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arXiv:2607.09339

PCPOP.jl: A Julia package for partially commutative polynomial optimization

Moisés Bermejo Morán, Abhishek Mishra

quant-phmath.OCSubmitted Jul 10, 2026• Announced Jul 13, 2026• 0 citations
v1 · Jul 10, 2026v2 · Jul 13, 2026
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Abstract

Here we present PCPOP, a Julia package for polynomial optimization that supports non-commutative optimization, tracial polynomial optimization, trace polynomial optimization and state polynomial optimization. PCPOP fully supports exact arithmetic computations and incorporates convenient functionalities such as algebraic reductions based on Gröbner basis methods, automatized symmetrization via Wedderburn decompositions, and Jordan algebra reductions. As a distinguished feature, PCPOP implements a specialized framework for polynomial computations in partially commutative variables that provides significant computational advantages for problems appearing in quantum information.

Comments
Implementation of the Partially-commutative polynomial optimization. Package available at https://github.com/abh1mishra/PCPOP.jl

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Here we present PCPOP, a Julia package for polynomial optimization that supports non-commutative optimization, tracial polynomial optimization, trace polynomial optimization and state polynomial optimization. PCPOP fully supports exact arithmetic computations and incorporates convenient functionalities such as algebraic reductions based on Gröbner basis methods, automatized symmetrization via Wedderburn decompositions, and Jordan algebra reductions. As a distinguished feature, PCPOP implements a specialized framework for polynomial computations in partially commutative variables that provides significant computational advantages for problems appearing in quantum information.
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Numerical investigations are an important research tool in quantum information theory. There already exists a wide range of computational tools implemented in various programming languages. However there is little effort in implementing this kind of tools in the Julia language. Julia is a modern numerical computation library with excellent support for vector and matrix algebra, extended type system that allows for implementation of elegant application interfaces and support for parallel and distributed computing. QI.jl is a new quantum information theory library implemented in Julia that provides functions for creating and analyzing quantum states, and for creating quantum operations in various representations. An additional feature of the library is a collection of functions for sampling random quantum states and operations such as unitary operations and generic quantum channels.