On estimating the entropy of shallow circuit outputs

TL;DR

Estimating the entropy of outputs from shallow quantum circuits is computationally hard, comparable to solving LWE problems, indicating that even simple quantum circuits can produce intractable entropy estimation tasks.

Contribution

This work proves the hardness of entropy estimation for shallow quantum circuits, linking it to LWE and exploring its complexity relative to general circuits.

Findings

Entropy estimation for shallow circuits is as hard as LWE.

Quantum circuits do not need to be complex to make entropy estimation difficult.

Log-depth circuits have intermediate hardness compared to general polynomial-size circuits.

Abstract

Estimating the entropy of probability distributions and quantum states is a fundamental task in information processing. Here, we examine the hardness of this task for the case of probability distributions or quantum states produced by shallow circuits. Specifically, we show that entropy estimation for distributions or states produced by either log-depth circuits or constant-depth circuits with gates of bounded fan-in and unbounded fan-out is at least as hard as the Learning with Errors (LWE) problem, and thus believed to be intractable even for efficient quantum computation. This illustrates that quantum circuits do not need to be complex to render the computation of entropy a difficult task. We also give complexity-theoretic evidence that this problem for log-depth circuits is not as hard as its counterpart with general polynomial-size circuits, seemingly occupying an intermediate hardness regime. Finally, we discuss potential future applications of our work for quantum gravity research by relating our results to the complexity of the bulk-to-boundary dictionary of AdS/CFT.