$8 for normal diffusion. A key distinction is between local and global dissipation. For local edge operators, dissipation tends to destroy coherence and drive $\omega\in[0,1]$9 toward $$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$0. For the global dissipation operator $$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$1, however, the mixed coherent-dissipative case satisfies

$$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$2

so the quadratic term dominates asymptotically for any $$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$3, and ballistic scaling survives even at large dissipation strength (Domino et al., 2016).

Weak stochastic gauge fields show that stochasticity does not imply diffusion in any universal sense. For the position- and spin-dependent gauge field, the averaged walk loses coherence after a short transient, crosses from ballistic to diffusive spreading, and its position distribution becomes Gaussian after many steps, with the paper giving $$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$4 steps as an example. In contrast, the weak stochastic electric field preserves Bloch oscillations despite temporal randomness. Using

$$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$5

and the weak-field approximation

$$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$6

the evolution is shown to be gauge-equivalent to a standard electric DTQW with renormalized phase

$$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$7

so the oscillation period becomes

$$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$8

For $$\dot{\rho} = -i(1-\omega)[H,\rho] + \omega \sum_k \left(L_k \rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right),$$9, the paper reports a longer period than in the ordinary electric walk, in agreement with simulation (Wójcik, 2024).

A common misconception is that a diffusive-looking probability profile necessarily certifies decoherence. A counterexample is a fully unitary one-dimensional quantum walk with time- and site-dependent coin angles $\omega=0$0, designed so that the probability distribution $\omega=0$1 matches a prescribed classical-looking law while coherence is retained. The same inverse construction also allows random walks with inhomogeneous biases to reproduce ballistic Hadamard-walk profiles (Montero, 2016).

3. Graph structure, directionality, and limiting behavior

Graph structure enters stochastic quantum walks through both the coherent and incoherent generators. In the local-interaction regime, each directed edge $\omega=0$2 contributes a separate Lindblad operator,

$\omega=0$3

or, in equivalent graph-based implementations, one uses $\omega=0$4. In the global-interaction regime, a single Lindblad operator collects all edges, typically as the adjacency matrix or its directed analogue. The global construction is technically compact but introduces the phenomenon called spontaneous moralization: vertices with a common child can become effectively connected even if the original digraph contains no such edge (Glos et al., 2017, Glos et al., 2018).

The limiting behavior depends strongly on this architectural choice. For local interaction, strongly connected digraphs are relaxing for arbitrary Hamiltonian $\omega=0$5, and weakly connected digraphs whose condensation graph has exactly one sink strongly connected component are also relaxing. When there are multiple sink components, relaxation can fail, and the asymptotics can depend on both graph connectivity and the coherent part. For global interaction on undirected graphs, the evolution is convergent for all $\omega=0$6 but not relaxing for graphs with more than one vertex. On directed graphs, global interaction may fail to converge because the Liouville-space generator can acquire nonzero purely imaginary eigenvalues, leading to persistent oscillations (Glos et al., 2017).

To restore directed-graph semantics, nonmoralizing global constructions enlarge the Hilbert space by splitting vertices into copies and choosing orthogonal blocks in the Lindblad operator. In QSWalk.jl this is implemented through an enlarged Hamiltonian $\omega=0$7, an enlarged Lindblad operator $\omega=0$8, and a local Hamiltonian $\omega=0$9, after which original-vertex probabilities are recovered by summing over copies,

$\omega=1$0

The limiting-state analysis shows that nonmoralization fixes the graph-structure defect, although it does not guarantee convergence of the full density matrix in every case (Glos et al., 2018, Glos et al., 2017).

4. Asymptotic diagnostics: scaling exponents, entropy, recurrence, and spectra

The asymptotic scaling exponent $\omega=1$1, defined by $\omega=1$2, is used as a transport classifier and as a coherence witness. In the analytical cases treated in the cited work, $\omega=1$3 indicates ballistic transport and $\omega=1$4 indicates normal diffusion. The distinction is especially sharp in the global-dissipation model, where transport remains ballistic for every $\omega=1$5 even though the state can be strongly mixed (Domino et al., 2016).

Entropy growth provides a complementary diagnostic. For dissipative quantum walks on networks, the von Neumann entropy

$\omega=1$6

shows an intermediate-time logarithmic regime

$\omega=1$7

In the classical CTRW limit this reproduces the information dimension $\omega=1$8. The paper studies a line of $\omega=1$9 nodes, for which $H$0 and hence $H$1, and a generation-$H$2 Sierpinski gasket with $H$3 nodes, for which

$H$4

For small but nonzero $H$5, the fitted slope is reported to approach roughly twice the classical value in both graph families (Schijven et al., 2014).

Recurrence has recently been studied for a discrete-time quantum stochastic walk on the line defined by the Kraus operators

$H$6

With monitored detection at the origin, the recurrence probability is

$H$7

A striking result is that adding a nonzero classical component can reduce recurrence below the purely unitary value, even though the balanced classical random walk at $H$8 is recurrent with certainty. The paper identifies a threshold coin angle

$H$9

such that for $L_k$0, $L_k$1 becomes non-monotone and initially decreases as $L_k$2 increases from $L_k$3. The special case $L_k$4 is exactly solvable and gives $L_k$5 for all $L_k$6 (Stefanak et al., 15 Jan 2025).

Spectral analysis of a quantized Sinai–Derrida model adds a further asymptotic perspective. On a ring, a Lindblad walk with stochastic bond disorder and coherent hopping exhibits a delocalization transition as the bias increases beyond a critical value, signaling under-damped relaxation. The same work emphasizes a non-monotonic dependence of the Lindbladian spectrum on the coherent hopping rate $L_k$7, and derives an effective rate

$L_k$8

which is used to argue that coherent hopping can enhance the effective disorder rather than smooth it out (Avnit et al., 2023).

5. Physical realizability, quantum simulation, and computational frameworks

The formal generality of the Lindblad equation does not imply unrestricted physical realizability. For number-conserving QSWs in the single-excitation picture, a microscopic weak-coupling Born-Markov-secular derivation imposes strong constraints. Local $L_k$9 or $\gamma_{nm}\neq\gamma_{mn}$0 system-bath couplings generically induce unwanted loss to the vacuum unless the effective coupling vanishes, in which case the incoherent walk dynamics also vanish. Pure-dephasing couplings $\gamma_{nm}\neq\gamma_{mn}$1 preserve excitation number, but reproducing the desired local transition rates while suppressing unwanted dephasing terms requires reservoir engineering and detailed knowledge of the system eigenstructure. The broader conclusion is that general implementations would require the complete solution of the underlying unitary dynamics and sophisticated reservoir engineering (Taketani et al., 2016).

Several simulation strategies circumvent this restriction by treating the walk as a target open-system evolution rather than as a literal environment-engineered process. For continuous-time QSWs, a quantum-trajectories-on-a-quantum-computer protocol simulates a broad class of walks for which the coherent Hamiltonian and the non-Hermitian decay operator commute in the single-excitation subspace,

$\gamma_{nm}\neq\gamma_{mn}$2

equivalently $\gamma_{nm}\neq\gamma_{mn}$3. In that setting the simulator needs coherence only for the average time between jumps,

$\gamma_{nm}\neq\gamma_{mn}$4

and only as many physical qubits or nodes as the largest coherently connected subgraph (Govia et al., 2016). For discrete-time QSWs, a trajectory-based protocol uses one ancilla per vertex, coherent system-ancilla initialization, ancilla measurement, and classical feed-forward to realize the ensemble Kraus map

$\gamma_{nm}\neq\gamma_{mn}$5

and the construction generalizes directly from a two-vertex example to arbitrary graph topology and connectivity (Schuhmacher et al., 2020).

A substantial computational ecosystem has also developed. QSWalk provides Mathematica routines for arbitrary directed and weighted graphs by vectorizing the density matrix into an $\gamma_{nm}\neq\gamma_{mn}$6-dimensional sparse superoperator and solving

$\gamma_{nm}\neq\gamma_{mn}$7

The package reports practical limits on a typical desktop of about $\gamma_{nm}\neq\gamma_{mn}$8 for dense graphs and $\gamma_{nm}\neq\gamma_{mn}$9 for sparse graphs, with memory around $\mathbb Z$0 GB and runtimes of minutes (Falloon et al., 2016). QSWalk.jl extends this approach in Julia and includes local, global, and nonmoralizing global regimes on arbitrary directed graphs (Glos et al., 2018). QSW_MPI targets massively parallel computers through sparse CSR storage, distributed matrix-exponential action, and MPI/OpenMP acceleration; the reported maximum speedup is around $\mathbb Z$1 for complete digraphs with MPI+OpenMP, while Erdős–Rényi digraphs achieved lower speedups, around $\mathbb Z$2 (Matwiejew et al., 2020).

6. Applications and current research directions

The application space is broad because stochastic quantum walks combine graph-based transport with tunable irreversibility. The continuous-time QSW literature explicitly connects them to artificial intelligence, biological processes, and quantum transport (Govia et al., 2016). In simulation software and case studies, they have been applied to line graphs, pure dephasing models, the seven-site FMO complex with a sink vertex representing exciton absorption, and quantum PageRank on directed graphs (Falloon et al., 2016).

Ranking on directed networks is a particularly developed application. In QSW-based PageRank, the stationary density matrix replaces the stationary probability vector of classical continuous-time random walks. Two schemes are emphasized: only incoherence and dephasing with incoherence, both built from the Google matrix $\mathbb Z$3. The cited numerical study reports that these QSW schemes best resolve degeneracies that are unresolvable via classical PageRank and do so with a convergence time comparable to that for classical PageRank, which is generally the minimum; for some networks, the convergence time is lower than the classical one and the ranking is almost degeneracy-free (Benjamin et al., 2022).

Machine-learning-inspired constructions use a quantum stochastic neural network in which neurons are graph vertices and learning acts on coherent couplings $\mathbb Z$4 and dissipative rates $\mathbb Z$5 in a GKLS generator. The reported tasks include function approximation, two-dimensional data classification, and sequence classification. In the cited toy models, a simple QSNN with five neurons is trained to determine whether a sequence of words is a sentence or not, and a QSNN with $\mathbb Z$6 neurons improves the accuracy of recognizing new types of inputs like verses. The same study reports that the coherent QSNN is more robust against both label noise and device noise than the decoherent QSNN (Wang et al., 2021).

Photonic implementations have used stochastic quantum walks for Haar-randomness generation. In an integrated femtosecond-laser-written photonic chip, a $\mathbb Z$7 waveguide array of total length $\mathbb Z$8 cm with segment length $\mathbb Z$9 mm and detuning amplitude $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$00 was used to realize a two-dimensional stochastic quantum walk. Averaging over $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$01 random settings, the output distributions converge toward the even distribution, which the paper interprets as evidence for Haar $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$02-design behavior; it also reports that the two-dimensional array outperforms a one-dimensional array of the same number of waveguides in the speed of convergence (Tang et al., 2021).

Another active direction adds stochastic resetting to graph walks. In a Lindblad interpolation between classical and quantum dynamics, resetting at rate $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$03 modifies the master equation to

$$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$04

and the resulting stationary or long-time average occupation statistics depend on both network topology and the quantum-to-classical interpolation parameter $$\dot{\rho}= -i\left[\hat{H},\rho\right] + \sum_{k}\gamma_k\left(\hat{L}_k\rho\hat{L}_k^{\dagger}-\frac{1}{2}\big\{\hat{L}_k^{\dagger}\hat{L}_k,\rho\big\}\right).$$05. The cited work reports that stochastic resetting can amplify the difference between classical and quantum occupation patterns, especially on heterogeneous networks such as Barabási–Albert graphs (Wald et al., 2020).

Taken together, these results show that stochastic quantum walks are not a single model but a technically connected family of open-system and randomized quantum-walk frameworks. Their unifying feature is the controlled coexistence of coherent transport and stochastic structure; their distinguishing results come from how that coexistence is implemented—through Lindblad jumps, stochastic averaging, random gauge fields, monitoring protocols, or graph-specific dissipative architectures.