  Linear reversible circuits represent a subclass of reversible circuits with
many applications in quantum computing. These circuits can be efficiently
simulated by classical computers and their size is polynomially bounded by the
number of qubits, making them a good candidate to deploy efficient methods to
reduce computational costs. We propose a new algorithm for synthesizing any
linear reversible operator by using an optimized version of the Gaussian
elimination algorithm coupled with a tuned LU factorization. We also improve
the scalability of purely greedy methods. Overall, on random operators, our
algorithms improve the state-of-the-art methods for specific ranges of problem
sizes: the custom Gaussian elimination algorithm provides the best results for
large problem sizes (n > 150) while the purely greedy methods provide quasi
optimal results when n < 30. On a benchmark of reversible functions, we manage
to significantly reduce the CNOT count and the depth of the circuit while
keeping other metrics of importance (T-count, T-depth) as low as possible.
