A SAT Encoding for Optimal Clifford Circuit Synthesis (2208.11713v1)

Abstract: Executing quantum algorithms on a quantum computer requires compilation to representations that conform to all restrictions imposed by the device. Due to device's limited coherence times and gate fidelities, the compilation process has to be optimized as much as possible. To this end, an algorithm's description first has to be synthesized using the device's gate library. In this paper, we consider the optimal synthesis of Clifford circuits -- an important subclass of quantum circuits, with various applications. Such techniques are essential to establish lower bounds for (heuristic) synthesis methods and gauging their performance. Due to the huge search space, existing optimal techniques are limited to a maximum of six qubits. The contribution of this work is twofold: First, we propose an optimal synthesis method for Clifford circuits based on encoding the task as a satisfiability (SAT) problem and solving it using a SAT solver in conjunction with a binary search scheme. The resulting tool is demonstrated to synthesize optimal circuits for up to $26$ qubits -- more than four times as many as the current state of the art. Second, we experimentally show that the overhead introduced by state-of-the-art heuristics exceeds the lower bound by $27\%$ on average. The resulting tool is publicly available at https://github.com/cda-tum/qmap.

• The paper presents a SAT encoding method that reformulates Clifford circuit synthesis as an iterative binary search to minimize gate count.
• It demonstrates scalability by efficiently synthesizing up to 26-qubit circuits within approximately 3 hours, surpassing brute force limits.
• Comparative analysis reveals only a 27% gate overhead above the lower bound, marking significant improvements over traditional heuristic methods.

Overview of "A SAT Encoding for Optimal Clifford Circuit Synthesis"

Quantum computing continues to drive advances in multiple fields, characterized by the capability to solve complex problems more efficiently than classical computing methods. Quantum circuits, expressed over qubits, are fundamental to the execution of quantum algorithms on quantum computers. The synthesis of these circuits involves transforming high-level algorithms into quantum gates compatible with a given quantum architecture. The paper, titled "A SAT Encoding for Optimal Clifford Circuit Synthesis," by Sarah Schneider, Lukas Burgholzer, and Robert Wille, addresses optimal synthesis specifically for Clifford circuits, a subset of quantum circuits crucial for error correction and the demonstration of various quantum phenomena.

Clifford Circuit Synthesis

Clifford circuits are defined by operations involving Clifford gates such as Hadamard (H), phase (S), and controlled-NOT (CNOT) gates. These form a non-universal subset yet are critical in quantum error correction protocols and stabilize certain quantum phenomena. Optimal circuit synthesis involves minimizing the number of gates, particularly the two-qubit gates which are error-prone and resource-intensive on contemporary quantum devices.

The paper proposes a method that leverages satisfiability (SAT) techniques to encode the problem of Clifford circuit synthesis. The challenge is that the Clifford group’s size grows exponentially, making a brute force search impractical even for circuits involving as few as six qubits. This research circumvents previous limitations by formulating the synthesis problem as a SAT problem and solving it with a SAT solver through an iterative binary search scheme.

Methodology and Contributions

The authors employ SAT encoding in a novel fashion to capture the functional requirements of Clifford circuit synthesis. The proposed encoding utilizes a binary search scheme to iteratively narrow down the number of gates needed to reach the target functionality defined by a stabilizer tableau. This approach allows the solver to efficiently explore the search space, which expands at a rate of $O(n^2)$ choices per gate, thus enabling a considerable reduction in computational load.

Significant highlights of the proposed method include:

1. Scalability: The SAT encoding handles up to 26-qubit Clifford circuits within a practical timeframe ($\approx$ 3 hours), a substantial improvement over existing methods where the limit is six qubits.
2. Efficiency and Accuracy: The experimental results exhibit a notably lower overhead from heuristic methods, averaging an excess of only 27% above the synthesized lower bound.
3. Comparative Analysis: The paper benchmarks its optimization approach against other contemporary synthesis methods, specifically those proposed by Aaronson, Gottesman, and Bravyi, demonstrating both runtime efficiency and minimal gate count advantages.

Implications and Future Directions

The implications of this research are profound for both the theoretical aspects and practical applications of quantum computing. By pushing the limits of optimal synthesis to a higher number of qubits, the research provides a more realistic baseline for evaluating heuristic methods in current and future quantum compilers. The adoption of SAT solvers for such optimizations showcases their versatility and encourages further exploration into hybrid classical-quantum computational techniques.

Future directions for this research involve refining the cost metrics to include other forms of optimization and exploring the potential to extract heuristics from the SAT-based solution schemes to improve practical quantum circuit synthesis approaches. Additionally, expanding the scope to accommodate non-Clifford circuits while maintaining optimality under SAT constraints remains a vital research horizon.

Conclusion

The paper makes a substantial contribution to quantum circuit synthesis by enabling the creation of optimally synthesized Clifford circuits at unprecedented scales. By utilizing an innovative SAT encoding approach, the authors extend the frontiers of what is computationally feasible and lay a foundation for future advancements in quantum computing synthesis methodologies. The work aligns with the broader goals of enhancing quantum technology—serving as a crucial step towards practical and efficient quantum applications.