Entanglement is a cornerstone of quantum information theory, yet efficiently generating entangled quantum states remains a significant challenge. In this work, we present a algebraic framework for constructing entangled states from systems of linear equations over the finite field ${\mathbb{F}}_d$, where $d$ is a prime number. We introduce the notion of an entanglement-producing system of linear equations and show how each can be systematically mapped to quantum circuits using only Hadamard and $d$-cnot gates with linear gate complexity in terms of the number of variables for the system. We formalize the criteria for when a system of equations produces genuine multipartite entanglement and provide constructive algorithms for synthesizing quantum circuits. Furthermore, we characterize that every entanglement-producing linear system is equivalent to structured simplex entanglement, which can be formulated in linear gate complexity in terms of the number of variables. By bridging abstract and linear algebraic techniques with practical circuit synthesis, this low-complexity approach provides a structured, efficient, and scalable entanglement generation.

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• Entanglement production
• Quantum algorithms & computation
• Quantum circuits
• Quantum computation
• Quantum engineering
• Quantum information processing
• Quantum information theory
• Quantum simulation