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Universal Sample Complexity Bounds in Quantum Learning Theory via Fisher Information matrix

Hyukgun Kwon, Seok Hyung Lie, Liang Jiang

Abstract

In this work, we show that the sample complexity (equivalently, the number of measurements) required in quantum learning theory within a general parametric framework, is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error and with high success probability under maximum likelihood estimation. The upper bound is governed by the supremum of the largest diagonal entry of the inverse Fisher information matrix, while the lower bound is characterized by any diagonal element evaluated at arbitrary parameter values. We then apply the general bounds to Pauli channel learning and to the estimation of Pauli expectation values in the asymptotic small-error regime, and recover the previously established sample complexity through considerably streamlined derivations. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value estimation without quantum memory. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators. We derive the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli expectation estimation with entangled probes. Finally, we highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, learning sample complexity is governed by the inverse Fisher information matrix.

Universal Sample Complexity Bounds in Quantum Learning Theory via Fisher Information matrix

Abstract

In this work, we show that the sample complexity (equivalently, the number of measurements) required in quantum learning theory within a general parametric framework, is fundamentally governed by the inverse Fisher information matrix. More specifically, we derive upper and lower bounds on the number of samples required to estimate the parameters of a quantum system within a prescribed small additive error and with high success probability under maximum likelihood estimation. The upper bound is governed by the supremum of the largest diagonal entry of the inverse Fisher information matrix, while the lower bound is characterized by any diagonal element evaluated at arbitrary parameter values. We then apply the general bounds to Pauli channel learning and to the estimation of Pauli expectation values in the asymptotic small-error regime, and recover the previously established sample complexity through considerably streamlined derivations. Furthermore, we identify the structural origin of exponential sample complexity in Pauli channel learning without entanglement and in Pauli expectation value estimation without quantum memory. We then extend the analysis to an error criterion based on the Euclidean distance between the true parameter values and their estimators. We derive the corresponding upper and lower bounds on the sample complexity, which are likewise characterized by the inverse Fisher information matrix. As an application, we consider Pauli expectation estimation with entangled probes. Finally, we highlight two fundamental contributions to quantum learning theory. First, we establish a systematic framework that determines the task-independent sample complexity under maximum-likelihood estimation. Second, we show that, in the small-error regime, learning sample complexity is governed by the inverse Fisher information matrix.
Paper Structure (57 sections, 13 theorems, 319 equations, 2 figures)

This paper contains 57 sections, 13 theorems, 319 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries of multi-parameter estimation
  3. Quantum parameter estimation
  4. Quantum metrology and mean squared error
  5. Quantum Learning Theory with $\ell_{k}$ distance
  6. Main result 1: Sample complexity of $\ell_{\infty}$-distance based learning
  7. Upper bound of $\ell_{\infty}$-distance learning
  8. Lower bound of $\ell_{\infty}$-distance learning
  9. Application to Pauli channel learning
  10. Pauli error rates and eigenvalues
  11. Entanglement-assisted scheme
  12. Separable scheme with single use of Pauli channel without ancilla
  13. Separable scheme with multiple uses of Pauli channel with unbounded classical register
  14. Application to Pauli expectation value estimation
  15. Quantum memory and Collective measurement
  16. ...and 42 more sections

Key Result

Theorem 1

The minimal sample complexity $M$ required to guarantees that the MLE satisfies is upper bounded as where and Here, $W_{0}$ is the Lambert $W$-function, $\|\mathbf{F}^{-1}_{\vb*{\theta}}\|_{\mathrm{op}}$ is operator norm of inverse of FIM, and $\mu_{R}$, $V_{H}$, $V_{R}$, $C$ and $\rho$ are positive coefficients that implicitly depend on $d$.

Figures (2)

  • Figure 1: Schematic of parameter estimation
  • Figure 2: Schematic of Pauli channel learning. Colors indicate different physical components: blue boxes represent the Pauli channel $\Lambda$, purple triangles denote entangled input and measurement, cyan triangles denote separable inputs and measurements, and green/pink blocks represent intermediate quantum control operations that do not generate entanglement. (a) Using a maximally entangled state and a Bell measurement. (b) A single use of the Pauli channel. (c) Allowing arbitrary quantum control between Pauli channel that does not generate the entanglement.

Theorems & Definitions (14)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • Corollary 3
  • Theorem 4
  • Corollary 4
  • Theorem 1
  • Theorem 2
  • ...and 4 more