The Complexity of Learning (Pseudo)random Dynamics of Black Holes and Other Chaotic Systems

TL;DR

This paper demonstrates that polynomially bounded quantum algorithms cannot accurately learn or predict the complex, pseudo-random dynamics of black holes or chaotic systems, highlighting the computational hardness of such tasks.

Contribution

It proves that bounded quantum algorithms cannot effectively reconstruct pseudo-random unitary dynamics, establishing the computational difficulty of learning black hole chaos.

Findings

Bounded quantum algorithms cannot predict pseudo-random unitary dynamics.

Learning black hole or chaotic system evolution is computationally hard.

The results hold even with access to complex observables and general quantum channels.

Abstract

It has been recently proposed that the naive semiclassical prediction of non-unitary black hole evaporation can be understood in the fundamental description of the black hole as a consequence of ignorance of high-complexity information. Validity of this conjecture implies that any algorithm which is polynomially bounded in computational complexity cannot accurately reconstruct the black hole dynamics. In this work, we prove that such bounded quantum algorithms cannot accurately predict (pseudo)random unitary dynamics, even if they are given access to an arbitrary set of polynomially complex observables under this time evolution; this shows that "learning" a (pseudo)random unitary is computationally hard. We use the common simplification of modeling black holes and more generally chaotic systems via (pseudo)random dynamics. The quantum algorithms that we consider are completely general, and their attempted guess for the time evolution of black holes is likewise unconstrained: it need not be a linear operator, and may be as general as an arbitrary (e.g. decohering) quantum channel.