Optimal Realization of a General Two-Qubit Quantum Gate

Since almost all quantum algorithms are represented in the quantum circuit model, the problem of finding a universal set of quantum gates is one the main problems in constructing quantum computers and implementing quantum algorithms. While it is rather easy to find a universal quantum basis, finding basis that satisfies some restrictions, due to implementation requirements, is far more challenging. For example, there are several small universal quantum bases for fault-tolerant computation (see [3] for details). In this paper we investigate the quantum basis consists of all one-qubit gates and CNOT as its only two-qubit gate. This basis first studied in [1], where they showed that this basis can implement any unitary n-qubit operation exactly. This basis is suitable for the case that we want to minimize the number of interactions between two qubits. The problem we are studding here is to implement an arbitrary unitary two-qubit operation with minimal number of applications of one-qubit and CNOT gates. Note that each one-qubit gate itself can be considered as a sequence of three gates of simple rotations along the y and z-axis. Therefore, the elementary gates are Ry(θ), Rz(α), and CNOT. Since quantum gates are so hard to achieve experimentally, minimizing the gate count will be of central importance in attaining near-term experimental milestones, such as the production of arbitrary entangled states. Our new construction requires at most 16 elementary one-qubit gates and 3 CNOTs which is less then any previously known construction. Moreover, using rewrite rules, we can often find even simpler circuits if they exist. Hence, our new construction brings certain state synthesis tasks within the grasp of experimentalists. In addition, as our quantum circuits for (arbitrary) n-qubit operations are always in terms