arXiv subjects

Merijn Moody

Publications and source records attributed to Merijn Moody.

5 recordsLinked to original sources

arXiv2026

A formal framework for higher-order spin models via hypergraphs, polymatroids, and the Tutte polynomial

We develop a rigorous mathematical framework for statistical mechanics models on hypergraphs, and give conditions for lifting the classical connection between Potts model partition functions and the Tutte polynomial from graphs to hypergraphs. We define hypergraphical models and their associated partition functions, and extend these to special classes of models induced by families of interaction functions. For boolean interaction families, whose interaction functions map to $\{0,1\}$, we show that the partition function is determined by a combinatorial rank function, and we establish sufficient conditions for a hypergraph deletion-contraction recurrence and for when the rank function defines a polymatroid. We illustrate the theory by applying it to three hypergraph interaction families: Parity Ising, Delta Potts, and And Ising. The induced hypergraphical models are not isomorphic to each other, but the first two reduce to the same graphical Ising models. We identify three polymatroids naturally associated with hypergraphs for these models: respectively, the binary matroid of the incidence matrix over $\mathbb{F}_2$, the hypergraphical polymatroid, and the boolean polymatroid. For graphs, the first two reduce to the classical graphical matroid, and their partition functions recover the multivariate Tutte polynomial. The Tutte polynomial thus admits at least two distinct generalizations for hypergraphs, both satisfying a deletion-contraction recurrence: the Tutte polynomial of the binary matroid of the hypergraph incidence matrix, and a multivariate version of the Poincar\'e polynomial of the hypergraphical polymatroid. The partition functions of And Ising models are likewise multivariate versions of the Poincar\'e polynomial, here of the boolean polymatroid. These examples illustrate the much greater range of hypergraphical models and underscore the need for the unifying theory.

math.CO↗

arXiv2026

A Hypergraph Tutte Polynomial

We introduce a Tutte polynomial for hypergraphs, $T_{\mathrm{HG}}$, together with $T_k$, a related Tutte polynomial for $k$-polymatroids. Both invariants admit deletion--contraction recursions that remain within their respective classes, and they are linked by the fact that $T_{\mathrm{HG}}$ specializes to $T_k$ on the associated polymatroid of any $(k+1)$-uniform hypergraph. We show that $T_{\mathrm{HG}}$ satisfies several desirable Tutte type properties, including multiplicativity and duality, while $T_k$ further admits a universality theorem, as well as a convolution product formula. In the uniform hypergraph setting, these latter results specialize back to $T_{\mathrm{HG}}$. We also relate $T_{\mathrm{HG}}$ to hypergraph extensions of the Potts and random cluster models. In particular, we study degree dependent random cluster and Potts partition functions, as well as Grimmett's many body Potts model, and compare their relationship with $T_{\mathrm{HG}}$ in both the general and uniform settings. Finally, we compare $T_{\mathrm{HG}}$ with the polymatroid Tutte polynomial $\mathcal{T}_P$ of Bernardi, K'alm'an, and Postnikov, showing that the two are incomparable in distinguishing power. As a consequence, we answer negatively a question raised by these authors by proving that the characteristic polynomial is not, in general, a specialization of $\mathcal{T}_P$.

math.CO↗

arXiv2026

QMaxCal: Path-Space Regularization for Open Quantum Control via Girsanov's Theorem

Reliable quantum control in the presence of decoherence requires policies that combat the effect of environmental noise on the controlled dynamics. Open quantum systems under continuous monitoring generate classical measurement records whose drift depends on the noise experienced by the system; the records of two evolutions sharing the same decoherence channels differ only in this drift, so Girsanov's theorem yields a closed-form, differentiable estimator of the KL divergence between their trajectory distributions. We instantiate this estimator with two physically motivated reference measures, yielding two regularizers that both drive the system toward states where the effects of decoherence are minimal: the Wiener KL (KL_W), which is empirically more effective under certain conditions on the noise model, and the drift-variance regularizer (R_DV), which works for all noise models. Both are qualitatively distinct from existing penalties on control fluence or smoothness: they penalize the observable consequences of control on the decoherence channels rather than the control amplitude itself. The regularizers outperform unregularized gradient-based and reinforcement-learning baselines across a range of open quantum systems -- including single- and multi-qubit benchmarks and a multi-qubit chain calibrated to a published snapshot of the IBM Kingston processor -- along several axes of evaluation: final-state fidelity, robustness to mismatch in the assumed noise model (gains grow from +17 pp at training noise to +27 pp under 2.5x noise mismatch), and occupation of forbidden states. The regularizers reduce infidelity by up to 50%, with ~16% gains on the calibrated IBM Kingston chain.

quant-ph↗

arXiv2026

Modeling Discrete Data with High-Order Vector Potts Models

Modeling high-dimensional data is challenging, yet essential to understanding many complex systems. Maximum entropy models such as Ising and Potts models have been used extensively to capture pairwise interactions from correlation patterns in data, allowing to infer graphical representations of complex systems from observations (e.g., from protein sequences or neural population activity). Recently, there has been growing interest in modeling higher-order correlation patterns involving simultaneously three or more variables. While progress has been made in binary data with high-order Ising models, we extend this framework to the more general case of discrete data. We introduce q-state spin models, a complete family of maximum entropy models that generalize the vector Potts model to include long-range and arbitrary high-order interactions. In the pairwise case, our models allow for more diverse interaction types compared to the standard vector Potts model. We discuss their statistical interpretation with examples and relate them to discrete Fourier analysis. Using a loop expansion of the partition function, we show that the statistical properties of spin models are fully captured by the algebraic structure of their interactions. We define gauge transformations under which this structure, and thus the partition function, remains invariant. Models equivalent under gauge transformations can be seen as different representations of the same abstract statistical model, despite generally having interactions of different orders, extending results from the binary case. For practical application to data analysis, we focus on a subset of models known in the binary case as Minimally Complex Models, generalizing them to discrete data. We obtain a closed-form expression for the marginal likelihood of these models, enabling fast model selection. We illustrate their use with simple real-world examples.

stat.ME↗

arXiv2025

Towards a Large Physics Benchmark

We introduce a benchmark framework developed by and for the scientific community to evaluate, monitor and steer large language model development in fundamental physics. Building on philosophical concepts of scientific understanding and creativity, we develop a scoring system in which each question is scored by an expert for its correctness, difficulty, and surprise. The questions are of three forms: (i) multiple-choice questions for conceptual understanding, (ii) analytical problems requiring mathematical derivation, and (iii) openended tasks requiring complex problem solving. Our current dataset contains diverse set of examples, including a machine learning challenge to classify high-energy physics events, such as the four top quark signal. To ensure continued relevance, we propose a living benchmark, where physicists contribute questions, for instance alongside new publications. We invite contributions via: http://www.physicsbenchmarks.org/. We hope that this benchmark will enable a targeted AI development that can make a meaningful contribution to fundamental physics research.

physics.data-an↗