# Fine-grained quantum computational supremacy

## TL;DR

This paper introduces a fine-grained approach to quantum supremacy, demonstrating that certain quantum models cannot be efficiently simulated classically within specific exponential time bounds under complexity conjectures.

## Contribution

It establishes fine-grained complexity bounds for classical simulation of specific quantum models, extending quantum supremacy results to exclude some exponential-time classical algorithms.

## Key findings

- Classical simulation of DQC1 and HC1Q models is infeasible within certain exponential time bounds.
- Universal quantum computing with Clifford and T gates cannot be classically simulated in sub-exponential T gate time.
- Results rely on conjectures in fine-grained complexity theory to exclude specific classical algorithms.

## Abstract

Output probability distributions of several sub-universal quantum computing models cannot be classically efficiently sampled unless some unlikely consequences occur in classical complexity theory, such as the collapse of the polynomial-time hierarchy. These results, so called quantum supremacy, however, do not rule out possibilities of super-polynomial-time classical simulations. In this paper, we study "fine-grained" version of quantum supremacy that excludes some exponential-time classical simulations. First, we focus on two sub-universal models, namely, the one-clean-qubit model (or the DQC1 model) and the HC1Q model. Assuming certain conjectures in fine-grained complexity theory, we show that for any $a>0$ output probability distributions of these models cannot be classically sampled within a constant multiplicative error and in $2^{(1-a)N+o(N)}$ time, where $N$ is the number of qubits. Next, we consider universal quantum computing. For example, we consider quantum computing over Clifford and $T$ gates, and show that under another fine-grained complexity conjecture, output probability distributions of Clifford-$T$ quantum computing cannot be classically sampled in $2^{o(t)}$ time within a constant multiplicative error, where $t$ is the number of $T$ gates.