# Anticoncentration theorems for schemes showing a quantum speedup

## TL;DR

This paper proves anticoncentration theorems for quantum schemes demonstrating exponential speedup, establishing foundational results that support the feasibility of physically meaningful quantum advantage demonstrations.

## Contribution

It introduces two anticoncentration theorems and hardness results for circuit-based and quench-type quantum schemes, enabling simpler, resource-efficient demonstrations of quantum speedup.

## Key findings

- Proved anticoncentration theorems for quantum schemes
- Established hardness results for quantum speedup schemes
- Designed resource-economical schemes in 1D and 2D architectures

## Abstract

One of the main milestones in quantum information science is to realise quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum computers, a state of affairs dubbed quantum speedup, or sometimes "quantum computational supremacy". The known schemes heavily rely on mathematical assumptions that are plausible but unproven, prominently results on anticoncentration of random prescriptions. In this work, we aim at closing the gap by proving two anticoncentration theorems and accompanying hardness results, one for circuit-based schemes, the other for quantum quench-type schemes for quantum simulations. Compared to the few other known such results, these results give rise to a number of comparably simple, physically meaningful and resource-economical schemes showing a quantum speedup in one and two spatial dimensions. At the heart of the analysis are tools of unitary designs and random circuits that allow us to conclude that universal random circuits anticoncentrate as well as an embedding of known circuit-based schemes in a 2D translation-invariant architecture.