Optimisation of time-ordered processes in the finite and asymptotic regime

TL;DR

This paper introduces relaxations for optimizing time-ordered processes in quantum systems, providing practical methods for automaton output prediction, entanglement detection, and adaptive protocol design, with insights into limits of computability.

Contribution

It presents new relaxations for complex quantum sequential optimization problems and demonstrates their effectiveness in various quantum information tasks.

Findings

Relaxations enable tractable analysis of quantum automata outputs

New protocol for many-body entanglement detection

Heuristics provide useful bounds on infinite-time optimization

Abstract

Many problems in quantum information theory can be formulated as optimizations over the sequential outcomes of dynamical systems subject to unpredictable external influences. Such problems include many-body entanglement detection through adaptive measurements, computing the maximum average score of a preparation game over a continuous set of target states and limiting the behavior of a (quantum) finite-state automaton. In this work, we introduce tractable relaxations of this class of optimization problems. To illustrate their performance, we use them to: (a) compute the probability that a finite-state automaton outputs a given sequence of bits; (b) develop a new many-body entanglement detection protocol; (c) let the computer invent an adaptive protocol for magic state detection. As we further show, the maximum score of a sequential problem in the limit of infinitely many time steps is in general incomputable. Nonetheless, we provide general heuristics to bound this quantity and show that they provide useful estimates in relevant scenarios.