Sharp complexity phase transitions generated by entanglement

TL;DR

This paper establishes a direct link between entanglement levels in quantum systems and their inherent computational complexity, revealing sharp phase transitions driven by system parameters in graph state simulations.

Contribution

It provides a quantitative characterization of how entanglement influences complexity and identifies phase transitions in simulating regular graph states based on their regularity.

Findings

Sharp complexity transition at k=3 in regular graph states

Duality between low and high regularity in simulation complexity

Entanglement correlates with computational hardness in quantum simulations

Abstract

Entanglement is one of the physical properties of quantum systems responsible for the computational hardness of simulating quantum systems. But while the runtime of specific algorithms, notably tensor network algorithms, explicitly depends on the amount of entanglement in the system, it is unknown whether this connection runs deeper and entanglement can also cause inherent, algorithm-independent complexity. In this work, we quantitatively connect the entanglement present in certain quantum systems to the computational complexity of simulating those systems. Moreover, we completely characterize the entanglement and complexity as a function of a system parameter. Specifically, we consider the task of simulating single-qubit measurements of $k$--regular graph states on $n$ qubits. We show that, as the regularity parameter is increased from $1$ to $n-1$, there is a sharp transition from an easy regime with low entanglement to a hard regime with high entanglement at $k=3$, and a transition back to easy and low entanglement at $k=n-3$. As a key technical result, we prove a duality for the simulation complexity of regular graph states between low and high regularity.