Optimizing the non-Clifford-count in unitary synthesis using Reinforcement Learning

In this paper we study the potential of using reinforcement learning (RL) in order to synthesize quantum circuits, while optimizing the T-count and CS-count, of unitaries that are exactly implementable by the Clifford+T and Clifford+CS gate sets, respectively. We have designed our RL framework to work with channel representation of unitaries, that enables us to perform matrix operations efficiently, using integers only. We have also incorporated pruning heuristics and a canonicalization of operators, in order to reduce the search complexity. As a result, compared to previous works, we are able to implement significantly larger unitaries, in less time, with much better success rate and improvement factor. Our results for Clifford+T synthesis on two qubit unitaries achieve close-to-optimal decompositions for up to 100 T gates, 5 times more than previous RL algorithms and to the best of our knowledge, the largest instances achieved with any method to date. Our RL algorithm is able to recover previously-known optimal linear complexity algorithm for T-count-optimal decomposition of 1 qubit unitaries. We illustrate significant reduction in the asymptotic T-count estimate of important primitives like controlled cyclic shift (43%), controlled adder (14.3%) and multiplier (14%), without adding any extra ancilla. For 2-qubit Clifford+CS unitaries, our algorithm achieves a linear complexity, something that could only be accomplished by a previous algorithm using SO(6) representation.