On the number of functions of <i>k</i>-valued logic which are polynomials modulo composite <i>k</i>

Selezneva, S. N.

doi:10.1515/dma-2017-0002

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…We can make use of a normal form defined below to simply generate the unique polynomials representing all polynomial functions. We will use the form given in [33] since it favours "local" terms over "non-local" ones . First let us introduce the composite degree a quantity which is used to restrict both degree of the monomials as well as the values of the coefficients of the unique polynomial representative we construct here.…”

Section: Discussionmentioning

confidence: 99%

“…Recall for example that a bi-variate polynomial P can be represented as: P (x, y) = c 00 + c 10 x + c 01 y + c 11 xy + • • • + c mn x m y n where c ij are the coefficients, x,y are the variables and m + n is the total degree of the polynomial. Note, that in general many polynomials can be associated to a single polynomial function, however there exist unique representations for any local dimension d and any number of variables n [32,33]. The uniqueness of the polynomial representative is guaranteed by restricting both the degree of the monomials as well as the value of the coefficients c ij ; these restriction depend on the normal form chosen.…”

Section: A Connecting the Dotsmentioning

confidence: 99%

“…n i=1 c p,m (x ei i )). Now following Theorem 1 in[33] we find: Remark 4. For d = p m with p a prime and 0 < m any polynomial function p (x 1 , .…”

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confidence: 91%

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Finite-Function-Encoding Quantum States

Appel,

Heilman,

Wertz

et al. 2020

Preprint

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We investigate the encoding of higher-dimensional logic into quantum states. To that end we introduce finite-function-encoding (FFE) states which encode arbitrary d-valued logic functions and investigate their structure as an algebra over the ring of integers modulo d. We point out that the polynomiality of the function is the deciding property for associating hypergraphs to states. Given a polynomial, we map it to a tensor-edge hypergraph, where each edge of the hypergraph is associated with a tensor. We observe how these states generalize the previously defined qudit hypergraph states, especially through the study of a group of finite-function-encoding Pauli stabilizers. Finally, we investigate the structure of FFE states under local unitary operations, with a focus on the bipartite scenario and its connections to the theory of complex Hadamard matrices.

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Section: Discussionmentioning

confidence: 99%

Section: A Connecting the Dotsmentioning

confidence: 99%

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Finite-Function-Encoding Quantum States

Appel,

Heilman,

Wertz

et al. 2020

Preprint

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Finite-Function-Encoding Quantum States

Appel

Heilman

Wertz

et al. 2022

Quantum

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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. \cite{2017PhRvA..95e2340S} 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.

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Checking the Polynomiality of Single-Variable Functions of $${k}$$-Valued Logic Composite Modulo $${k}$$

Selezneva

2024

MoscowUniv.Comput.Math.Cybern.

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On the number of functions of k-valued logic which are polynomials modulo composite k

Abstract: A function of

Cited by 3 publications

References 7 publications

Finite-Function-Encoding Quantum States

Finite-Function-Encoding Quantum States

Finite-Function-Encoding Quantum States

Checking the Polynomiality of Single-Variable Functions of $${k}$$-Valued Logic Composite Modulo $${k}$$

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