Implementation of Leaking Quantum Walks on a Photonic Processor

Technical Summary: Implementation of Leaking Quantum Walks on a Photonic Processor

Problem Statement
Quantum walks (QWs) serve as fundamental models for coherent quantum dynamics and are essential resources for quantum information processing, simulation, and algorithmic speedups. While QWs have been implemented across various platforms, including trapped ions and bulk optics, integrated photonic circuits are increasingly recognized as a versatile platform due to their phase stability and scalability. However, real-world photonic systems inevitably suffer from particle losses due to non-ideal component efficiencies. While the inclusion of absorbing sites within a QW is a critical scenario for simulating open quantum systems and energy transport (e.g., in biological complexes like the Fenna-Matthews-Olson complex), it has received relatively little experimental attention. Existing literature contains limited numerical studies and few experimental implementations regarding absorbing boundaries. Consequently, the specific effects of controlled, localized absorption on the coherence and dynamics of QWs remain largely unexplored.

Methodology
This work addresses the gap in understanding absorbing boundaries by combining theoretical simulations with experimental implementation on a universal photonic quantum processor.

• Theoretical Framework: The authors modeled a discrete-time quantum walk (DTQW) on a finite one-dimensional lattice ($2M=8$ sites) using a coined model. The evolution involves a Hadamard coin operator followed by a conditional shift. To simulate leakage, asymmetric boundary conditions were imposed: one edge acts as a hard, perfectly reflective boundary, while the opposite edge functions as a partially absorbing (leaking) boundary. The leakage is modeled via a tunable transmission parameter $r^2$ in the beam splitter transformation at the boundary, where $r^2=0$ represents a lossless reflection and $0 < r^2 < 1$ represents varying degrees of leakage. The study analyzed the time evolution of the walker's mean position ($\langle x \rangle$) and variance ($\sigma^2$) over 100 steps for different injection sites (near the leaking edge vs. near the reflective edge) and leakage strengths ($r^2=0.2$ for low leakage, $r^2=0.8$ for high leakage).

• Experimental Implementation: The experiments were conducted using the QuiX Quantum Alquor20, a reconfigurable photonic processor featuring 20 input/output ports and a mesh of 190 tunable Mach-Zehnder interferometers (MZIs) fabricated on stoichiometric silicon nitride. The processor was programmed to implement the unitary transformation corresponding to the leaking QW model.

  • Setup: A coherent attenuated beam at $\lambda = 942$ nm served as the walker.
  • Configuration: 8 of the 20 available modes were utilized to simulate the lattice. Mode 1 was configured as the leaking boundary, and Mode 8 as the reflective boundary.
  • Procedure: Photons were injected into specific input modes (2, 3, 6, 7) representing positions near the leaking or reflective edges. The system was evolved for $N=4$ to $20$ steps. Output intensities were measured using photodiodes to reconstruct the spatial probability distributions. Calibration was performed to normalize intensity distributions.

Key Contributions

1. Experimental Realization of Leaking Boundaries: The paper presents one of the few experimental demonstrations of QWs with absorbing boundaries, utilizing a programmable photonic processor to introduce tunable, mode-dependent losses.
2. Systematic Analysis of Leakage Effects: The study systematically investigates how the leakage probability and the initial injection site (distance from the lossy edge) influence the walker's dynamics, specifically focusing on interference patterns, coherence, and spreading behavior.
3. Validation of Theoretical Models: The work provides a direct comparison between numerical simulations and experimental data, validating the theoretical model of non-unitary QW evolution in a controlled photonic environment.

Results

• Low-Leakage Regime ($r^2=0.2$): The dynamics closely resemble those of an ideal, lossless QW. The symmetry between opposite injection sites is largely preserved, and the variance exhibits oscillatory behavior indicative of coherent spreading and relocalization. Losses in this regime result primarily in a moderate attenuation of amplitudes without qualitatively altering the walk's structure.
• High-Leakage Regime ($r^2=0.8$): Significant deviations from the lossless case are observed. The symmetry between injection sites is broken, and the oscillation frequencies and amplitudes of the mean position and variance are modified.
  • Proximity Effect: The impact of leakage is most pronounced when the walker is injected near the leaking boundary (e.g., input site 2). In this scenario, strong leakage leads to faster propagation of the mean position and enhanced spreading of the wavefunction compared to the lossless or weak-leakage cases.
  • Distance Effect: When the walker is injected far from the leaking boundary (e.g., input site 7), the dynamics remain largely governed by coherent propagation for the initial steps (up to ~15 steps), as the wavefunction has not yet reached the lossy edge. Even in the strong-leakage regime, the early-time dynamics for distant injection sites are nearly identical to the lossless case.
• Experimental Agreement: The experimental measurements of mean position and variance show good agreement with the theoretical simulations across all tested configurations, confirming the processor's ability to accurately model non-unitary quantum dynamics.

Significance and Claims
The authors claim that their results demonstrate that controlled losses can effectively modify interference patterns and coherence in QWs without fully destroying quantum features. The study establishes that the presence of absorbing boundaries acts as a tunable resource for engineering on-chip QWs. Specifically, the work suggests that:

• Controlled decoherence can be engineered to probe the dynamics of open quantum systems.
• The introduction of absorbing sites offers a pathway for simulating energy transport in complex networks, such as those found in biological systems (e.g., the FMO complex), where excitons interact with an environment.
• The ability to tune leakage provides a method for exploring quantum-computational methods that may take advantage of decoherence, rather than treating it solely as a detrimental error source.

The paper concludes that while strong leakage induces decoherence and alters propagation, the system retains quantum characteristics, making it a viable platform for studying open quantum system dynamics and optimizing transport processes in the presence of noise.