Algebraic synthesis of time-optimal unitaries in SU(2) with alternating controls

We present an algebraic framework to study the time-optimal synthesis of arbitrary unitaries in SU (2), when the control set is restricted to rotations around two non-parallel axes in the Bloch sphere. Our method bypasses commonly used control-theoretical techniques and easily imposes necessary conditions on timeoptimal sequences. In a straightforward fashion, we prove that time-optimal sequences are solely parametrized by three rotation angles and derive general bounds on those angles as a function of the relative rotation speed of each control and the angle between the axes. Results are substantially different whether both clockwise and counterclockwise rotations about the given axes are allowed, or only clockwise rotations. In the first case, we prove that any finite time-optimal sequence is composed at most of five control concatenations, while for the more restrictive case, we present scaling laws on the maximum length of any finite time-optimal sequence. The bounds we find for both cases are stricter than previously published ones and severely constrain the structure of time-optimal sequences, allowing for an efficient numerical search of the time-optimal Michele Allegra and Börge Hemmerling have contributed equally to this work. B Clarice D. Aiello [email protected] 1 Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2 Dipartimento di Fisica, Università Degli Studi di Torino and INFN, Sezione di Torino, 10125 Turin, Italy 3 Department of Physics, Harvard University, Cambridge, MA 02138, USA 4 Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 5 Research Laboratory of Electronics and Department of Nuclear Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA