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arXiv:2602.05566 · Physical Review A

Simulation of boson sampling with optical feedback

Yu. A. Biriukov, I. V. Dyakonov

quant-phSubmitted Feb 5, 2026• Announced Feb 6, 2026• 1 citation

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.

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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.

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The boson sampling problem has brought a lot of attention in the quantum information field because it is not efficiently solvable with a classical computer; nonetheless it can be implementable with linear optical interferometers with single-boson sources. Recently, we introduced a more general problem, the multi-boson correlation sampling problem, which allows to take advantage of the muti-mode spectral distribution of the bosonic sources together with time-correlated measurements in order to achieve a sampling not only over the output channels of the interferometer but also over the joint detection times. This problem was analyzed for both single-photon sources and thermal sources. In this work, we demonstrate that it is possible to use single qubit bosonic sources in order to sample not only in the described "space-time" degree of freedom but also over all the possible exponential number of multi-qubit output states.

Boson Sampling is the problem of sampling from the same output probability distribution as a collection of indistinguishable single photons input into a linear interferometer. It has been shown that, subject to certain computational complexity conjectures, in general the problem is difficult to solve classically, motivating optical experiments aimed at demonstrating quantum computational ``supremacy.'' There are a number of challenges faced by such experiments, including the generation of indistinguishable single photons. We provide a quantum circuit that simulates Boson Sampling with arbitrarily distinguishable particles. This makes clear how distinguishabililty leads to decoherence in the standard quantum circuit model, allowing insight to be gained. At the heart of the circuit is the quantum Schur transform, which follows from a representation theoretic approach to the physics of distinguishable particles in first quantization. The techniques are quite general and have application beyond Boson Sampling.

The original formulation of the boson sampling problem assumed that little or no photon collisions occur. However, modern experimental realizations rely on setups where collisions are quite common, i.e., the number of photons M injected into the circuit is close to the number of detectors N. Here we present a classical algorithm that simulates a bosonic sampler: it calculates the probability of a given photon distribution at the interferometer outputs for a given distribution at the inputs. This algorithm is most effective in cases with multiple photon collisions, and in those cases, it outperforms known algorithms.

Boson-sampling problems are specific cases of multi-photon quantum interference in which the complexity of sampling from the output distribution of multiple photons on multiple modes arises from its relation to certain matrix functions that cannot be efficiently computed in polynomial time using the best-known classical algorithms. Specifically, gaussian boson sampling (GBS) is a protocol that employs Gaussian states as inputs to a linear interferometer, and whose output samples can be used to accelerate graph search and optimization problems [1]. To date, GBS experiments have been reported by using path or time-bin encoding of the optical modes.