Efficient algorithm for multi-qudit twirling

Max Planck Institute for Quantum Optics, Hans-Kopfermann-Strase 1, D-85748 Garching, Germany(Dated: February 1, 2008)We present an efficient algorithm for twirling a multi-qudit quantum state. The algorithm canbe used for approximating the twirling operation in an ensemble of physical systems in which thesystems cannot be individually accessed. It can also be used for computing the twirled densitymatrix on a classical computer. The method is based on a simple non-unitary operation involvinga random unitary. When applying this basic building block iteratively, the mean squared error ofthe approximation decays exponentially. In contrast, when averaging over random unitary matricesthe error decreases only algebraically. We present evidence that the unitaries in our algorithm cancome from a very imperfect random source or can even be chosen deterministically from a set ofcyclically alternating matrices. Based on these ideas we present a quantum circuit realizing twirlingefficiently.