Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits

We present an algorithm, along with its implementation, to approximate single-qubit unitaries using quantum circuits consisting of Clifford and T gates. In addition to meeting the known logarithmic lower bounds on the number of gates required to approximate a unitary by a quantum circuit, we give computational evidence that a very close to the best, or the best existing approximation for unitaries of the type diag1, exp(i*phi) were indeed found. In particular, the quality of our approximation is determined by the ability of the PARI software package to find the solution of a certain type of Diophantine equation, and the choice of internal parameter Delta determining the size of a computer search. We have furthermore structured our search to guarantee that our near-optimal approximations can be found for reasonable error parameters—currently, down to 10^-17, allowing to execute very long quantum circuits. We discuss how to improve our implementation further to handle even smaller errors (by a few orders of magnitude) that would enable the high-precision synthesis of even larger quantum algorithms.