Encoder Circuit Optimization for Nonbinary Quantum Error Correction Codes in Prime Dimensions: An Algorithmic Framework

Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits&#x2014;quantum information elements that can occupy superpositions of the basis <inline-formula><tex-math notation="LaTeX">$|0\rangle$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$|1\rangle$</tex-math></inline-formula> states&#x2014;recent advances have demonstrated the practical implementation of higher dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher order states are less stable, and thus remain coherent for a shorter duration than the basis <inline-formula><tex-math notation="LaTeX">$|0\rangle$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$|1\rangle$</tex-math></inline-formula> states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (<italic>d</italic>-level quantum systems). Previous works have typically established generating gate sets for higher dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13% &#x2013;44% reduction in encoder circuit gate count for the qutrit (<italic>d</italic> &#x003D; 3) <inline-formula><tex-math notation="LaTeX">$[[9,5,3]]_{3}$</tex-math></inline-formula>, <inline-formula><tex-math notation="LaTeX">$[[5,1,3]]_{3}$</tex-math></inline-formula>, and <inline-formula><tex-math notation="LaTeX">$[[7,1,3]]_{3}$</tex-math></inline-formula> codes, and a 9% &#x2013;21% reduction for the ququint (<italic>d</italic> &#x003D; 5) <inline-formula><tex-math notation="LaTeX">$[[10,6,3]]_{5}$</tex-math></inline-formula> code when compared to prior work. We also achieved circuit depth reductions up to 42%.