Gaussian matrix product states cannot efficiently describe critical systems

TL;DR

Gaussian fermionic matrix product states are insufficient for efficiently approximating critical systems, as they require superpolynomial bond dimension, unlike non-Gaussian MPS which can do so with polynomial resources.

Contribution

The paper demonstrates that Gaussian MPS cannot efficiently approximate critical states, highlighting the limitations of Gaussian ansatz in tensor network simulations.

Findings

GfMPS require superpolynomial bond dimension for critical states

Non-Gaussian MPS can approximate critical states with polynomial bond dimension

Numerical evidence suggests bond dimension is subexponential for critical systems

Abstract

Gaussian fermionic matrix product states (GfMPS) form a class of ansatz quantum states for 1d systems of noninteracting fermions. We show, for a simple critical model of free hopping fermions, that: (i) any GfMPS approximation to its ground state must have bond dimension scaling superpolynomially with the system size, whereas (ii) there exists a non-Gaussian fermionic MPS approximation to this state with polynomial bond dimension. This proves that, in general, imposing Gaussianity at the level of the tensor network may significantly alter its capability to efficiently approximate critical Gaussian states. We also provide numerical evidence that the required bond dimension is subexponential, and thus can still be simulated with moderate resources.