Recurrence in discrete-time quantum stochastic walks

Technical Summary: Recurrence in Discrete-Time Quantum Stochastic Walks

Problem Statement
The paper investigates the phenomenon of recurrence—the probability that a walker returns to its starting position—in discrete-time quantum stochastic walks (DTQSW) on a line. While classical random walks (RW) on a 1D line are recurrent (return with certainty), unitary quantum walks (QW) can be transient (return with probability $<1$) due to destructive interference. The authors explore the behavior of a DTQSW model that interpolates between a unitary QW and a classical balanced RW. A central question is whether introducing classical randomness (stochasticity) into the quantum evolution always enhances recurrence or if it can, counterintuitively, suppress it. The study specifically addresses the challenge of accurately determining asymptotic recurrence probabilities, which are difficult to estimate via direct numerical simulation due to slow convergence and memory constraints associated with density matrix evolution.

Methodology
The authors model the DTQSW as an open quantum system governed by a completely positive trace-preserving (CPTP) map. The evolution is defined by three Kraus operators:

1. $\hat{E}_0 = \sqrt{1-p}\,\hat{U}$: Represents the unitary quantum walk step with probability $1-p$.
2. $\hat{E}_{1,2} = \sqrt{p/2}\,\hat{I}_c \otimes \hat{T}^{(\dagger)}$: Represent incoherent classical steps (left/right shifts) with total probability $p$.

To analyze recurrence, the authors employ the "monitored recurrence" approach, where the system is subjected to a partial projective measurement at the origin at each step. If the walker is found at the origin, the process stops; otherwise, the amplitude at the origin is erased, and the evolution continues.

To overcome the limitations of direct simulation (which requires iterating the CPTP map on a density matrix, leading to quadratic memory scaling and slow convergence), the authors utilize the theory of first-return generating functions (FR-functions). They derive the Stieltjes function for the system using Fourier transformation in momentum space. By evaluating the reduced FR-function $F(z)$ at values of $z$ close to 1 (specifically $z=0.99999$), they approximate the asymptotic recurrence probability $R$ with significantly higher accuracy and effective step counts ($t_{eff} \approx 10^5$) than direct iteration allows.

The study also compares this model to an alternative DTQSW where the classical component is a correlated random walk (CRW), which involves measuring the coin state before and after the step, thereby inducing decoherence.

Key Contributions and Results

1. Suppression of Recurrence Probability:
   The primary finding is that introducing classical randomness ($p > 0$) into a unitary QW can decrease the recurrence probability compared to the pure unitary case ($p=0$), despite the fact that the pure classical limit ($p=1$) is recurrent.

   • For coin angles $\theta$ in the range $(\theta^*, \pi/2)$, where $\theta^* \approx 0.2892\pi$, the recurrence probability $R(p)$ is non-monotonic. It initially decreases as $p$ increases from 0, forming a "dip," before eventually rising to 1 as $p \to 1$.
   • This behavior is identified as a robust feature of the asymptotic limit, not a transient effect of finite simulation time.

2. Role of Interference and Decoherence:
   The suppression of recurrence is attributed to the interplay between quantum interference and classical randomness. In the dip region, the addition of classical steps disrupts the specific interference patterns of the unitary walk in a way that reduces the likelihood of return, even though the classical walk itself is recurrent.

   • In contrast, for the alternative model interpolating between a QW and a correlated RW (where the coin state is measured, causing decoherence), the recurrence probability is a step function: $R(p) = 1$ for any $p > 0$. The authors conclude that any amount of coin-state decoherence enforces classical recurrence behavior.

3. Numerical Framework:
   The paper provides a robust numerical framework using generating functions to evaluate recurrence in open quantum systems. This method allows for the investigation of asymptotic properties that are inaccessible via standard time-evolution simulations, demonstrating that the observed suppression is a genuine asymptotic feature.

Significance
The paper claims that these results challenge the intuitive notion that adding classical noise to a quantum system simply bridges the gap between quantum and classical behaviors monotonically. Instead, it demonstrates that specific regimes of quantum-classical interplay can yield performance characteristics (in this case, reduced recurrence) that are distinct from both the pure quantum and pure classical limits.

The authors suggest that this behavior implies that classical randomness may be beneficial for certain tasks involving discrete-time quantum walks and partial measurements, such as subroutines in quantum algorithms, by potentially suppressing unwanted returns or altering transport properties. The work highlights that the specific mechanism of introducing stochasticity (e.g., incoherent shifts vs. coin decoherence) is crucial in determining the recurrence properties of the walk.

The study remains modest regarding experimental realization, noting that while photonic implementations and superconducting qubits with mid-circuit readout offer potential pathways, current noise levels in NISQ devices present significant challenges to demonstrating these effects over the necessary number of steps.