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Online learning of a panoply of quantum objects

Akshay Bansal, Ian George, Soumik Ghosh, Jamie Sikora, Alice Zheng

TL;DR

This work develops a general online-learning framework for learning unknown quantum objects by placing them in compact convex subsets of positive semidefinite matrices and interacting with convex loss functions via co-objects. It proves a sublinear regret bound, R_T <= 4 B C D sqrt(A T), under Lipschitz losses and trace-bounded object sets, and instantiates it for quantum states, effects, channels, interactive measurements, Gram matrices, and more. To handle variable traces and trace-function derivatives, the authors introduce a generalized Pinsker inequality and establish the Fréchet derivative of the Phi_E functional, enabling a robust Bregman-divergence-based analysis. The results yield concrete regret guarantees across a panoply of quantum objects, with avenues for tighter bounds under specialized loss structures like logarithmic losses.

Abstract

In many quantum tasks, there is an unknown quantum object that one wishes to learn. An online strategy for this task involves adaptively refining a hypothesis to reproduce such an object or its measurement statistics. A common evaluation metric for such a strategy is its regret, or roughly the accumulated errors in hypothesis statistics. We prove a sublinear regret bound for learning over general subsets of positive semidefinite matrices via the regularized-follow-the-leader algorithm and apply it to various settings where one wishes to learn quantum objects. For concrete applications, we present a sublinear regret bound for learning quantum states, effects, channels, interactive measurements, strategies, co-strategies, and the collection of inner products of pure states. Our bound applies to many other quantum objects with compact, convex representations. In proving our regret bound, we establish various matrix analysis results useful in quantum information theory. This includes a generalization of Pinsker's inequality for arbitrary positive semidefinite operators with possibly different traces, which may be of independent interest and applicable to more general classes of divergences.

Online learning of a panoply of quantum objects

TL;DR

This work develops a general online-learning framework for learning unknown quantum objects by placing them in compact convex subsets of positive semidefinite matrices and interacting with convex loss functions via co-objects. It proves a sublinear regret bound, R_T <= 4 B C D sqrt(A T), under Lipschitz losses and trace-bounded object sets, and instantiates it for quantum states, effects, channels, interactive measurements, Gram matrices, and more. To handle variable traces and trace-function derivatives, the authors introduce a generalized Pinsker inequality and establish the Fréchet derivative of the Phi_E functional, enabling a robust Bregman-divergence-based analysis. The results yield concrete regret guarantees across a panoply of quantum objects, with avenues for tighter bounds under specialized loss structures like logarithmic losses.

Abstract

In many quantum tasks, there is an unknown quantum object that one wishes to learn. An online strategy for this task involves adaptively refining a hypothesis to reproduce such an object or its measurement statistics. A common evaluation metric for such a strategy is its regret, or roughly the accumulated errors in hypothesis statistics. We prove a sublinear regret bound for learning over general subsets of positive semidefinite matrices via the regularized-follow-the-leader algorithm and apply it to various settings where one wishes to learn quantum objects. For concrete applications, we present a sublinear regret bound for learning quantum states, effects, channels, interactive measurements, strategies, co-strategies, and the collection of inner products of pure states. Our bound applies to many other quantum objects with compact, convex representations. In proving our regret bound, we establish various matrix analysis results useful in quantum information theory. This includes a generalization of Pinsker's inequality for arbitrary positive semidefinite operators with possibly different traces, which may be of independent interest and applicable to more general classes of divergences.
Paper Structure (32 sections, 34 theorems, 127 equations, 2 algorithms)

This paper contains 32 sections, 34 theorems, 127 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Online learning and its application to quantum
  3. Online learning of quantum states and related work
  4. Our setting
  5. Main results
  6. Challenge #1: Variable traces.
  7. Solution #1: A generalized Pinsker's inequality.
  8. Challenge #2: Trace functional differentiation.
  9. Solution #2: Fréchet derivative.
  10. Regret bounds for learning quantum objects
  11. Example #1: Quantum channels.
  12. Example #2: Collections of inner products of pure states.
  13. A panoply of other quantum examples.
  14. Limitations and future work
  15. Regret analysis
  16. ...and 17 more sections

Key Result

Lemma 1.1

algo:generalRFTL guarantees where $D$ is the diameter of $\mathcal{K}$ relative to the function $R ([)]{}$, i.e., $D^2 = \max_{\varphi, \varphi' \in \mathcal{K}} \{R ([)]{\varphi} - R ([)]{\varphi'} \}$.

Theorems & Definitions (57)

  • Lemma 1.1: hazan2016introduction Lemma 5.3
  • Theorem 1.2: Sublinearity of regret
  • Theorem 1.3: Generalized Pinsker's inequality
  • Remark 1.4
  • Lemma 1.5
  • Lemma 1.6
  • Lemma 1.7
  • Theorem 1.8: Informal, see \ref{['sec:quantumApplications']} for definitions and formal statements
  • Lemma 1.9
  • Definition 2.1
  • ...and 47 more