Efficient variational synthesis of quantum circuits with coherent multi-start optimization

We consider the problem of the variational quantum circuit synthesis into a gate set con-sisting of the CNOT gate and arbitrary single-qubit (1q) gates, with the primary objective being the minimization of the CNOT count. First, we note that along with the discrete architecture search, suffering from the combi-natorial explosion of complexity, optimization over 1q gates can also be a crucial roadblock due to the omnipresence of local minimums (well known in the context of variational quan-tum algorithms but apparently underappreci-ated in the context of the variational compil-ing). Taking the issue seriously, we make an extensive search over the initial conditions an essential part of our approach. Another key idea we propose is to use parametrized two-qubit (2q) controlled phase gates, which can interpolate between the identity gate and the CNOT gate, and allow a continuous relaxation of the discrete architecture search, which can be executed jointly with the optimization over 1q gates. This coherent optimization of the architecture together with 1q gates appears to work surprisingly well in practice, sometimes even outperforming optimization over 1q gates alone (for fixed optimal architectures). As il-lustrative examples and applications we derive 8 CNOT and T depth 3 decomposition of the 3q Toffoli gate on the nearest-neighbor topol-ogy, rediscover best known decompositions of the 4q Toffoli gate on all 4q topologies in-cluding a 1 CNOT gate improvement on the star-shaped topology, and propose decompo-sition of the 5q Toffoli gate on the nearest -neighbor topology with 48 CNOT gates. We also benchmark the performance of our ap-proach on a number of 5q quantum circuits from the ibm_qx_mapping database, showing that it is highly competitive with the exist-ing software. The algorithm developed in this work is available as a Python package CPFlow .