The paper analyzes quantum sampling in interacting light–matter networks that combine fermionic and bosonic inputs, with the stated aim of obtaining output expressions beyond determinants and permanents. In the pure limits, fermionic sampling reduces to determinants and bosonic sampling to permanents; mixed statistics can instead be represented by matrix functions that are neither. For the specified measurement projections, the output probability is the squared modulus of a generalized immanant—a calculated function of the relevant network submatrix—and the construction is stated to cover fermions, bosons and anyons. The model uses a linear network with hybrid modes, allows at most one detected excitation per mode, and focuses on matched incident and detected excitation counts.
A small example makes the distinction concrete
The first application keeps the construction small: it has M = 3 modes and N = 3 excitations, using fermionic, bosonic and mixed mode characters. In the two-fermion/one-boson case, the resulting function is neither a permanent, a determinant nor the common standard immanant—the usual character-defined matrix-function family used as a baseline. Its residual component is described as linearly independent and orthogonal.
The larger model adds a mixed term
The paper tests the same idea in an asymmetric five-mode example with two bosonic and three fermionic modes. The overlap contains determinant and permanent contributions, but also an additional mixed matrix function called immFB. Unlike a standard immanant, immFB is not character-defined.
The reported vector decomposition for immFB contains determinant and standard-immanant components but no permanent component. Its relative residual norm is 0.88, meaning most of the vector norm remains outside those projections. The authors do not give a specific complexity result for the residual or the other coefficient; to their knowledge, those terms have not been studied in the literature.
Quasiparticles leave permutation territory
A quasiparticle proof of concept, motivated by the Jaynes–Cummings model, uses N = 2 excitations in M = 2 modes with F = B = 1. Its sampling expression includes two constant maps, ρ(i) = 1 and ρ(i) = 2. Each map sends every input index to one fixed value, so neither is a permutation, and the resulting expression is described as exceeding general immanants.
For generalized quasiparticle sampling with M > 2 modes and N > 2 quasiparticles, the paper contrasts N! permutation terms with N^N all-function terms. It says the latter count grows faster than the term counts in permanent, determinant and immanant expressions. That counting statement alone is not a proof of computational hardness.
What remains open
The authors interpret these constructions as giving hybrid samplers a form of parastatistical functionality inaccessible to systems governed only by bosonic or fermionic exchange symmetries, with possible relevance to complexity analysis. The conclusion remains tied to the specified network and measurement assumptions, and no experimental implementation or empirical validation is reported. The work therefore leaves both the physical realization of the samplers and the complexity of the residual components for further study.
The supplied document is an arXiv version 1 preprint, identified as arXiv:2608.25788v1 and headed 26 Aug 2026. The authors acknowledge support from the Deutsche Forschungsgemeinschaft through TRR 142, Projects A04 and C10, under Grant 231447078.
Paper data and sources
Original title: Quantum sampling in hybrid light-matter systems with mixed statistics
Authors: Franziska Barkhausen, Laura Ares, Stefan Schumacher, Jan Sperling
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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