# Quantum Statistical Memory Advantage Reveals Predictive Structure in Chaotic Invariant Measures

arXiv: https://arxiv.org/abs/2606.13422 · HTML: https://mlsanity.com/paper/2606.13422/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 51 references that are in the map.
- First posted on arXiv 11 Jun 2026.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation](https://mlsanity.com/paper/2106.04399/) (2021)
- [WeatherBench: A benchmark dataset for data-driven weather forecasting](https://mlsanity.com/paper/2002.00469/) (2020)
- [A rigorous and robust quantum speed-up in supervised machine learning](https://mlsanity.com/paper/2010.02174/) (2020)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Predicting Many Properties of a Quantum System from Very Few Measurements](https://mlsanity.com/paper/2002.08953/) (2020)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Quantum advantage in learning from experiments](https://mlsanity.com/paper/2112.00778/) (2021)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [A quantum-inspired classical algorithm for recommendation systems](https://mlsanity.com/paper/1807.04271/) (2018)
- [DySLIM: Dynamics Stable Learning by Invariant Measure for Chaotic Systems](https://mlsanity.com/paper/2402.04467/) (2024)
- [Quantum-Informed Machine Learning for Predicting Spatiotemporal Chaos with Practical Quantum Advantage](https://mlsanity.com/paper/2507.19861/) (2025)

## Abstract (arXiv)

Long-horizon prediction of a chaotic system is governed by fidelity to its invariant measure, and which parts of that measure a learned predictor needs is a physical question not known in advance. We establish a quantum statistical memory advantage for interrogating it. A quantum statistical prior (Q-Prior), trained once on classical data, stores an efficiently preparable k-point marginal in polynomially many circuit parameters against exponentially many for explicit tabulation. Collective Bell measurements on two copies then estimate any post hoc Pauli-expectation magnitude at a copy cost independent of register size, with a worst-case exponential separation from single-copy protocols. A single Bell dataset supports an entire family of candidate observables at a cost growing only logarithmically in the family size, so an exponentially large candidate space stays open for later analysis. We implement the protocol on IQM superconducting processors with two-copy registers of up to 54 physical-qubit chips. We apply this to turbulent channel flow and ERA5 forecasting, resolving invariant structure by statistical sector and order. Turbulent phase correlations persist through eighth order, while most predictive gains arise from low-order constraints; in ERA5, planetary-wave phase coherence complements covariance regularisation, identifying distinct low-order sectors with predictive value. Quantum statistical memory is therefore both a near-term computational resource and an instrument for identifying which invariant structures matter for prediction.