Simulation of boson sampling with optical feedback

TL;DR

The paper addresses boson sampling in a linear-optical circuit with optical feedback, where L looped output modes return to inputs, enabling time-multiplexed, open-system dynamics. It develops three complementary modeling frameworks—Kraus-operator formalism, partial-trace evolution, and correlation-tensor methods—to compute output distributions and the stationary state, proving that for most random interferometers a unique stationary state exists as a fixed point of the associated quantum channel. It provides a theorem linking the existence and uniqueness to the spectral properties of $U_{LL}$ and demonstrates stabilization times and density-matrix structures numerically, including methods to reconstruct the stationary state from correlation tensors. The work suggests that such looped-boson-sampling architectures can achieve quantum-simulation capabilities with significantly fewer spatial resources, while highlighting open questions about computational hardness and practical implementation in the presence of loss and finite truncation.

Abstract

This work presents a theoretical model of boson sampling with optical feedback, in which a subset of the interferometer's output modes is looped back into the input modes. If the bosons are injected periodically into the input modes of the interferometer and optical feedback lines' length match the period of injection, it allows for interference between bosons injected at the consequent time iterations. We propose several methods methods for computing the output photon distributions in both output spacial and temporal modes, including not only standard spatiotemporal mode-unfolding technique, but also the Kraus-operator formalism, and a correlation-tensor-based approach. The two latter approaches help us to reveal that for random interferometers this system evolves to a unique stationary state over time. Because of the existence of the stationary state, we introduce new computational problem \textit{Stationary Distribution Boson Sampling} which appears to be harder than conventional boson sampling problem and contains it as a special case when there are no optical feedback lines.

Figures (8)

• Figure 1: (Left) General scheme of a boson sampler with optical feedback channels and transfer matrix $U$. (Right) The same interferometer, but with all spatiotemporal modes mapped to consider the system as a standard boson sampler with total transfer matrix $U_{total}$.
• Figure 2: Distribution of system stabilization times for a two-mode system with one feedback channel in the absence of losses. The distribution is obtained from a sample of 30,000 random matrices.
• Figure 3: Characteristics of the stationary density matrix in the looped modes for a system with 2 modes, one of which is looped. (Left) Photon number distribution with the closest approximation by a thermal and a coherent state. (Right) View of the average stationary density matrix for this system. Bar height represents the modulus of matrix elements, color represents the phase. Since the stationary state is established in one mode, each cell corresponds to a specific photon number.
• Figure 4: Three-mode system with two feedback channels and a random input state. (Left) Random input state in the basis from 0 to 1 photon in 1 mode. (Right) Stationary state, which no longer has a block-diagonal structure due to the presence of off-diagonal elements in the input density matrix. Unlike Fig. \ref{['fig:photon_distrib']}, the axes here represent not the photon number but the state indices in the basis $\mathcal{H}_L$.
• Figure 5: Stationary density matrix in a 3-mode system with 2 feedback channels. Unlike the case with one loop, this density matrix has off-diagonal elements corresponding to superpositions of states with fixed photon numbers. Unlike Fig. \ref{['fig:photon_distrib']}, the axes here represent not the photon number but the state indices in the basis $\mathcal{H}_L$.