Contraction of fermionic operator circuits and the simulation of strongly correlated fermions

TL;DR

This paper introduces a framework for contracting fermionic operator circuits efficiently, enabling the simulation of strongly correlated fermionic systems using tensor network methods like MERA, TTN, and PEPS.

Contribution

It extends known qudit operator circuit techniques to fermionic systems, allowing for efficient contraction with minimal additional computational cost.

Findings

Fermionic circuits can be contracted with the same efficiency as qudit circuits.

The method generalizes tensor network simulations from spin to fermionic systems.

Additional costs are mainly due to mode reordering during contraction.

Abstract

A fermionic operator circuit is a product of fermionic operators of usually different and partially overlapping support. Further elements of fermionic operator circuits (FOCs) are partial traces and partial projections. The presented framework allows for the introduction of fermionic versions of known qudit operator circuits (QUOC), important for the simulation of strongly correlated d-dimensional systems: The multiscale entanglement renormalization ansatz (MERA), tree tensor networks (TTN), projected entangled pair states (PEPS), or their infinite-size versions (iPEPS etc.). After the definition of a FOC, we present a method to contract it with the same computation and memory requirements as a corresponding QUOC, for which all fermionic operators are replaced by qudit operators of identical dimension. A given scheme for contracting the QUOC relates to an analogous scheme for the corresponding fermionic circuit, where additional marginal computational costs arise only from reordering of modes for operators occurring in intermediate stages of the contraction. Our result hence generalizes efficient schemes for the simulation of d-dimensional spin systems, as MERA, TTN, or PEPS to the fermionic case.