On Reducing the Amount of Samples Required for Training of QNNs: Constraints on the Linear Structure of the Training Data

Alexander Mandl; Johanna Barzen; Frank Leymann; Daniel Vietz

Home/Archive/arXiv

arXiv · 2309.13711

On Reducing the Amount of Samples Required for Training of QNNs: Constraints on the Linear Structure of the Training Data

Alexander Mandl·Johanna Barzen·Frank Leymann·Daniel Vietz

View original source ↗Read source document ↗Export metadata ↓

Abstract

Training classical neural networks generally requires a large number of training samples. Using entangled training samples, Quantum Neural Networks (QNNs) have the potential to significantly reduce the amount of training samples required in the training process. However, to minimize the number of incorrect predictions made by the resulting QNN, it is essential that the structure of the training samples meets certain requirements. On the one hand, the exact degree of entanglement must be fixed for the whole set of training samples. On the other hand, training samples must be linearly independent and non-orthogonal. However, how failing to meet these requirements affects the resulting QNN is not fully studied. To address this, we extend the proof of the QNFL theorem to (i) provide a generalization of the theorem for varying degrees of entanglement. This generalization shows that the average degree of entanglement in the set of training samples can be used to predict the expected quality of the QNN. Furthermore, we (ii) introduce new estimates for the expected accuracy of QNNs for moderately entangled training samples that are linear dependent or orthogonal. Our analytical results are (iii) experimentally validated by simulating QNN training and analyzing the quality of the QNN after training.

Explore related subjects

quant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Alexander Mandl, Johanna Barzen, Frank Leymann, Daniel Vietz. 2023-12-10. On Reducing the Amount of Samples Required for Training of QNNs: Constraints on the Linear Structure of the Training Data. https://arxiv.org/abs/2309.13711

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Measurement-Induced Local Dephasing Generates Symmetrically Located Entangled Sites in a Fermionic Tight-Binding Lattice

We investigate an odd-sized fermionic open tight-binding chain subjected to stochastic projective measurements at its central site, effectively inducing localized dephasing. Focusing initially on the single-particle regime, we demonstrate that when the system is prepared in an even-parity state, the dynamics under central-site dephasing drive it toward a nontrivial steady state, which we characterize through both analytical and numerical approaches. Remarkably, this steady state exhibits long-range quantum correlations in the form of symmetrically positioned, pairwise entangled sites across the chain. We further show that the degree of pairwise mode entanglement can be significantly enhanced by increasing the particle number, provided the system is initialized within a specific symmetry sector associated with an underlying strong symmetry operator. Our results identify a minimal measurement-induced route for generating symmetry-selected long-range pairwise mode entanglement, with possible implications for quantum communication and distributed quantum information processing.

quant-ph↗

arXiv2026

Role of scrambling and noise in temporal information processing with quantum systems

Scrambling quantum systems have attracted attention as effective substrates for temporal information processing. Here we consider a quantum reservoir processing framework that captures a broad range of physical computing models with quantum systems. We examine the scalability and memory retention of the model with scrambling reservoirs modelled by high-order unitary designs in both noiseless and noisy settings. In the former regime, we show that measurement readouts become exponentially concentrated with increasing reservoir size, yet strikingly do not worsen with the reservoir iterations. Thus, while repeatedly reusing a small scrambling reservoir with quantum data might be viable, scaling up the problem size deteriorates generalization unless one can afford an exponential shot overhead. In contrast, the memory of early inputs and initial states decays exponentially in both reservoir size and reservoir iterations. In the noisy regime, we also prove that memory decays exponentially in time for local noisy channels. These results required us to introduce new proof techniques for bounding concentration in temporal quantum models. Beyond this extreme scrambling regime, we numerically demonstrate that exponential concentration can still exist even with a physical reservoir such as an Ising model whenever the reservoir operates in a quantum-chaotic phase. In contrast, physical reservoirs in a many-body localized phase and at the edge of chaos appear to not suffer from such phenomena

quant-ph↗

arXiv2026

Superpositions of Quantum Gaussian Processes

We generalise the Gaussian formalism of Continuous Variable (CV) systems to describe their entanglement with Discrete Variable (DV) systems, leading to superpositions of CV Gaussian states. A new class of CV-DV entangled states, named Gaussian-Branched Cat States (GBCSs), yields an analytical formalism to describe quantum hybrid systems. GBCSs are fully characterised by their superposed phase-space parameters: sets of generalised complex first moments and covariance matrices, along with the DV reduced density matrix (phases and contrasts). These states arise in all the instances where Gaussian dynamics, operations, and measurements are performed conditionally on a DV state. The time evolution of the GBCS phase-space parameters allows one -- via a new set of equations in closed form -- to analytically treat a large set of unitary and open dynamics, generated by Gaussian Hamiltonians labelled by DV eigenvalues. Conditional operations, such as displacements and rotations, and Gaussian measurements (homodyne/heterodyne) jointly with DV projectors, can be both described as maps on GBCS's parameters. A phase-space perturbation theory is given to extend the analysis to non-orthogonal DV super-operators, e.g. DV decay. We showcase our general formalism with two paradigmatic examples of experimental modelling: (i) a dispersively coupled qubit to a driven parametric amplifier; (ii) a levitated nanoparticle undergoing Stern-Gerlach matter-wave interferometry in a diffusive environment. Both examples highlight the generation of novel Wigner negativities through qubit measurements.

quant-ph↗