22 March 2023
14:00 till
15:00
[DMO] Barbara Terhal & Maarten Stroeks: Fermionic Optimization
A basic problem in physics is to determine the lowest eigenvalue, or `ground state energy', of a many-body Hamiltonian. For fermionic Hamiltonians, modeling electrons in solids or chemistry, this problem is the minimization of a low-degree polynomial in non-commutative Majorana variables (forming a Clifford algebra). Since determining the lowest eigenvalue is a (QMA)-hard problem, it has been of interest to upper and lower bound the eigenvalue by optimizing over simple classes of states resp. relaxing the problem to a SDP.
We prove that for sparse fermionic Hamiltonians the upperbound over so-called Gaussian fermionic states, a well-known efficiently described class of quantum states, achieves a constant approximation ratio, meaning that the scaling with the number of variables is the same as for the true lowest eigenvalue. This is in contrast to recent work showing that for a class of random dense fermionic Hamiltonians, with high probability, there is no constant approximation ratio. We discuss some further results and open questions in this area.
References: 1. Herasymenko, Stroeks, Helsen, Terhal, https://arxiv.org/abs/2211.16518 ; 2. Hastings, O'Donnell, https://dl.acm.org/doi/10.1145/3519935.3519960
