# Quantum advantage with shallow circuits

## TL;DR

This paper demonstrates that shallow quantum circuits can outperform classical circuits in solving specific problems, establishing a quantum advantage with constant-depth quantum circuits for a new problem called the 2D Hidden Linear Function problem.

## Contribution

The paper introduces the 2D Hidden Linear Function problem and proves that it can be efficiently solved by constant-depth quantum circuits but requires logarithmic depth classical circuits.

## Key findings

- Quantum circuits solve the 2D Hidden Linear Function problem with certainty.
- Classical probabilistic circuits need logarithmic depth to solve the problem.
- Quantum advantage is established with shallow circuits for a specific computational task.

## Abstract

We prove that constant-depth quantum circuits are more powerful than their classical counterparts. To this end we introduce a non-oracular version of the Bernstein-Vazirani problem which we call the 2D Hidden Linear Function problem. An instance of the problem is specified by a quadratic form q that maps n-bit strings to integers modulo four. The goal is to identify a linear boolean function which describes the action of q on a certain subset of n-bit strings. We prove that any classical probabilistic circuit composed of bounded fan-in gates that solves the 2D Hidden Linear Function problem with high probability must have depth logarithmic in n. In contrast, we show that this problem can be solved with certainty by a constant-depth quantum circuit composed of one- and two-qubit gates acting locally on a two-dimensional grid.