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Gaussian matrix product states cannot efficiently describe critical systems

MPS-Authors
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Franco Rubio,  Adrián
Theory, Max Planck Institute of Quantum Optics, Max Planck Society;
MCQST - Munich Center for Quantum Science and Technology, External Organizations;

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Cirac,  J. Ignacio
Theory, Max Planck Institute of Quantum Optics, Max Planck Society;
MCQST - Munich Center for Quantum Science and Technology, External Organizations;

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Citation

Franco Rubio, A., & Cirac, J. I. (2022). Gaussian matrix product states cannot efficiently describe critical systems. Physical Review B, 106: 235136. doi:10.1103/PhysRevB.106.235136.


Cite as: https://hdl.handle.net/21.11116/0000-000A-6318-C
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.