Grover search and the no-signaling principle

TL;DR

This paper explores the connection between the no-signaling principle and the limits of quantum computational speedup, showing that violating one implies violating the other within certain theoretical models.

Contribution

It demonstrates that in specific deviations from quantum mechanics, the no-signaling principle and the hardness of NP problems are fundamentally linked.

Findings

Superluminal signaling resources scale polynomially with Grover speedup resources.

No-signaling principle is equivalent to the inability to efficiently solve NP-hard problems in analyzed models.

Violations of one property imply violations of the other in the studied theories.

Abstract

Two of the key properties of quantum physics are the no-signaling principle and the Grover search lower bound. That is, despite admitting stronger-than-classical correlations, quantum mechanics does not imply superluminal signaling, and despite a form of exponential parallelism, quantum mechanics does not imply polynomial-time brute force solution of NP-complete problems. Here, we investigate the degree to which these two properties are connected. We examine four classes of deviations from quantum mechanics, for which we draw inspiration from the literature on the black hole information paradox. We show that in these models, the physical resources required to send a superluminal signal scale polynomially with the resources needed to speed up Grover's algorithm. Hence the no-signaling principle is equivalent to the inability to solve NP-hard problems efficiently by brute force within the classes of theories analyzed.