[Paper Review] Data-driven discovery of statistically relevant information in quantum simulators

Roberto Verdel, Vittorio Vitale+8 more|arXiv (Cornell University)|Jul 19, 2023

Quantum many-body systems71 references4 citations

TL;DR

This paper introduces a theory-agnostic, data-driven framework for identifying statistically relevant observables and effective degrees of freedom in quantum simulators using non-parametric unsupervised learning. By applying principal component analysis entropy, information imbalance, and intrinsic dimension estimation to experimental snapshots of a spinor Bose-Einstein condensate, the method reveals time-dependent universal dynamics and ranks operators by relevance without assumptions or controlled dimensionality reduction.

ABSTRACT

Quantum simulators offer powerful means to investigate strongly correlated quantum matter. However, interpreting measurement outcomes in such systems poses significant challenges. Here, we present a theoretical framework for information extraction in synthetic quantum matter, illustrated for the case of a quantum quench in a spinor Bose-Einstein condensate experiment. Employing non-parametric unsupervised learning tools that provide different measures of information content, we demonstrate a system-agnostic approach to identify dominant degrees of freedom. This enables us to rank operators according to their relevance, akin to effective field theory. To characterize the corresponding effective description, we then explore the intrinsic dimension of data sets as a measure of the complexity of the dynamics. This reveals a simplification of the data structure, which correlates with the emergence of time-dependent universal behavior in the studied system. Our assumption-free approach can be immediately applied in a variety of experimental platforms.

Motivation & Objective

• To develop a theory-agnostic method for extracting relevant physical information from large-scale quantum simulator data without assuming underlying models.
• To address the challenge of identifying dominant degrees of freedom in strongly correlated quantum systems, especially after quantum quenches.
• To quantify the complexity of many-body dynamics via intrinsic dimension estimation, linking data structure to emergent universal behavior.
• To provide a systematic, assumption-free approach to rank observables by their information content, analogous to effective field theory.
• To enable immediate application across diverse experimental quantum platforms by relying only on measured data snapshots.

Proposed method

• Employ principal component analysis (PCA) on many-body snapshots to compute spectral entropy as a measure of information content and correlation structure.
• Use information imbalance between subsets and the full feature space to assess how well a subset predicts the full data manifold.
• Estimate the intrinsic dimension of data manifolds using the TWO-NN method, which relies on the empirical cumulative distribution of nearest-neighbor ratios.
• Apply linear fitting to the empirical cumulative distribution of nearest-neighbor ratios to estimate the intrinsic dimension $I_d$.
• Implement subsampling with the delete-$d$ Jackknife estimator to robustly estimate statistical errors due to limited experimental realizations.
• Combine multiple metrics (PCA entropy, information imbalance, intrinsic dimension) to identify the most relevant observables and dynamical regimes.

Experimental results

Research questions

• RQ1 Which observables in a quantum simulator carry the most statistically relevant information about the underlying many-body dynamics?
• RQ2 How can we identify dominant degrees of freedom in a quantum many-body system after a quantum quench without prior theoretical assumptions?
• RQ3 What is the relationship between the intrinsic dimension of data manifolds and the emergence of universal, time-dependent behavior in quantum systems?
• RQ4 Can unsupervised, non-parametric learning tools reliably reveal effective field-theory-like descriptions from raw experimental data?
• RQ5 How can we quantify the complexity of quantum simulator data and detect simplifications in the dynamics without model-dependent assumptions?

Key findings

• The PCA entropy identifies the most informative observables: the joint data set combining $n_{1,+1}, n_{1,-1}, n_{2,+2}, n_{2,-2}$ has the lowest entropy, indicating maximal information content.
• Information imbalance analysis shows that subsets combining features from both relevant pairs ($\{n_{1,+1},n_{1,-1}\}$ and $\{n_{2,+2},n_{2,-2}\}$) have $\Delta(A\to B) \sim 0$, indicating they capture nearly all information in the full feature space.
• The intrinsic dimension of the data manifold exhibits a long, stable plateau after an initial drop, indicating a timescale after which dynamics simplify and may be described by universal scaling.
• The intrinsic dimension $I_d$ is estimated via linear fitting of $\{\ln(\mu), -\ln[1-F_{\mathrm{emp}}(\mu)]\}$, with consistent Pareto-like behavior over a significant range of $\ln(\mu)$, validating the TWO-NN method.
• Subsampling with $b=30$ and $q=100$ yields stable error estimates, confirming robustness of statistical inference despite limited realizations ($N_r=225$).
• The framework successfully identifies two relevant pairs of observables and reveals a transition to simpler, universal dynamics, consistent with effective field theory intuition, without assuming any physical model.