Source-linked AI summary
Entropy scaling and simulability by Matrix Product States
Norbert Schuch, Michael M. Wolf, Frank Verstraete, J. Ignacio Cirac
TL;DR
The paper asks which block-entropy scaling laws determine efficient approximation by Matrix Product States. It establishes criteria for von Neumann and Rényi entropies, showing that a strict von Neumann area law alone is insufficient and applying the results to time-evolution simulation.
Problem
The relation between entropy scaling and efficient MPS approximability remains unclear, despite the common belief that a bounded von Neumann entropy, or area law, is sufficient.
Method
The paper relates block-entropy scaling to MPS approximability through entropy-based bounds and constructs examples covering determined and undetermined scaling regimes.
Results
At most logarithmic Sα for α < 1 implies approximability, while faster-than-logarithmic Sα for α > 1 or linear von Neumann entropy rules it out; constant von Neumann entropy remains undetermined.
Takeaways & Limitations
A strict von Neumann area law does not by itself guarantee MPS approximability, and the paper gives evidence that some translationally invariant time evolutions may resist efficient MPS simulation.
Takeaways & Limitations
More refined criteria may require other figures of merit, including smooth Rényi entropies, and the translationally invariant bounded-von-Neumann-entropy case remains open.
Abstract
from arXiv · showhide
We investigate the relation between the scaling of block entropies and the efficient simulability by Matrix Product States (MPS), and clarify the connection both for von Neumann and Renyi entropies (see Table I). Most notably, even states obeying a strict area law for the von Neumann entropy are not necessarily approximable by MPS. We apply these results to illustrate that quantum computers might outperform classical computers in simulating the time evolution of quantum systems, even for completely translational invariant systems subject to a time independent Hamiltonian.