Jens Eisert

This is not really a paper, but rather a note that stresses that if one implements a #variationalprinciple like in #VQA for translationally invariant #Hamiltonians, one better obtains energy densities that scale better than a small constant.

https://scirate.com/arxiv/2301.06142

A note on lower bounds to variational problems with guarantees

Variational methods play an important role in the study of quantum many body problems, both in the flavour of classical variational principles based on tensor networks as well as of quantum variational principles in near-term quantum computing. This brief pedagogical note stresses that for translationally invariant lattice Hamiltonians, one can easily derive efficiently computable lower bounds to ground state energies that can and should be compared with variational principles providing upper bounds. As small technical results, it is shown that (i) the Anderson bound and a (ii) common hierarchy of semi-definite relaxations both provide approximations with performance guarantees that scale like a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semi-definite relaxations inspired by the marginal problem.

SciRate
Jan 18, 2023 at 5:47amWeb
Show threadJens EisertJan 18, 2023
As one can easily get classical estimates of ground state energies up to rigorously guaranteed constant errors, at constant computational classical effort. This admittedly basic insight places some stringent demands on the quality of quantum prescriptions.
Show threadJens EisertJan 18, 2023
As technical results, it is shown that (i) the #Andersonbound and a (ii) common hierarchy of #semidefiniterelaxations both provide approximations with performance guarantees that scale like a constant in the energy density for cubic lattices. (iii) Also, the Anderson bound is systematically improved as a hierarchy of semi-definite relaxations inspired by the marginal problem.
Show threadJens EisertJan 18, 2023
Concerning the latter, I was actually unaware of a beautiful paper by Ilya Kull, Norbert Schuch, Ben Dive, and Miguel Navascués https://arxiv.org/abs/2212.03014 that present an elaborate relaxation of the ground state problem, different from but related to the above latter statement.
Lower Bounding Ground-State Energies of Local Hamiltonians Through the Renormalization Group

Given a renormalization scheme, we show how to formulate a tractable convex relaxation of the set of feasible local density matrices of a many-body quantum system. The relaxation is obtained by introducing a hierarchy of constraints between the reduced states of ever-growing sets of lattice sites. The coarse-graining maps of the underlying renormalization procedure serve to eliminate a vast number of those constraints, such that the remaining ones can be enforced with reasonable computational means. This can be used to obtain rigorous lower bounds on the ground state energy of arbitrary local Hamiltonians, by performing a linear optimization over the resulting convex relaxation of reduced quantum states. The quality of the bounds crucially depends on the particular renormalization scheme, which must be tailored to the target Hamiltonian. We apply our method to 1D translation-invariant spin models, obtaining energy bounds comparable to those attained by optimizing over locally translation-invariant states of $n\gtrsim 100$ spins. Beyond this demonstration, the general method can be applied to a wide range of other problems, such as spin systems in higher spatial dimensions, electronic structure problems, and various other many-body optimization problems, such as entanglement and nonlocality detection.

arXiv.org
Show threadJens EisertJan 18, 2023
Overall, this is a small reminder that quantum variational principles have to be pretty accurate to hope to outperform classical methods.