An Exact Polynomial Task-Risk Bridge for Time-Multiplexed Photonic Quantum Reservoirs

Abstract:Time-multiplexed (TDM) photonic reservoirs are usually judged by their scores on a few datasets, which say little about a chip's overall performance, and designed by enumerating candidate circuits and simulating each one, which is inefficient and gives no guarantee of finding a good structure. For a time-unrolled passive linear network with data-modulated gates or sources and a ridge readout, we derive an exact bridge from the task to the prediction risk in three steps. (i) Task features: the finite-shot risk depends on the time series only through finitely many statistics of its training windows and their correlations with the target, selected by the encoding and the optical paths. (ii) Encoding: every output moment is a polynomial in the features of the data-modulated sites, with degree and support bounded by tuples of optical paths. Gate encoding reads the characteristic function and creates interactions across time lags; squeezing encoding reads the moment-generating function and displacement encoding low-order moments, and with quadrature receivers they yield only additive and linear models, respectively, for every topology. (iii) Readout: for homodyne, heterodyne and photon-number receivers the risk closes exactly, and its dependence on the number of shots is an explicit sum over signal-to-noise modes; threshold clicks admit finite-dictionary approximations with certified error. Parts of the network that share no light carry independent states, which a receiver combines only through measurement events that join them. Checks on seven datasets confirm every exact statement to machine precision. The three steps thus yield a useful space of candidate TDM architectures, pruned by exact statements rather than by trial simulation, which can in future guide the selection of TDM chips for target requirements.

Quantum Physics (quant-ph); Optics (physics.optics)

arXiv:2609.35243 [quant-ph]

(or arXiv:2609.35243v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.35243 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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