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arXiv:2509.25587 · IEEE Transactions on Quantum Engineering

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

Aditya Sodhani, Keshab K. Parhi

Published Jan 1, 2026• 0 citations
arXivHTMLPDF

Abstract

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—quantum information elements that can occupy superpositions of the basis$|0\rangle$and$|1\rangle$states—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$|0\rangle$and$|1\rangle$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 (d-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 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)$[[9,5,3]]_{3}$,$[[5,1,3]]_{3}$, and$[[7,1,3]]_{3}$codes, and a 9–21% reduction for the ququint (d = 5)$[[10,6,3]]_{5}$code when compared to prior work. We also achieved circuit depth reductions up to 42%.

DOI
10.1109/tqe.2026.3669054

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A Qutrit [ ⁣[6,2,2] ⁣]3[\![ 6,2,2 ]\!]_{3}[[6,2,2]]3​ Quantum Error-Correcting Code with a Transversal Binary AND Gate
Quantum computation and error correction in higher dimensional systems offer a promising route to more efficient and robust codes and circuits, compared to qubits. In this work, we present a novel qutrit[ ⁣[6,2,2] ⁣]3[\![ 6,2,2 ]\!]_{3}[[6,2,2]]3​quantum error correcting code with a transversal implementation of the binary AND gate. The key insight in our approach is that we can interpret a symmetric T-depth one circuit decomposition — composed of a CX circuit,T\mathrm{T}TandT†\mathrm{T}^{\dagger}T†gates, followed by the CX circuit in reverse — as a distance 1 CSS code. We thereafter increase the code distance by augmenting the code circuit with additional stabilizers while preserving the “built-in” transversal logical binary AND. As a result, we can leverage qutrit Clifford+T circuit synthesis to combine the advantages of quantum error correction and qudits, and moreover protect against leakage errors to higher levels.
The practical implementation of quantum algorithms on noisy intermediate-scale quantum devices encounters operational limitations due to decoherence and other sources of noise inherent in real hardware. To mitigate these errors while preserving the original functionality of the algorithm, shorter quantum circuits are therefore preferred. This motivates the development of effective quantum circuit optimization algorithms. Learning-based approaches have emerged as a leading candidate, yet existing autonomous agents remain inefficient, spending most of their training capacity rediscovering elementary reductions that deterministic rule-based methods already handle reliably. To address this challenge, we propose a reinforcement learning framework that embeds a deterministic Commutation-and-Reduction (CR) algorithm directly into the training environment. After every agent action, the CR algorithm automatically resolves elementary commutations and cancellations, enabling the agent to focus its learning capacity on the non-trivial optimizations where reinforcement learning adds real value. Empirical evaluation across two gate sets, the universal Clifford+T basis and the CNOT+Pauli basis, shows that RL+CR produces shorter circuits than a standard RL agent at all tested scales. We demonstrate that RL trained on smaller quantum circuits can be applied to larger quantum circuits. On 20-qubit Clifford+T circuits, five times larger than the training circuits, RL+CR removes twice as many gates as standard RL. This work provides a robust approach that could accelerate the compilation and optimization processes for future fault-tolerant and utility-scale quantum systems.
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