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Adaptive Online Learning of Quantum States

Xinyi Chen, Elad Hazan, Tongyang Li, Zhou Lu, Xinzhao Wang, Rui Yang

TL;DR

This paper presents adaptive and dynamic regret bounds for online shadow tomography, which are polynomial in the number of qubits and sublinear in the number of measurements.

Abstract

The problem of efficient quantum state learning, also called shadow tomography, aims to comprehend an unknown $d$-dimensional quantum state through POVMs. Yet, these states are rarely static; they evolve due to factors such as measurements, environmental noise, or inherent Hamiltonian state transitions. This paper leverages techniques from adaptive online learning to keep pace with such state changes. The key metrics considered for learning in these mutable environments are enhanced notions of regret, specifically adaptive and dynamic regret. We present adaptive and dynamic regret bounds for online shadow tomography, which are polynomial in the number of qubits and sublinear in the number of measurements. To support our theoretical findings, we include numerical experiments that validate our proposed models.

Adaptive Online Learning of Quantum States

TL;DR

This paper presents adaptive and dynamic regret bounds for online shadow tomography, which are polynomial in the number of qubits and sublinear in the number of measurements.

Abstract

The problem of efficient quantum state learning, also called shadow tomography, aims to comprehend an unknown -dimensional quantum state through POVMs. Yet, these states are rarely static; they evolve due to factors such as measurements, environmental noise, or inherent Hamiltonian state transitions. This paper leverages techniques from adaptive online learning to keep pace with such state changes. The key metrics considered for learning in these mutable environments are enhanced notions of regret, specifically adaptive and dynamic regret. We present adaptive and dynamic regret bounds for online shadow tomography, which are polynomial in the number of qubits and sublinear in the number of measurements. To support our theoretical findings, we include numerical experiments that validate our proposed models.
Paper Structure (26 sections, 22 theorems, 62 equations, 4 figures, 1 table, 5 algorithms)

This paper contains 26 sections, 22 theorems, 62 equations, 4 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Adaptive online learning.
  3. Contributions.
  4. Preliminaries
  5. Notations.
  6. Minimizing Dynamic Regret
  7. Mistake bound of the path length setting.
  8. Dynamic regret given dynamical models.
  9. Minimizing Adaptive Regret
  10. $k$-shift regret bound.
  11. Mistake bound of the $k$-shift setting.
  12. Numerical Experiments
  13. Comparison to non-adaptive algorithms.
  14. Adaptive regret bound on the $k$-shift setting.
  15. Dynamic regret bound on the path length setting.
  16. ...and 11 more sections

Key Result

Theorem 1

Assume the path length $\mathcal{P}=\sum_{t=1}^{T-1} \|\varphi_t-\varphi_{t+1}\|_{\color{black}1}\ge 1$ and the loss $\ell_{t}$ is convex, L-Lipschitz, and maps to $[0,1]$, the dynamic regret of Algorithm algo:md+exp is bounded by $O(L \sqrt{T(n+\log (T)) \mathcal{P}})$.

Figures (4)

  • Figure 1: Comparison between Algorithm \ref{['algo:adaptive']} and non-adaptive RFTL in the $k$-shift setting. The regret attained by our adaptive Algorithm \ref{['algo:adaptive']} is denoted by "CBCE" (the blue curve with purple shadow), and the regret generated by the non-adaptive algorithm RFTL is denoted by the orange curve with red shadow. The lower solid line stands for the average value of regret over 10 random experiments and the upper line stands for the maximum.
  • Figure 2: Comparison between our algorithms and non-adaptive RFTL in the path length setting. Our algorithms are denoted by "DOMD" / "CBCE", and the non-adaptive algorithm RFTL with different learning rates $\eta$ are denoted by the value of $\eta$.
  • Figure 3: Adaptive regret bounds in the $k$-shift setting.
  • Figure 4: Dynamic regret bound on the path length setting.

Theorems & Definitions (39)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Lemma 1
  • Theorem 2
  • Corollary 4
  • Definition 1: Gateaux differentiability
  • Definition 2: Directional derivatives
  • Lemma 2: Lemma VI.4 in Ref. gateaux
  • ...and 29 more