High-Precision Multi-Qubit Clifford+T Synthesis by Unitary Diagonalization

Abstract: Resource-efficient and high-precision approximate synthesis of quantum circuits expressed in the Clifford+T gate set is vital for Fault-Tolerant quantum computing. Efficient optimal methods are known for single-qubit $R_Z$ unitaries, otherwise the problem is generally intractable. Search-based methods, like simulated annealing, empirically generate low resource cost approximate implementations of general multi-qubit unitaries so long as low precision (Hilbert-Schmidt distances of $\epsilon \geq 10^{-2}$) can be tolerated. These algorithms build up circuits that directly invert target unitaries. We instead leverage search-based methods to first approximately diagonalize a unitary, then perform the inversion analytically. This lets difficult continuous rotations be bypassed and handled in a post-processing step. Our approach improves both the implementation precision and run time of synthesis algorithms by orders of magnitude when evaluated on unitaries from real quantum algorithms. On benchmarks previously synthesizable only with analytical techniques like the Quantum Shannon Decomposition, diagonalization uses an average of 95% fewer non-Clifford gates.