Fermionic Quantum Many-Body Systems: A Quantum Information Approach

Abstract

These lecture notes show how Quantum Information Theory can help us to gain deeper insight into the physics of fermionic quantum many-body systems. In the first part, we start from fermionic correlations as they appear in recent experiments with cold gases and develop a pairing theory for fermionic particles applicable to the current experimental setups. In the second part we introduce and discuss a new class of variational states, the fermionic Projected Entangled Pair States (fPEPS) that allow for an efficient approximation of ground and thermal states of local fermionic Hamiltonians.

Algorithm for initializing a generalized fermionic Gaussian state on a quantum computer

“…chapter 2.3 in [41]. In fact, equation (32) can be seen to be identical to equation (23), by noting that we can write…”

“…In order to be able to compute equation (32), one requires explicit expressions for A [α] p † q , A [α] p † q † , and A [α] pq , which are generalizations of equations ( 10)-( 12). While A [α] and A [α] p † q are given in [25], we derive explicit formulas for the missing two expressions in appendix A, where we also present an alternative derivation for the exemplary expression A [α] p † q † rs using the implicit formula in equation (23), which is shown to be identical to the expression obtained from equation (32). The findings are summarized below,…”

“…This can be traced back to the fact that they typically only describe the ground state of non-interacting systems, i.e. quadratic Hamiltonians [23,24]. Recently, Shi et al demonstrated that certain families of non-Gaussian fermionic and bosonic states can be computed efficiently using a variational wave function that includes non-factorizable correlations [25], effectively extending the family of efficiently computable states beyond Gaussian states to so-called generalized Gaussian states.…”

Field Theory of Many-Body Lindbladian Dynamics

“…Diagonalizing Fst , one finds 2N eigenvalues of Ȟst that determine the population numbers of ρ st in its diagonal basis [62]:…”

“…Let us also mention the notable topological classification of Lindbladian fermions [53,54,55] and various results pertaining to both bosonic and fermionic Gaussian states [56,57,58,59,60,61,62,63]. These techniques have been variously applied to the study numerous problems, including the Bogoliubov spectrum of driven-dissipative condensates [29], exact solutions of nonlinear integrable systems [64], and the study topological properties of various low-dimensional dissipative systems [65,66,67,68,69].…”