# Finite-Function-Encoding Quantum States

Paul Appel1,2, Alexander J. Heilman3, Ezekiel W. Wertz3, David W. Lyons3, Marcus Huber1,2, Matej Pivoluska4,5, and Giuseppe Vitagliano1,2

1Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna, Austria
2Institute for Quantum Optics and Quantum Information – IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
3Mathematics and Physics, Lebanon Valley College, 101 North College Avenue, Annville, Pennsylvania, 17003, United States of America
4Institute of Computer Science, Masaryk University, Botanická 68a, 60200 Brno, Czech Republic
5Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 5807/9, 845 11 Karlova Ves, Slovakia

### Abstract

We introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions, i.e., multivariate functions over the ring of integers modulo $d$, and investigate some of their structural properties. We also point out some differences between polynomial and non-polynomial function encoding states: The former can be associated to graphical objects, that we dub tensor-edge hypergraphs (TEH), which are a generalization of hypergraphs with a tensor attached to each hyperedge encoding the coefficients of the different monomials. To complete the framework, we also introduce a notion of finite-function-encoding Pauli (FP) operators, which correspond to elements of what is known as the generalized symmetric group in mathematics. First, using this machinery, we study the stabilizer group associated to FFE states and observe how qudit hypergraph states introduced in Ref. [1] admit stabilizers of a particularly simpler form. Afterwards, we investigate the classification of FFE states under local unitaries (LU), and, after showing the complexity of this problem, we focus on the case of bipartite states and especially on the classification under local FP operations (LFP). We find all LU and LFP classes for two qutrits and two ququarts and study several other special classes, pointing out the relation between maximally entangled FFE states and complex Butson-type Hadamard matrices. Our investigation showcases also the relation between the properties of FFE states, especially their LU classification, and the theory of finite rings over the integers.

Logical functions on binary systems can be conveniently encoded into phases of multi-qubit quantum systems. All such boolean functions are polynomials of finite degree and can thus be visualised as hypergraphs, giving rise to graph and hypergraph states. We explore the question what happens when considering logical functions with d-valued integers as inputs. Encoding those in higher-dimensional quantum systems gives rise to a rich family of states with interesting properties.

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