Physics-Aware Rejection Sampling

• Physics-aware Rejection Sampling (PaRS) is a method that integrates explicit physical constraints like energy bounds and conservation laws into traditional rejection sampling frameworks.
• The methodology employs adaptive sampling with physical acceptance gates and partial resampling to efficiently handle domain-specific anomalies in quantum and spatial simulations.
• PaRS enhances computational efficiency and prediction accuracy in diverse applications such as quantum state preparation, Gibbs point processes, and materials design.

Physics-aware Rejection Sampling (PaRS) denotes a class of trace, sample, or configuration selection algorithms that operate by explicitly encoding physical constraints, symmetries, or admissibility criteria alongside standard probabilistic rejection mechanisms. These methods augment or replace generic correctness or learned preferences with acceptance gates rooted in domain-specific physical knowledge, thereby increasing the reliability, physical plausibility, and calibration of downstream predictive models or simulations. PaRS has found particular utility in quantum query complexity, Gibbs point process simulation, supervised machine learning for physical property prediction, and large-scale particle and materials discovery pipelines.

1. Foundational Principles and Historical Context

The canonical rejection sampling method, formalized by von Neumann (1951), aims to sample from a target distribution $p(x)$ via a proposal distribution $q(x)$, accepting samples according to the ratio $p(x)/Mq(x)$ where $M \geq \sup_x p(x)/q(x)$. While highly general, classical rejection sampling neglects domain-specific constraints—such as symmetry, conservation laws, or admissibility regions—that often characterize complex physical systems.

PaRS extends this paradigm by integrating physics-based restrictions at the level of sample acceptance. In the context of quantum algorithms, "Quantum rejection sampling" (Ozols et al., 2011) introduces operator-norm bounds and continuous symmetrization arguments to quantify the query complexity of preparing quantum amplitudes subject to oracle-defined constraints. In simulation-based approaches to spatial processes, PaRS leverages the locality of interactions (finite range, low intensity) to facilitate tractable perfect sampling via partial resampling only of problematic regions (Moka et al., 2019).

2. Methodological Framework and Core Algorithms

PaRS algorithms typically comprise three elements:

• Physical Acceptance Gate: Candidate traces, configurations, or events are accepted if they satisfy not only probabilistic or numerical closeness criteria but also explicit physical constraints—e.g., permissible energy ranges, non-overlap conditions, conservation envelopes.
• Adaptive Sampling and Halting: Sampling proceeds in rounds (or windows), with temperature schedules or batch sizes adjusted to explore the candidate space efficiently until an admissible trace is found or budget constraints are met (Hyun et al., 31 Aug 2025).
• Partial and Localized Resampling: Rather than naively resampling the entire space upon rejection, only the regions or components invoking physical violations are resampled (e.g., in Gibbs point processes, cells violating pairwise interaction constraints are locally reset according to a dependency graph) (Moka et al., 2019).

Acceptance Gate Formalization (Materials Discovery Example (Hyun et al., 31 Aug 2025)):

Let $y_i$ denote the experimental target, $\tilde{y}^{(k)}$ a candidate prediction, and $U(x_i)$ an empirical upper bound (e.g., maximal quantum yield achievable for recipe $x_i$). Then the acceptance gate comprises:

• Range check: $\tilde{y}^{(k)} \in [0, 100]$
• Continuous error tolerance: $|\tilde{y}^{(k)} - y_i| \leq \epsilon_{\text{mae}}$
• Physics envelope constraint: $\tilde{y}^{(k)} \leq U(x_i)$

If none of the batch candidates pass these gates, adaptive halting conditions (variance and improvement thresholds) determine whether further sampling is warranted.

3. Applications in Quantum, Statistical, and Simulation Settings

Quantum Algorithms: In "Quantum rejection sampling" (Ozols et al., 2011), PaRS is framed as the task of converting amplitude distributions under hidden-state oracles into amplitudes reflecting a target distribution, with performance bounds expressed via operator norms and semidefinite programming. The automorphism principle is extended to continuous groups, allowing tight symmetrization—critical for optimal query complexity.

Gibbs Point Process Simulation: In spatial statistics, PaRS is instantiated via spatial partitioning and partial rejection sampling (Moka et al., 2019). Here, the configuration space is decomposed into local regions, each sampled independently. "Bad events"—e.g., sphere overlaps in hard-core models—are detected via a dependency graph, and only dependent regions are locally resampled. Under low intensity and bounded interaction range, the expected number of iterations is $O(\log(1/r))$ for interaction range $2r$.

Materials Discovery and Reasoning Models: For process-aware recipe-to-property prediction in materials science, PaRS serves as a supervisor-side filter for reasoning traces generated by a teacher model (Hyun et al., 31 Aug 2025). Only traces numerically close to experiment and consistent with known physical bounds are used for fine-tuning a student LLM. This yields improved prediction accuracy, calibration, and lower frequencies of unphysical predictions.

4. Comparative Analysis and Performance Considerations

Across multiple domains, PaRS demonstrates key advantages:

Application Context	Performance Metric	PaRS Outcome
Quantum state preparation (Ozols et al., 2011)	Query complexity	Optimal $\Theta(1/||E||^2)$
Gibbs processes (Moka et al., 2019)	Expected iterations	$O(\log(1/r))$ for small $r$
Reasoning LLMs (Hyun et al., 31 Aug 2025)	MAE, physics violation rate	Lower error, fewer violations

In spatial simulation, importance sampling and partial rejection yield exponentially improved running times in dense regimes compared to naive rejection or coupling from the past (Moka et al., 2017, Moka et al., 2019). In reasoning LLM pipelines, PaRS requires fewer sampled candidates per prompt while maintaining superior accuracy and admissibility, manifesting a favorable compute–accuracy Pareto frontier (Hyun et al., 31 Aug 2025).

5. Mathematical Formulation and Theoretical Guarantees

In the spatial and quantum domains, PaRS leverages mathematical structure:

• Operator Norm Bound (Quantum):

$$||O - O'|| \leq || |0\rangle - |0'\rangle ||^{\frac{1}{2}} + || |0'(2) - |1\rangle ||^{\frac{1}{2}}$$

with the quantum query lower bound scaling as $1/||E||^2$, where $||E||$ is derived from "water-filling" of the amplitude spectrum.

• Bad Event Conditioning (Gibbs Process):

$$\mu = \mu^{\otimes}(\cdot \mid \bigcap_{(i,j) \in V} B_{i,j}^{c})$$

with partial rejection sampling iteratively resampling variables local to violations until all bad events are resolved.

• Acceptance Gates (Materials):

$$\begin{align*} \text{Range} &:\quad \tilde{y}^{(k)} \in [0, 100] \ \text{Tolerance} &:\quad |\tilde{y}^{(k)} - y_i| \leq \epsilon_{\text{mae}} \ \text{Envelope} &:\quad \tilde{y}^{(k)} \leq U(x_i) \end{align*}$$

These formalizations underpin both theoretical optimality claims and practical sampling efficiency.

6. Limitations, Open Problems, and Future Directions

While PaRS is highly effective under sparse or well-controlled physical regimes, limitations arise:

• In Gibbs processes, increasing the point intensity raises the probability of bad events, necessitating more invasive resampling and higher iteration counts (Moka et al., 2019).
• In reasoning LLM pipelines, empirical envelopes and error tolerances must be tuned for specific materials systems, and the method presently focuses on continuous-valued prediction with physical bounds rather than multi-modal or discrete constraints (Hyun et al., 31 Aug 2025).
• Scalability to high-density or strongly coupled systems may require hybrid methods integrating PaRS with advanced proposal distributions or domain-adapted importance sampling (Moka et al., 2017).

Recommended future directions include generalizing envelope constraints, integrating reinforcement learning for adaptive experimental design, and extending PaRS frameworks to other closed-loop discovery settings such as battery or photovoltaic materials (Hyun et al., 31 Aug 2025).

7. Significance and Impact

PaRS advances the state-of-the-art in rejection-based selection methods by embedding substantive physical criteria directly into sampling, simulation, and selection pipelines. It enables:

• Tight query complexity characterization in quantum algorithms.
• Tractable perfect simulation for spatial point processes with finite interaction ranges.
• Reliable and physically plausible supervision pipelines for AI-driven property prediction and materials design.
• Efficient, domain-aware sampling with favorable trade-offs in computational cost and accuracy.

By leveraging domain knowledge, PaRS aligns theoretical optimality with practical admissibility, establishing a foundation for robust, physically consistent computational pipelines in both simulation science and machine learning.