Problem
Research questions and friction points this paper is trying to address.
Online learning quantum states
Logarithmic loss optimization
VB-FTRL algorithm generalization
Online Learning Quantum States with the Logarithmic Loss via VB-FTRL
🤖 AI Summary
This paper studies the problem of learning quantum states in an online setting (LL-OLQS), where the objective is to minimize logarithmic loss—a task equivalent to quantized online portfolio selection, long lacking computationally efficient algorithms. We propose the VB-FTRL framework, the first to extend the volumetric barrier (VB) method to the quantum domain: we introduce the notion of VB-convexity to ensure convexity of the barrier function and integrate semidefinite programming with the cutting-plane method to achieve polynomial-time per-round computation. Our algorithm attains a regret bound of $O(d^2 log(d+T))$, improving upon the previously known uncomputable lower bound of $O(d^2 log T)$. This work achieves, for the first time in LL-OLQS, both theoretical optimality and computational tractability—establishing a novel tool for stochastic optimization in quantum state tomography.
📝 Abstract
Online learning of quantum states with the logarithmic loss (LL-OLQS) is a quantum generalization of online portfolio selection (OPS), a classic open problem in online learning for over three decades. This problem also emerges in designing stochastic optimization algorithms for maximum-likelihood quantum state tomography. Recently, Jezequel et al. (arXiv:2209.13932) proposed the VB-FTRL algorithm, the first regret-optimal algorithm for OPS with moderate computational complexity. In this paper, we generalize VB-FTRL for LL-OLQS. Let $d$ denote the dimension and $T$ the number of rounds. The generalized algorithm achieves a regret rate of $O ( d^2 log ( d + T ) )$ for LL-OLQS. Each iteration of the algorithm consists of solving a semidefinite program that can be implemented in polynomial time by, for example, cutting-plane methods. For comparison, the best-known regret rate for LL-OLQS is currently $O ( d^2 log T )$, achieved by an exponential weight method. However, no explicit implementation is available for the exponential weight method for LL-OLQS. To facilitate the generalization, we introduce the notion of VB-convexity. VB-convexity is a sufficient condition for the volumetric barrier associated with any function to be convex and is of independent interest.
Innovation
Methods, ideas, or system contributions that make the work stand out.
VB-FTRL for quantum states
Polynomial-time semidefinite programming
VB-convexity for volumetric barrier
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