Information-theoretic Hardness of Out-of-time-order Correlators

TL;DR

This paper demonstrates that certain quantum properties related to out-of-time-order correlators (OTOCs) are efficiently learnable only with direct access to OTOCs, revealing fundamental limitations of reconstructing them from time-ordered correlators.

Contribution

It establishes the exponential hardness of reconstructing OTOCs from time-ordered correlators, providing a theoretical foundation for their use in quantum simulations.

Findings

Reconstruction of OTOCs from time-ordered correlators can be exponentially inefficient.

Efficient learning of certain quantum properties requires direct access to OTOCs.

Provides a framework distinguishing measurement protocols as classes of adaptive quantum learning algorithms.

Abstract

We establish that there are properties of quantum many-body dynamics which are efficiently learnable if we are given access to out-of-time-order correlators (OTOCs), but which require exponentially many operations in the system size if we can only measure time-ordered correlators. This implies that any experimental protocol which reconstructs OTOCs solely from time-ordered correlators must be, in certain cases, exponentially inefficient. Our proofs leverage and generalize recent techniques in quantum learning theory. Along the way, we elucidate a general definition of time-ordered versus out-of-time-order experimental measurement protocols, which can be considered as classes of adaptive quantum learning algorithms. Moreover, our results provide a theoretical foundation for novel applications of OTOCs in quantum simulations.