In the near term, programming quantum computers will remain severely limited by low quantum volumes. Therefore, it is desirable to implement quantum circuits with the fewest resources possible. For the common Clifford+T circuits, most research is focused on reducing the number of T gates, since they are an order of magnitude more expensive than Clifford gates in quantum error corrected encoding schemes. However, this optimization sometimes leads to more 2-qubit gates, which, even though they are less expensive in terms of fault-tolerance, contribute significantly to the overall circuit cost. Approaches based on the ZX-calculus have recently gained some popularity in the field, but reduction of 2-qubit gates is not their focus. In this work, we present an alternative for improving 2-qubit gate count of a quantum circuit with the ZX-calculus by using heuristics in ZX-diagram simplification. Our approach maintains the good reduction of the T gate count provided by other strategies based on ZX-calculus, thus serving as an extension for other optimization algorithms. Our results show that combining the available ZX-calculus-based optimizations with our algorithms can reduce the number of 2-qubit gates by as much as 40% compared to current approaches using ZX-calculus. Additionally, we improve the results of the best currently available optimization technique of Nam et. al for some circuits by up to 15%.

翻译：短期内，量子计算机的编程能力仍将受到低量子体积的严重限制。因此，以最少资源实现量子电路至关重要。对于常见的Clifford+T电路，多数研究聚焦于减少T门数量——因为在量子纠错编码方案中，T门的成本比Clifford门高一个数量级。然而，这种优化有时会导致双量子比特门数量增加，尽管这些门在容错性方面成本较低，但它们仍显著贡献于整体电路成本。基于ZX演算的方法近年来在该领域逐渐流行，但减少双量子比特门并非其核心目标。本文提出一种替代方案，通过在ZX图简化中采用启发式策略，基于ZX演算优化量子电路的双量子比特门数量。该方法能够保持其他基于ZX演算策略带来的良好T门减少效果，从而作为其他优化算法的扩展。实验结果表明，将现有基于ZX演算的优化与我们的算法结合，相较于当前基于ZX演算的方法，可将双量子比特门数量降低高达40%。此外，对于部分电路，我们进一步将Nam等人当前最优优化技术的结果提升了15%。