Polynomial-time T-depth Optimization of Clifford+T circuits via Matroid Partitioning (1303.2042v2)

Abstract: Most work in quantum circuit optimization has been performed in isolation from the results of quantum fault-tolerance. Here we present a polynomial-time algorithm for optimizing quantum circuits that takes the actual implementation of fault-tolerant logical gates into consideration. Our algorithm re-synthesizes quantum circuits composed of Clifford group and T gates, the latter being typically the most costly gate in fault-tolerant models, e.g., those based on the Steane or surface codes, with the purpose of minimizing both T-count and T-depth. A major feature of the algorithm is the ability to re-synthesize circuits with additional ancillae to reduce T-depth at effectively no cost. The tested benchmarks show up to 65.7% reduction in T-count and up to 87.6% reduction in T-depth without ancillae, or 99.7% reduction in T-depth using ancillae.

• The paper presents a polynomial-time algorithm that minimizes T-depth and T-count in Clifford+T circuits by leveraging matroid partitioning.
• Empirical results demonstrate up to 65.7% reduction in T-count and up to 87.6% reduction in T-depth without ancillae, and up to 99.7% with ancillae.
• The approach scales to large circuits with thousands of T gates, offering significant benefits for fault-tolerant quantum computations.

Polynomial-time T-depth Optimization of Clifford+T Circuits via Matroid Partitioning

The paper presents an algorithm that addresses the challenge of T-depth optimization in quantum circuits constructed from Clifford and T gates—a problem of significant interest due to the fault-tolerance properties and high cost of T gates in quantum computation. Specifically, it introduces a polynomial-time method to optimize quantum circuits with respect to both T-count and T-depth, leveraging matroid partitioning to achieve significant reductions in quantum resource consumption while maintaining fault tolerance.

Summary of Core Contributions

1. Optimization Algorithm:
   • The authors propose a polynomial-time algorithm that effectively minimizes the T-count and T-depth of Clifford+T quantum circuits. This algorithm is particularly applicable to circuits used in fault-tolerant quantum computation, where T gates represent a substantial overhead in terms of error correction and resource requirement.
   • The algorithm optimizes circuits by calculating the overall phase it imparts when decomposed into known fault-tolerant gates, then determining how to reorder operations to achieve the minimum T-depth.
2. Empirical Results:
   • Experimental findings demonstrate a substantial reduction of up to 65.7% in T-count and up to 87.6% reduction in T-depth without additional ancillae, alongside a reduction of up to 99.7% in T-depth when ancillae are permitted.
   • These improvements were consistently observed across a variety of benchmarks, including arithmetic operations and reversible circuits, highlighting the robustness and effectiveness of the approach.
3. Theoretical Framework:
   • The paper bases its optimization strategy on the theoretical foundation of matroid partitioning, efficiently solving the problem of finding minimal partitions of phase adjustments that can be executed in parallel stages.
   • By translating circuit phases into a matroid problem, it utilizes known polynomial-time algorithms from the field of combinatorics, thus effectively bridging the gap between abstract mathematical concepts and practical quantum computing challenges.
4. Algorithm Scalability:
   • By effectively handling large quantum circuits, such as those used in GF($2^m$) multipliers with up to 192 qubits and over 28,000 T gates, the algorithm demonstrates practical scalability, making it suitable for complex quantum systems without inducing prohibitive computational overhead.

Implications and Future Work

The proposed method offers clear advantages for the development and refinement of quantum code designed to run on near-term quantum computers. As the field progresses towards larger and more complicated instances of quantum circuits, the reduction of T-depth helps in enhancing the fidelity and speed of quantum computations by minimizing error rates associated with fault-tolerant T gates.

Moreover, the capability to balance the trade-off between ancillae utilization and circuit depth is particularly relevant for quantum computing architectures constrained by physical qubits. As such, the optimization algorithm positions itself as an influential tool for tailoring quantum circuits according to available hardware resources.

Looking forward, future research directions could include further exploration of polynomial transformations related to phase functions for additional gate sets, or the integration of such optimization processes with hybrid quantum-classical algorithms. Additionally, refining synthesis methodologies for linear reversible functions within the optimization framework presents another promising research avenue.

The paper provides a noteworthy advancement in optimizing quantum circuits essential for mitigating the cost of fault tolerance, a crucial step towards practical and scalable quantum computing.