Optimal Design of Two-Qubit Quantum Circuits

Current quantum computing hardware is unable to sustain quantum coherent operations for more than a handful of gate operations. Consequently, if near-term experimental milestones, such as synthesizing arbitrary entangled states, or performing fault-tolerant operations, are to be met, it will be necessary to minimize the number of elementary quantum gates used. In order to demonstrate non-trivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an oprimal quantum circuit for a general two-qubit gate that requires at most 3 CNOT gates and 15 elementary onequbit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requires at most 2 CNOTs and 12 onequbit gates. We then prove that these constructions are optimal with respect to the family of CNOT, y-rotation, z-rotation, and phase gates.