  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%.
