Quadratic Form Expansions for Unitaries

Abstract

Abstract.We introduce techniques to analyze unitary operations in terms of quadratic form expansions, a form similar to a sum over paths in the computational basis when the phase contributed by each path is described by a quadratic form over R. We show how to relate such a form to an entangled resource akin to that of the one-way measurement model of quantum computing. Using this, we describe various conditions under which it is possible to efficiently implement a unitary operation U , either when provided a quadratic form expansion for U as input, or by finding a quadratic form expansion for U from other input data.

Extended phase map decompositions for unitaries

“…and we call this representation the Phase map decomposition of a given unitary operator implemented in the one-way pattern (de Beaudrap et al 2006;de Beaudrap et al 2008).…”

“…The positive branch of a one-way pattern can be expressed in terms of a phase map decomposition RΦP (de Beaudrap et al 2006;de Beaudrap et al 2008), which we then further analyse to obtain the primary structure of the matrix M that represents RΦP in the computational basis. The positive branch of a one-way pattern can be expressed in terms of a phase map decomposition RΦP (de Beaudrap et al 2006;de Beaudrap et al 2008), which we then further analyse to obtain the primary structure of the matrix M that represents RΦP in the computational basis.…”

“…Our approach is based on the phase map decomposition (de Beaudrap et al 2006;de Beaudrap et al 2008) and leads to some preliminary results on the connection between the column structure of a given unitary and the angles of measurements in a pattern that implements it. Extended phase map decompositions for unitaries.…”

“…Our approach is based on the phase map decomposition (de Beaudrap et al 2006;de Beaudrap et al 2008) and leads to some preliminary results on the connection between the column structure of a given unitary and the angles of measurements in a pattern that implements it. Our approach is based on the phase map decomposition (de Beaudrap et al 2006;de Beaudrap et al 2008) and leads to some preliminary results on the connection between the column structure of a given unitary and the angles of measurements in a pattern that implements it.…”

“…We will give a complete structural characterisation of the map implemented by the positive branch of a one-way pattern. The positive branch of a one-way pattern can be expressed in terms of a phase map decomposition RΦP (de Beaudrap et al 2006;de Beaudrap et al 2008), which we then further analyse to obtain the primary structure of the matrix M that represents RΦP in the computational basis. The columns of M can be written as…”