Hay from the haystack: explicit examples of exponential quantum circuit complexity

TL;DR

This paper constructs explicit examples of quantum states and unitaries with constant description length but exponential circuit complexity, highlighting the difficulty of identifying such complex states.

Contribution

It introduces infinite families of quantum states and unitaries with exponential circuit complexity despite having simple descriptions, using transcendence degree properties.

Findings

Most quantum states have exponential circuit complexity.

Explicit constructions of simple-description yet complex states are provided.

Exponential complexity applies to approximate generation of these states.

Abstract

The vast majority of quantum states and unitaries have circuit complexity exponential in the number of qubits. In a similar vein, most of them also have exponential minimum description length, which makes it difficult to pinpoint examples of exponential complexity. In this work, we construct examples of constant description length but exponential circuit complexity. We provide infinite families such that each element requires an exponential number of two-qubit gates to be generated exactly from a product and where the same is true for the approximate generation of the vast majority of elements in the family. The results are based on sets of large transcendence degree and discussed for tensor networks, diagonal unitaries, and maximally coherent states.