arXiv subjects

Annie E. Paine

Publications and source records attributed to Annie E. Paine.

13 recordsLinked to original sources

arXiv2026

Effective-Hamiltonian Quantum Solvers for Differential Equations: Alternative Constructions and Function Encodings

Differential equations can be encoded as ground-state problems by constructing a positive-semidefinite effective Hamiltonian whose minimum-energy state represents the solution. We extend this framework in several directions. First, we show how multiple nonzero initial, boundary, and data conditions can be incorporated into the homogeneous formulation $A | f\rangle=0$ using a known nonzero reference condition. We then analyse an alternative formulation based on $A |f\rangle = | b\rangle$, with Hamiltonian $H_b=A^\dagger(I-|b\rangle\langle b|)A$, which incorporates source terms and nonzero constraints through an augmented system. For nonlinear differential equations, we examine the unphysical ground-state degeneracy introduced by tensor-product representations and consider both Hamiltonian constraints and ansatz-level restrictions for targeting physically valid product states. Finally, we develop grid-value amplitude-encoded versions of both Hamiltonian constructions and compare them with spectral coefficient encoding. Through linear and nonlinear examples, we assess how the choice of Hamiltonian formulation and function representation affects solution recovery, spectral gap, degeneracy, and readout. These results broaden the applicability of ground-state-based quantum differential-equation solvers while clarifying their principal practical trade-offs.

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arXiv2026

Conservative quantum offline model-based optimization

Offline model-based optimization (MBO) refers to the task of optimizing a black-box objective function using only a fixed set of prior input-output data, without any active experimentation. Recent work has introduced quantum extremal learning (QEL), which leverages the expressive power of variational quantum circuits to learn accurate surrogate functions by training on a few data points. However, as widely studied in the classical machine learning literature, predictive models may incorrectly extrapolate objective values in unexplored regions, leading to the selection of overly optimistic solutions. In this paper, we propose integrating QEL with conservative objective models (COM) - a regularization technique aimed at ensuring cautious predictions on out-of-distribution inputs. The resulting hybrid algorithm, COM-QEL, builds on the expressive power of quantum neural networks while safeguarding generalization via conservative modeling. Empirical results on benchmark optimization tasks demonstrate that COM-QEL reliably finds solutions with higher true objective values compared to the original QEL, validating its superiority for offline design problems.

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arXiv2026

A unified quantum computing quantum Monte Carlo framework through structured state preparation

We extend Quantum Computing Quantum Monte Carlo (QCQMC) beyond ground-state energy estimation by systematically constructing the quantum circuits used for state preparation. Replacing the original Variational Quantum Eigensolver (VQE) prescription with task-adapted unitaries, we show that QCQMC can address excited-state spectra via Variational Fast Forwarding and the Variational Unitary Matrix Product Operator (VUMPO), combinatorial optimization via a symmetry-preserving VQE ansatz, and finite-temperature observables via Haar-random unitaries. Benchmarks on molecular, condensed-matter, nuclear-structure, and graph-optimization problems demostrate that the QMC diffusion step consistently improves the energy accuracy of the underlying state-preparation method across all tested domains. For weakly correlated systems, VUMPO achieves near-exact energies with significantly shallower circuits by offloading optimization to a classical tensor-network pre-training step, while for strongly correlated systems, the QMC correction becomes essential. We further provide a proof-of-concept demonstration that Haar-random basis state preparation within QCQMC yields finite-temperature estimates from pure-state dynamics.

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arXiv2026

Weak forms offer strong regularisations: how to make physics-informed (quantum) machine learning more robust

Physics-informed (PI) methodologies have surged to become a pillar route to solve Differential Equations (DEs), sustained by the growth of machine learning methods in scientific contexts. The main proposition of PI is to minimise variationally a loss function, formally ensuring that a neural surrogate of the solution has the DE locally satisfied. The nature of such formulation encouraged the exploration of equivalent quantum algorithms, where the surrogate solution is expressed by variational quantum architectures. The locality of typical loss functions emphasises the DE to hold at an ensemble of points sampled in the domain, but encounters issues when generalising beyond such points, or when propagating boundary conditions. Issues which affect classical and quantum PI algorithms alike. The quest to fill this gap in robustness and accuracy against mainstream DE solvers has led to a plethora of proposals in various directions. In particular, classical DE solvers have long employed the weak form - an integral based approach aiming at imposing a global condition on the solution - prioritising a good average behaviour instead of ``overfitting'' select points. Here, we propose and explore to combine contributions from both local and global loss functions in PI routines, to exploit the advantages and mitigate the weaknesses of both. We showcase this intuition in a variety of problems focusing on differentiable quantum architectures, and demonstrating in particular how orchestrating such hybrid loss formulation with domain decomposition can offer a strong advantage over local-only strategies.

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arXiv2025

Vortex Detection from Quantum Data

Quantum solutions to differential equations represent quantum data -- states that contain relevant information about the system's behavior, yet are difficult to analyze. We propose a toolbox for reading out information from such data, where customized quantum circuits enable efficient extraction of flow properties. We concentrate on the process referred to as quantum vortex detection (QVD), where specialized operators are developed for pooling relevant features related to vorticity. Specifically, we propose approaches based on sliding windows and quantum Fourier analysis that provide a separation between patches of the flow field with vortex-type profiles. First, we show how contour-shaped windows can be applied, trained, and analyzed sequentially, providing a clear signal to flag the location of vortices in the flow. Second, we develop a parallel window extraction technique, such that signals from different contour positions are coherently processed to avoid looping over the entire solution mesh. We show that Fourier features can be extracted from the flow field, leading to classification of datasets with vortex-free solutions against those exhibiting Lamb-Oseen vortices. Our work exemplifies a successful case of efficiently extracting value from quantum data and points to the need for developing appropriate quantum data analysis tools that can be trained on them.

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arXiv2025

Quantum algorithm for solving nonlinear differential equations based on physics-informed effective Hamiltonians

We propose a distinct approach to solving linear and nonlinear differential equations (DEs) on quantum computers by encoding the problem into ground states of effective Hamiltonian operators. Our algorithm relies on constructing such operators in the Chebyshev space, where an effective Hamiltonian is a sum of global differential and data constraints. Once the effective Hamiltonian is formed, solutions of differential equations can be obtained using the ground state preparation techniques (e.g. imaginary-time evolution and quantum singular value transformation), bypassing variational search. Unlike approaches based on discrete grids, the algorithm enables evaluation of solutions beyond fixed grid points and implements constraints in the physics-informed way. Our proposal inherits the best traits from quantum machine learning-based DE solving (compact basis representation, automatic differentiation, nonlinearity) and quantum linear algebra-based approaches (fine-grid encoding, provable speed-up for state preparation), offering a robust strategy for quantum scientific computing in the early fault-tolerant era.

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arXiv2024

What can we learn from quantum convolutional neural networks?

Quantum machine learning (QML) shows promise for analyzing quantum data. A notable example is the use of quantum convolutional neural networks (QCNNs), implemented as specific types of quantum circuits, to recognize phases of matter. In this approach, ground states of many-body Hamiltonians are prepared to form a quantum dataset and classified in a supervised manner using only a few labeled examples. However, this type of dataset and model differs fundamentally from typical QML paradigms based on feature maps and parameterized circuits. In this study, we demonstrate how models utilizing quantum data can be interpreted through hidden feature maps, where physical features are implicitly embedded via ground-state feature maps. By analyzing selected examples previously explored with QCNNs, we show that high performance in quantum phase recognition comes from generating a highly effective basis set with sharp features at critical points. The learning process adapts the measurement to create sharp decision boundaries. Our analysis highlights improved generalization when working with quantum data, particularly in the limited-shots regime. Furthermore, translating these insights into the domain of quantum scientific machine learning, we demonstrate that ground-state feature maps can be applied to fluid dynamics problems, expressing shock wave solutions with good generalization and proven trainability.

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arXiv2023

Physics-Informed Quantum Machine Learning: Solving nonlinear differential equations in latent spaces without costly grid evaluations

We propose a physics-informed quantum algorithm to solve nonlinear and multidimensional differential equations (DEs) in a quantum latent space. We suggest a strategy for building quantum models as state overlaps, where exponentially large sets of independent basis functions are used for implicitly representing solutions. By measuring the overlaps between states which are representations of DE terms, we construct a loss that does not require independent sequential function evaluations on grid points. In this sense, the solver evaluates the loss in an intrinsically parallel way, utilizing a global type of the model. When the loss is trained variationally, our approach can be related to the differentiable quantum circuit protocol, which does not scale with the training grid size. Specifically, using the proposed model definition and feature map encoding, we represent function- and derivative-based terms of a differential equation as corresponding quantum states. Importantly, we propose an efficient way for encoding nonlinearity, for some bases requiring only an additive linear increase of the system size $\mathcal{O}(N + p)$ in the degree of nonlinearity $p$. By utilizing basis mapping, we show how the proposed model can be evaluated explicitly. This allows to implement arbitrary functions of independent variables, treat problems with various initial and boundary conditions, and include data and regularization terms in the physics-informed machine learning setting. On the technical side, we present toolboxes for exponential Chebyshev and Fourier basis sets, developing tools for automatic differentiation and multiplication, implementing nonlinearity, and describing multivariate extensions. The approach is compatible with, and tested on, a range of problems including linear, nonlinear and multidimensional differential equations.

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arXiv2023

Quantum Chebyshev Transform: Mapping, Embedding, Learning and Sampling Distributions

We develop a paradigm for building quantum models in the orthonormal space of Chebyshev polynomials. We show how to encode data into quantum states with amplitudes being Chebyshev polynomials with degree growing exponentially in the system size. Similar to the quantum Fourier transform which maps computational basis space into the phase (Fourier) basis, we describe the quantum circuit for the mapping between computational and Chebyshev spaces. We propose an embedding circuit for generating the orthonormal Chebyshev basis of exponential capacity, represented by a continuously-parameterized shallow isometry. This enables automatic quantum model differentiation, and opens a route to solving stochastic differential equations. We apply the developed paradigm to generative modeling from physically- and financially-motivated distributions, and use the quantum Chebyshev transform for efficient sampling of these distributions in extended computational basis.

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arXiv2022

Quantum Kernel Methods for Solving Differential Equations

We propose several approaches for solving differential equations (DEs) with quantum kernel methods. We compose quantum models as weighted sums of kernel functions, where variables are encoded using feature maps and model derivatives are represented using automatic differentiation of quantum circuits. While previously quantum kernel methods primarily targeted classification tasks, here we consider their applicability to regression tasks, based on available data and differential constraints. We use two strategies to approach these problems. First, we devise a mixed model regression with a trial solution represented by kernel-based functions, which is trained to minimize a loss for specific differential constraints or datasets. Second, we use support vector regression that accounts for the structure of differential equations. The developed methods are capable of solving both linear and nonlinear systems. Contrary to prevailing hybrid variational approaches for parametrized quantum circuits, we perform training of the weights of the model classically. Under certain conditions this corresponds to a convex optimization problem, which can be solved with provable convergence to global optimum of the model. The proposed approaches also favor hardware implementations, as optimization only uses evaluated Gram matrices, but require quadratic number of function evaluations. We highlight trade-offs when comparing our methods to those based on variational quantum circuits such as the recently proposed differentiable quantum circuits (DQC) approach. The proposed methods offer potential quantum enhancement through the rich kernel representations using the power of quantum feature maps, and start the quest towards provably trainable quantum DE solvers.

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arXiv2022

Protocols for Trainable and Differentiable Quantum Generative Modelling

We propose an approach for learning probability distributions as differentiable quantum circuits (DQC) that enable efficient quantum generative modelling (QGM) and synthetic data generation. Contrary to existing QGM approaches, we perform training of a DQC-based model, where data is encoded in a latent space with a phase feature map, followed by a variational quantum circuit. We then map the trained model to the bit basis using a fixed unitary transformation, coinciding with a quantum Fourier transform circuit in the simplest case. This allows fast sampling from parametrized distributions using a single-shot readout. Importantly, simplified latent space training provides models that are automatically differentiable, and we show how samples from distributions propagated by stochastic differential equations (SDEs) can be accessed by solving stationary and time-dependent Fokker-Planck equations with a quantum protocol. Finally, our approach opens a route to multidimensional generative modelling with qubit registers explicitly correlated via a (fixed) entangling layer. In this case quantum computers can offer advantage as efficient samplers, which perform complex inverse transform sampling enabled by the fundamental laws of quantum mechanics. On a technical side the advances are multiple, as we introduce the phase feature map, analyze its properties, and develop frequency-taming techniques that include qubit-wise training and feature map sparsification.

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arXiv2021

Quantum Quantile Mechanics: Solving Stochastic Differential Equations for Generating Time-Series

We propose a quantum algorithm for sampling from a solution of stochastic differential equations (SDEs). Using differentiable quantum circuits (DQCs) with a feature map encoding of latent variables, we represent the quantile function for an underlying probability distribution and extract samples as DQC expectation values. Using quantile mechanics we propagate the system in time, thereby allowing for time-series generation. We test the method by simulating the Ornstein-Uhlenbeck process and sampling at times different from the initial point, as required in financial analysis and dataset augmentation. Additionally, we analyse continuous quantum generative adversarial networks (qGANs), and show that they represent quantile functions with a modified (reordered) shape that impedes their efficient time-propagation. Our results shed light on the connection between quantum quantile mechanics (QQM) and qGANs for SDE-based distributions, and point the importance of differential constraints for model training, analogously with the recent success of physics informed neural networks.

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arXiv2021

Solving nonlinear differential equations with differentiable quantum circuits

We propose a quantum algorithm to solve systems of nonlinear differential equations. Using a quantum feature map encoding, we define functions as expectation values of parametrized quantum circuits. We use automatic differentiation to represent function derivatives in an analytical form as differentiable quantum circuits (DQCs), thus avoiding inaccurate finite difference procedures for calculating gradients. We describe a hybrid quantum-classical workflow where DQCs are trained to satisfy differential equations and specified boundary conditions. As a particular example setting, we show how this approach can implement a spectral method for solving differential equations in a high-dimensional feature space. From a technical perspective, we design a Chebyshev quantum feature map that offers a powerful basis set of fitting polynomials and possesses rich expressivity. We simulate the algorithm to solve an instance of Navier-Stokes equations, and compute density, temperature and velocity profiles for the fluid flow in a convergent-divergent nozzle.

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