A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits (1206.0758v3)

Abstract: We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a meet-in-the-middle technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method we implemented this algorithm and found factorizations of the commonly used quantum logical operations into elementary gates in the Clifford+T set. In particular, we report a decomposition of the Toffoli gate over the set of Clifford and T gates. Our decomposition achieves a total T-depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate. Due to the size of the search space the algorithm is only practical for small parameters, such as the number of qubits, and the number of gates in an optimal implementation.

• The paper demonstrates that the meet-in-the-middle algorithm significantly enhances circuit synthesis, achieving a 40% T-depth reduction for the Toffoli gate.
• The methodology leverages the Clifford+T gate set and systematic search strategies to decompose quantum operations efficiently.
• The results underscore the algorithm's potential for near-term quantum devices by reducing circuit depth and supporting scalable quantum algorithm development.

A Meet-in-the-Middle Algorithm for Depth-Optimal Quantum Circuit Synthesis

The paper presents a meet-in-the-middle algorithm designed to synthesize depth-optimal quantum circuits. This approach significantly enhances computational efficiency compared to brute force methods. Leveraging the Clifford+$T$ gate set, the algorithm facilitates efficient decomposition of quantum operations, with the authors particularly demonstrating an optimized Toffoli gate synthesis, achieving a remarkable 40% reduction in $T$-depth compared to existing methods.

Key Contributions

1. Meet-in-the-Middle Technique: The algorithm allows for depth-optimal circuit synthesis through a meet-in-the-middle strategy. The method involves generating circuits up to a certain depth and systematically searching for combinations that implement the desired unitary operation.
2. Numerical Results: Implementations of common gates, such as the Toffoli gate, exhibit substantial improvements. Notably, the Toffoli gate decomposition reduces the $T$-depth to 3, a considerable advancement over previous approaches.
3. Scalability: Although limited to small parameters, the meet-in-the-middle algorithm efficiently handles circuit decompositions involving a few qubits, highlighting its practical relevance in near-term quantum computers.
4. Algorithmic Flexibility: The algorithm is adaptable to various cost metrics beyond gate depth, including metrics focusing on minimizing the sequence of non-Clifford gates.
5. Ancilla Usage: The paper demonstrates that using ancilla qubits can further optimize circuits, reducing both the depth and the number of required gate stages.

Implications for Quantum Computing

The potential impact of this research is multi-faceted. Practically, the efficient synthesis of quantum circuits with reduced depth can lead to faster execution times, crucial for near-term quantum devices constrained by error rates and coherence times. Theoretically, this work contributes to understanding circuit complexity, particularly in optimizing depth, which is essential for developing scalable quantum algorithms.

Future Directions

1. Approximate Synthesis: Extending the algorithm to handle approximate synthesis could immensely benefit the compilation of algorithms requiring high precision with finite gate sets.
2. Large-Scale Circuit Optimization: Investigating how precomputed databases of optimal circuits can be integrated into general optimization frameworks may offer pathways for handling larger circuits.
3. Advanced Data Structures: The exploration of more efficient data structures for unitary representation and manipulation could further enhance performance, especially concerning memory utilization and search efficiency.

This work marks a significant stride in quantum circuit synthesis, particularly under practical constraints imposed by current quantum computing technology. As quantum computers continue to scale, algorithms like this will be pivotal in exploiting quantum mechanical resources effectively.