arXiv subjects

Eric A. F. Reinhardt

Publications and source records attributed to Eric A. F. Reinhardt.

5 recordsLinked to original sources

arXiv2026

A Quantum Roadmap for Softmax Attention: Exact Born-Rule Analogs for Softmax Attention on the Probability Simplex

The attention mechanism forms the foundation of many modern AI models such as the Transformer. In one subclass of problems where attention is used, inputs and outputs are bound to the probability simplex so that all outputs sum to one. In this setting, softmax attention admits an exact, component-by-component quantum realization. Attention scores are Hadamard-test statistics on block-encoded projections of amplitude-encoded inputs. The exponential softmax is the interior of a cosine-squared family generated by Born-rule measurement under an exact bijection, whose boundary expresses sparse attention with exact zeros at finite parameter values. The softmax temperature is a repetition count where post-selected measurement rounds realize discretized inverse temperature exactly. Value aggregation is a deterministic column-loading channel that dilates the column-stochastic value matrix. The gated residual is the preparation angle of a single ancilla, with the additive identity at a mixing angle of π/2. Every learnable parameter is a rotation-gate angle. The composed layer is exact in the infinite-shot limit with one measure-and-reload step per attention score; a fully-coherent variant is ε-approximate via quantum singular value transformation in the infinite depth limit. The algebraic core is machine-checked in Lean 4.

quant-ph↗

arXiv2026

Resource-Efficient QUBO Formulation for Anchored Currency Arbitrage

Currency arbitrage (CA) involves trading currencies in cycles to exploit discrepancies in market valuations. Quadratic unconstrained binary optimization (QUBO) involves minimizing a quadratic cost (energy) function of binary variables. Previous works have explored the use of QUBO to solve CA problems. We build on these previous works by introducing realistic constraints such as beginning cycles from a held currency and accounting for per-transaction trading fees. We show that this formulation requires fewer logical variables (qubits) than previous QUBO encodings in the literature. We derive provably sufficient penalty weights for its constraint terms. We also introduce an exact anchor-gauge reweighting of the exchange rates that compresses the QUBO coefficient range from the rate scale to the arbitrage scale, addressing the finite analog precision of annealing hardware. We demonstrate the efficacy of this formulation using classical simulated annealing against an exact Held-Karp baseline on the same CPU and show that it can effectively find profitable cycles and account for trading fees. Finally, we benchmark faithful implementations of five prior QUBO encodings at matched sampler budgets and show that the proposed encoding is the only one to recover the exact fee-adjusted optimum.

quant-ph↗

arXiv2026

GNN For Muon Particle Momentum estimation

Due to a high rate of overall data generation relative to data generation of interest, the CMS experiment at the Large Hadron Collider uses a combination of hardware- and software-based triggers to select data for capture. Accurate momentum calculation is crucial for improving the efficiency of the CMS trigger systems, enabling better classification of low- and high- momentum particles and reducing false triggers. This paper explores the use of Graph Neural Networks (GNNs) for the momentum estimation task. We present two graph construction methods and apply a GNN model to leverage the inherent graph structure of the data. In this paper firstly, we show that the GNN outperforms traditional models like TabNet in terms of Mean Absolute Error (MAE), demonstrating its effectiveness in capturing complex dependencies within the data. Secondly we show that the dimension of the node feature is crucial for the efficiency of GNN.

physics.data-an↗

arXiv2025

Sinusoidal Approximation Theorem for Kolmogorov-Arnold Networks

The Kolmogorov-Arnold representation theorem states that any continuous multivariable function can be exactly represented as a finite superposition of continuous single variable functions. Subsequent simplifications of this representation involve expressing these functions as parameterized sums of a smaller number of unique monotonic functions. These developments led to the proof of the universal approximation capabilities of multilayer perceptron networks with sigmoidal activations, forming the alternative theoretical direction of most modern neural networks. Kolmogorov-Arnold Networks (KANs) have been recently proposed as an alternative to multilayer perceptrons. KANs feature learnable nonlinear activations applied directly to input values, modeled as weighted sums of basis spline functions. This approach replaces the linear transformations and sigmoidal post-activations used in traditional perceptrons. Subsequent works have explored alternatives to spline-based activations. In this work, we propose a novel KAN variant by replacing both the inner and outer functions in the Kolmogorov-Arnold representation with weighted sinusoidal functions of learnable frequencies. Inspired by simplifications introduced by Lorentz and Sprecher, we fix the phases of the sinusoidal activations to linearly spaced constant values and provide a proof of its theoretical validity. We also conduct numerical experiments to evaluate its performance on a range of multivariable functions, comparing it with fixed-frequency Fourier transform methods and multilayer perceptrons (MLPs). We show that it outperforms the fixed-frequency Fourier transform and achieves comparable performance to MLPs.

stat.ML↗

arXiv2025

SineKAN: Kolmogorov-Arnold Networks Using Sinusoidal Activation Functions

Recent work has established an alternative to traditional multi-layer perceptron neural networks in the form of Kolmogorov-Arnold Networks (KAN). The general KAN framework uses learnable activation functions on the edges of the computational graph followed by summation on nodes. The learnable edge activation functions in the original implementation are basis spline functions (B-Spline). Here, we present a model in which learnable grids of B-Spline activation functions are replaced by grids of re-weighted sine functions (SineKAN). We evaluate numerical performance of our model on a benchmark vision task. We show that our model can perform better than or comparable to B-Spline KAN models and an alternative KAN implementation based on periodic cosine and sine functions representing a Fourier Series. Further, we show that SineKAN has numerical accuracy that could scale comparably to dense neural networks (DNNs). Compared to the two baseline KAN models, SineKAN achieves a substantial speed increase at all hidden layer sizes, batch sizes, and depths. Current advantage of DNNs due to hardware and software optimizations are discussed along with theoretical scaling. Additionally, properties of SineKAN compared to other KAN implementations and current limitations are also discussed

cs.LG↗