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Statistical inference for quantum singular models

Hiroshi Yano, Yota Maeda, Naoki Yamamoto

TL;DR

This study bases its study on classical singular learning theory and seeks to extend it within the framework of Bayesian quantum state estimation, to define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods.

Abstract

Deep learning has seen substantial achievements, with numerical and theoretical evidence suggesting that singularities of statistical models are considered a contributing factor to its performance. From this remarkable success of classical statistical models, it is naturally expected that quantum singular models will play a vital role in many quantum statistical tasks. However, while the theory of quantum statistical models in regular cases has been established, theoretical understanding of quantum singular models is still limited. To investigate the statistical properties of quantum singular models, we focus on two prominent tasks in quantum statistical inference: quantum state estimation and model selection. In particular, we base our study on classical singular learning theory and seek to extend it within the framework of Bayesian quantum state estimation. To this end, we define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods. The key idea of the proof is the introduction of a quantum analog of the likelihood function using classical shadows. Consequently, we construct an asymptotically unbiased estimator of the quantum generalization loss, the quantum widely applicable information criterion (QWAIC), as a computable model selection metric from given measurement outcomes.

Statistical inference for quantum singular models

TL;DR

This study bases its study on classical singular learning theory and seeks to extend it within the framework of Bayesian quantum state estimation, to define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods.

Abstract

Deep learning has seen substantial achievements, with numerical and theoretical evidence suggesting that singularities of statistical models are considered a contributing factor to its performance. From this remarkable success of classical statistical models, it is naturally expected that quantum singular models will play a vital role in many quantum statistical tasks. However, while the theory of quantum statistical models in regular cases has been established, theoretical understanding of quantum singular models is still limited. To investigate the statistical properties of quantum singular models, we focus on two prominent tasks in quantum statistical inference: quantum state estimation and model selection. In particular, we base our study on classical singular learning theory and seek to extend it within the framework of Bayesian quantum state estimation. To this end, we define quantum generalization and training loss functions and give their asymptotic expansions through algebraic geometrical methods. The key idea of the proof is the introduction of a quantum analog of the likelihood function using classical shadows. Consequently, we construct an asymptotically unbiased estimator of the quantum generalization loss, the quantum widely applicable information criterion (QWAIC), as a computable model selection metric from given measurement outcomes.

Paper Structure

This paper contains 27 sections, 37 theorems, 210 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Statistical inference in quantum state models
  4. Singular learning theory
  5. Main results
  6. Generalization and training losses in quantum state estimation
  7. Asymptotic behaviors of GnQ and TnQ
  8. Quantum Widely Applicable Information Criterion
  9. Concrete examples
  10. Analytic and algebraic calculation
  11. Numerical simulation
  12. Conclusion and open questions
  13. Assumptions and Fundamental conditions
  14. Assumptions in classical learning theory
  15. Fundamental conditions
  16. ...and 12 more sections

Key Result

Theorem 3.1

The generalization loss $G_n^{Q}$ and training loss $T_n^{Q}$ can be expanded as follows:

Figures (8)

  • Figure 1: Our setting in quantum state estimation.
  • Figure 2: Resolution of singularities of a parameter space. Technically, obtaining the normal crossing representation requires blowups to be repeated; however, to better convey the essence of the desingularization, a single application of the blowup is depicted in this figure.
  • Figure 3: Resolution of the singularity of $y^2 = x^3$, called the monoidal transformation hartshorne2013algebraic.
  • Figure 4: Numerical results for a regular model.
  • Figure 5: $C_n^Q$ for a singular model.
  • ...and 3 more figures

Theorems & Definitions (93)

  • Definition 2.1: Definition \ref{['def:regular for classical']}
  • Theorem 3.1: Basic theorem; informal version of Theorem \ref{['thm:q_basic']}
  • Theorem 3.2: Regular asymptotic expansion; informal version of Theorem \ref{['thm:q_expectations for regular cases']}
  • Theorem 3.3: Singular asymptotic expansion; informal version of Theorem \ref{['thm:q_expansion formulas for the expectations for singular cases']}
  • Theorem 3.4: Asymptotic unbiasedness of QWAIC; informal version of Theorems \ref{['thm:q_QWAIC for regular cases']} and \ref{['thm:q_QWAIC is unbiased estimator for singular cases']}
  • Example 4.1: A regular case; a classical state model
  • Example 4.2: a singular case; a classical state model
  • Example 4.3: A singular case; a quantum state model
  • Definition A.1: watanabe2018mathematical
  • Remark A.2
  • ...and 83 more