CNOT-Optimal Clifford Synthesis as SAT

2025-04-01

AI Summary

This work addresses the challenge of simultaneously minimizing CNOT count and depth in Clifford circuit synthesis—a longstanding bottleneck in quantum compilation. We propose the first SAT-based exact synthesis method achieving strict optimality for both metrics. Our key contributions are: (1) the first integration of Clifford group normal form constraints into a SAT encoding, guaranteeing globally optimal CNOT count; (2) a two-stage SAT formulation that separately and sequentially optimizes CNOT count and CNOT depth; and (3) native support for hardware-specific connectivity constraints and qubit relabeling. Experimental evaluation shows that, on fully connected architectures, our method reduces CNOT count by 32.1% and depth by 48.1% compared to TKET; under restricted topologies, it further improves upon Qiskit by reducing CNOT count by 30.3% and depth by 35.9%. By bridging the performance–scalability gap between heuristic and exact approaches, this work establishes a new state-of-the-art in rigorous Clifford synthesis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Clifford circuit optimization is an important step in the quantum compilation pipeline. Major compilers employ heuristic approaches. While they are fast, their results are often suboptimal. Minimization of noisy gates, like 2-qubit CNOT gates, is crucial for practical computing. Exact approaches have been proposed to fill the gap left by heuristic approaches. Among these are SAT based approaches that optimize gate count or depth, but they suffer from scalability issues. Further, they do not guarantee optimality on more important metrics like CNOT count or CNOT depth. A recent work proposed an exhaustive search only on Clifford circuits in a certain normal form to guarantee CNOT count optimality. But an exhaustive approach cannot scale beyond 6 qubits. In this paper, we incorporate search restricted to Clifford normal forms in a SAT encoding to guarantee CNOT count optimality. By allowing parallel plans, we propose a second SAT encoding that optimizes CNOT depth. By taking advantage of flexibility in SAT based approaches, we also handle connectivity restrictions in hardware platforms, and allow for qubit relabeling. We have implemented the above encodings and variations in our open source tool Q-Synth. In experiments, our encodings significantly outperform existing SAT approaches on random Clifford circuits. We consider practical VQE and Feynman benchmarks to compare with TKET and Qiskit compilers. In all-to-all connectivity, we observe reductions up to 32.1% in CNOT count and 48.1% in CNOT depth. Overall, we observe better results than TKET in the CNOT count and depth. We also experiment with connectivity restrictions of major quantum platforms. Compared to Qiskit, we observe up to 30.3% CNOT count and 35.9% CNOT depth further reduction.

Problem

• Optimizing CNOT count in Clifford circuits using SAT encoding
• Improving CNOT depth via parallel plans in SAT approaches
• Addressing hardware connectivity constraints with flexible SAT methods

Innovation

• SAT encoding for CNOT count optimality
• Parallel plans optimize CNOT depth
• Handles hardware connectivity restrictions