arXiv subjects

Hans-Martin Rieser

Publications and source records attributed to Hans-Martin Rieser.

7 recordsLinked to original sources

arXiv2026

Quantum Tensor Network Learning with DMRG

Tensor Networks are a relatively new machine learning approach. The architectures proposed initially are inspired by approaches from quantum many-body physics simulations. One common layout is the matrix product state (MPS) also known as a tensor train optimized with gradient descent techniques. We introduce a global normalization condition, so that the MPS represents a quantum state. We investigate two optimization methods that find the locally optimal tensors and compare them regarding their effectiveness. One is based on gradient descent and the other on an adaptation of DMRG.

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arXiv2026

Entanglement is Half the Story: Post-Selection vs. Partial Traces

While tensor networks have their traditional application in simulating quantum systems, in the recent decade they have gathered interest as machine learning models. We combine the experience from both fields and derive how quantum constraints placed on a tensor network manifest a change in capabilities. To this end, we employ a method of inference of classical tensor networks on a quantum computer to define a hybrid architecture. This hybrid tensor network is a practical unified framework for it's classical and quantum tensor network edge cases. We identify post-selection as the important property on which this interpolation hinges. The amount of post-selection corresponds to the level to which quantum constraints are enforced on the tensor network. On this basis, we propose a new hyperparameter which controls the transition between the hybrid and the quantum tensor network. In the comparison of classical and quantum tensor networks it complements the bond dimension. Quantum machine learning is improved by using the hyperparameter to allocate the practically limited post-selection to the quantum model in a trainable manner.

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arXiv2026

Theory and interpretability of Quantum Extreme Learning Machines: a Pauli-transfer matrix approach

Quantum reservoir computers (QRCs) have emerged as a promising approach to quantum machine learning, since they utilize the natural dynamics of quantum systems for data processing and are simple to train. Here, we consider $n$-qubit quantum extreme learning machines (QELMs) with initial-state encoding and continuous-time reservoir dynamics. We apply the Pauli transfer matrix (PTM) formalism to theoretically analyze the influence of encoding, reservoir dynamics, and measurement operations (including temporal multiplexing) on the QELM performance. This formalism reveals the complete set of (nonlinear) features generated by the encoding, and shows how the subsequent quantum channels linearly transform these Pauli features before they are probed by the chosen measurement operators. Optimizing such a QELM can therefore be cast as a decoding problem in which one shapes the channel-induced transformations such that task-relevant features become available to the regressor, effectively reversing the information scrambling of a unitary. Operator spreading under unitary evolution determines decodability of Pauli features, which underlies the nonlinear processing capacity of the reservoir. When paired with certain observables, structured Hamiltonians can reduce model expressivity, as reflected in a low readout rank. We trace this effect to Hamiltonian symmetries and derive asymptotic rank estimates for symmetry-resolved observable families. The PTM formalism yields a nonlinear vector (auto-)regression model as an interpretable classical representation of a QELM. As a specific application, we focus on forecasting nonlinear dynamical systems and show that a QELM trained on such trajectories learns a surrogate-approximation to the underlying flow map.

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arXiv2026

Kernel-based optimization of measurement operators for quantum reservoir computers

Finding optimal measurement operators is crucial for the performance of quantum reservoir computers (QRCs), since they employ a fixed quantum feature map. We formulate the training of both stateless (quantum extreme learning machines, QELMs) and stateful (memory dependent) QRCs in the framework of kernel ridge regression. We thus extend the kernel viewpoint of supervised quantum models to recurrent QRCs by deriving an exact Hilbert--Schmidt kernel representation of the optimal readout observable on history space. This approach renders an optimal measurement operator that minimizes prediction error for a given reservoir and training dataset. For large qubit numbers, this method is more efficient than the conventional training of QRCs. We discuss efficiency and practical implementation strategies, including Pauli basis decomposition and operator diagonalization, to adapt the optimal observable to hardware constraints. To demonstrate the effectiveness of this approach, we present numerical experiments on image classification and time series prediction tasks, including chaotic and strongly non-Markovian systems. The developed method can also be applied to other quantum machine learning models.

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arXiv2025

A Hybrid Quantum Solver for Gaussian Process Regression

Gaussian processes are widely known for their ability to provide probabilistic predictions in supervised machine learning models. Their non-parametric nature and flexibility make them particularly effective for regression tasks. However, training a Gaussian process model using standard methods requires matrix inversions with a cubic time complexity, which poses significant computational challenges for inference on larger datasets. Quantum algorithms, such as the HHL algorithm, have been proposed as solutions that overcome the need for classical matrix inversions by efficiently solving linear systems of equations using quantum computers. However, to gain a computational advantage over classical algorithms, these algorithms require fault-tolerant quantum computers with a large number of qubits, which are not yet available. The Variational Quantum Linear Solver is a hybrid quantum-classical algorithm that solves linear systems of equations by optimizing the parameters of a variational quantum circuit using a classical computer. This method is especially suitable for noisy intermediate-scale quantum computers, as it does not require many qubits. It can be used to compute the posterior distribution of a Gaussian process by reformulating the matrix inversion into a set of linear systems of equations. We empirically demonstrate that using the Variational Quantum Linear Solver to perform inference for Gaussian process regression delivers regression quality comparable to that of classical methods.

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arXiv2025

Hybrid quantum tensor networks for aeroelastic applications

We investigate the application of hybrid quantum tensor networks to aeroelastic problems, harnessing the power of Quantum Machine Learning (QML). By combining tensor networks with variational quantum circuits, we demonstrate the potential of QML to tackle complex time series classification and regression tasks. Our results showcase the ability of hybrid quantum tensor networks to achieve high accuracy in binary classification. Furthermore, we observe promising performance in regressing discrete variables. While hyperparameter selection remains a challenge, requiring careful optimisation to unlock the full potential of these models, this work contributes significantly to the development of QML for solving intricate problems in aeroelasticity. We present an end-to-end trainable hybrid algorithm. We first encode time series into tensor networks to then utilise trainable tensor networks for dimensionality reduction, and convert the resulting tensor to a quantum circuit in the encoding step. Then, a tensor network inspired trainable variational quantum circuit is applied to solve either a classification or a multivariate or univariate regression task in the aeroelasticity domain.

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arXiv2023

Tensor networks for quantum machine learning

Once developed for quantum theory, tensor networks have been established as a successful machine learning paradigm. Now, they have been ported back to the quantum realm in the emerging field of quantum machine learning to assess problems that classical computers are unable to solve efficiently. Their nature at the interface between physics and machine learning makes tensor networks easily deployable on quantum computers. In this review article, we shed light on one of the major architectures considered to be predestined for variational quantum machine learning. In particular, we discuss how layouts like MPS, PEPS, TTNs and MERA can be mapped to a quantum computer, how they can be used for machine learning and data encoding and which implementation techniques improve their performance.

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