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arXiv:2608.25788

Quantum sampling in hybrid light-matter systems with mixed statistics

Franziska Barkhausen, Laura Ares, Stefan Schumacher, Jan Sperling

quant-phmath-phSubmitted Aug 26, 2026• Announced Aug 27, 2026• 0 citations
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Abstract

Boson sampling is one of the most prominent methods to verify quantum advantage, harnessing the computational complexity of the permanent of a network matrix. Similar quantum sampling problems utilize other expressions for the output probabilities, such as the fermion-based analog applying determinants. In this work, we study sampling setups of interacting light-matter systems, combining fermion and boson sampling. The mixed network input consists of fermionic and bosonic excitations. Sampling the output probability of the modes then yields contributions that are neither purely determinant nor permanent but general immanants. Specifically, we prove that these immanant contributions are linearly independent from determinants and permanents, rendering our mixed sampling network an ideal candidate for light-matter quantum application beyond pure fermion and boson sampling. Furthermore, quantum sampling with statistics that are neither bosonic nor fermionic results in expressions which even exceed immanants, further extending the functionality of quantum samplers.

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Boson sampling is one of the most prominent methods to verify quantum advantage, harnessing the computational complexity of the permanent of a network matrix. Similar quantum sampling problems utilize other expressions for the output probabilities, such as the fermion-based analog applying determinants. In this work, we study sampling setups of interacting light-matter systems, combining fermion and boson sampling. The mixed network input consists of fermionic and bosonic excitations. Sampling the output probability of the modes then yields contributions that are neither purely determinant nor permanent but general immanants. Specifically, we prove that these immanant contributions are linearly independent from determinants and permanents, rendering our mixed sampling network an ideal candidate for light-matter quantum application beyond pure fermion and boson sampling. Furthermore, quantum sampling with statistics that are neither bosonic nor fermionic results in expressions which even exceed immanants, further extending the functionality of quantum samplers.
Boson Sampling is a prominent candidate for the demonstration of quantum computational advantage but it remains unclear whether a large-scale boson sampler can find useful computational applications. The challenge is that to use a boson sampler to estimate, for example, a physical observable, it is necessary to coarse grain the outcome distribution, owing to the exponential size of the outcome space and the anti-concentration properties of the Boson Sampling distribution. In this work, we analyse the complexity of a specific type of coarse-graining of Boson Sampling distributions based on linear functions of the output photon occupation numbers, which we refer to as linear statistics. We present an efficient classical algorithm for approximating linear statistics of boson samplers within additive error for different kinds of input states, such as Fock states and squeezed states. This allows us to unify in the same framework recent results on efficient quantum-inspired classical algorithms for simulating molecular vibronic spectra, or efficient classical approximations of coarse-grained distributions based on detector binning. Additionally, we show how our algorithm can be used to classically evaluate a proposed one-way function based on Boson Sampling, while other proposals for cryptographic applications escape our classical simulation techniques. We leave open the question of classical simulability of non-linear statistics and connect it to the problem of computing transition amplitudes of linear-optical circuits with one layer of interactions.
Boson sampling has emerged as an important tool to demonstrate the difference between quantum and classical computers and has attracted the interest of experimentalists and theoreticians. In this work we study boson sampling from general, single-mode states using a scheme that can generate any such state by combining Gaussian states and photon number measurements. We derive a formula that can be used to calculate the output photon number probabilities of these states after they travel through a linear interferometer. This extends the Boson sampling protocol to the widest array of possible single-mode states and from this we show that the complexity scaling of all such states is similar and hence there is no complexity advantage of using complex input states over simpler ones.
This thesis presents results under the topics of quantum correlations and quantum information processing using linear-optical networks. The first part of this thesis concerns with quantum correlations, specially in Gaussian states, which play a central role as resource in quantum protocols. The second part is about quantum information processing using linear-optical networks that are of great practical interest due to their simple physical realization. In the first part, we present a measurement-based method for verifying nonzero quantum discord, a measure of quantum correlations, in bipartite systems that can be used for both continuous and discrete-variable systems. Based on this method, we show that the only bipartite Gaussian states with zero quantum discord are product states. We also present a simple and efficient experimental method for verifying quantum correlations in Gaussian states using homodyne measurements. Moreover, we introduce an operational measure for quantifying quantum correlations in Gaussian states that is purely based on Gaussian measurements. We then illustrate the operational significance of the measure in terms of a Gaussian quantum protocol. In the second part, we present results on linear quantum-optics experiments, specifically boson sampling, an intermediate model of quantum computation that can be implemented using single-photon states, linear-optical networks, and photodetectors. We first present an efficient experimental method for characterization of linear-optical networks, which is essential for practical implementation of boson sampling. We show that using coherent states one can directly measure all the nontrivial phases and moduli of the matrix describing the network. Next, we consider the problem of boson sampling with Gaussian states. We derive a general formula for calculating the output photon-counting probabilities. We show that for input thermal states, sampling from the output probability distribution can be efficiently classically simulated. Using this, we find that permanents of positive-semidefinite Hermitian matrices can be approximated to within a multiplicative error in BPPNP. We also discuss boson sampling with squeezed vacuum states. Finally, we provide a sufficient condition for the efficient classical simulation of the general sampling problem using linear-quantum optics, which involves a multimode input quantum state to a linear-optical network and set of measurements at the output. This condition is based on the phase-space quasiprobability distributions, and suggests that negativity is a quantum resource. We apply this condition to implementations of boson sampling using single-photon or spontaneous parametric down-conversion sources. We show that above some threshold for loss and noise in the experiment, boson sampling is classically simulatable.
Boson sampling is a key candidate for demonstrating quantum advantage and has already yielded significant advances in quantum simulation, machine learning, and graph theory. In this work, a unification and extension of distinct forms of boson sampling is developed. The devised protocol merges discrete-variable scattershot boson sampling with continuous-variable Gaussian boson sampling. Therefore, it is rendered possible to harness the complexity of more interesting states, such as squeezed photons, in advanced sampling protocols. A generating function formalism is developed for the joint description of multiphoton and multimode light undergoing Gaussian transformations. The resulting analytical tools enable one to explore interfaces of different photonic quantum-information-processing platforms. A numerical simulation of unified sampling is carried out, benchmarking its performance, complexity, and scalability. Entanglement is characterized to exemplify the generation of quantum correlations from the nonlinear interactions of a unified sampler.
Gaussian boson sampling (GBS) seeks quantum advantage by sampling photon-number patterns generated with squeezed inputs and passive linear optics. Many proposed GBS applications instead target function computation by applying functions to mode-resolved photon-number outcomes---a natural form of experimental postprocessing that produces classical outputs. Sampling hardness alone, however, does not determine the complexity of these tasks. By analyzing the irreducible decomposition of fixed-photon-number operator spaces, we prove that the expectation value of every such function in the average case over passive linear-optical networks can be classically evaluated for inputs with finite squeezing strength. We also provide a classical algorithm that estimates this value to inverse-polynomial additive error. The result provides new theoretical tools for analyzing linear-optical quantum systems, helps clarify the origin of current GBS hardness evidence, and inspires new applications of GBS with genuine quantum advantages.
We propose boson sampling from a system of coupled photons and Bose-Einstein condensed atoms placed inside a multi-mode cavity as a simulation process testing the quantum advantage of quantum systems over classical computers. Consider a two-level atomic transition far-detuned from photon frequency. An atom-photon scattering and interatomic collisions provide interactions that create quasiparticles and excite atoms and photons into squeezed entangled states, orthogonal to the atomic condensate and classical field driving the two-level transition, respectively. We find a joint probability distribution of atom and photon numbers within a quasi-equilibrium model via a hafnian of an extended covariance matrix. It shows a sampling statistics that is ♯P-hard for computing, even if only photon numbers are sampled. Merging cavity-QED and quantum-gas technologies into a hybrid boson sampling setup has the potential to overcome the limitations of separate, photon or atom, sampling schemes and reveal quantum advantage.