# Automated construction of $U(1)$-invariant matrix-product operators from graph representations

## TL;DR

This paper introduces an automated method to construct $U(1)$-invariant matrix-product operators from finite-states machine representations, simplifying the process and reducing computational costs in tensor network algorithms.

## Contribution

The authors develop an algorithmic scheme that automates the construction of $U(1)$-invariant MPOs from FSM representations, handling quantum number shifts and phase factors efficiently.

## Key findings

- Automated MPO construction reduces computational complexity.
- Exact MPO representations for Hamiltonian variances are generated.
- Method applicable to various $U(1)$-invariant operators.

## Abstract

We present an algorithmic construction scheme for matrix-product-operator (MPO) representations of arbitrary $U(1)$-invariant operators whenever there is an expression of the local structure in terms of a finite-states machine (FSM). Given a set of local operators as building blocks, the method automatizes two major steps when constructing a $U(1)$-invariant MPO representation: (i) the bookkeeping of auxiliary bond-index shifts arising from the application of operators changing the local quantum numbers and (ii) the appearance of phase factors due to particular commutation rules. The automatization is achieved by post-processing the operator strings generated by the FSM. Consequently, MPO representations of various types of $U(1)$-invariant operators can be constructed generically in MPS algorithms reducing the necessity of expensive MPO arithmetics. This is demonstrated by generating arbitrary products of operators in terms of FSM, from which we obtain exact MPO representations for the variance of the Hamiltonian of a $S=1$ Heisenberg chain.