Fundamental building blocks for unitary matrices and quantum logic gates

A unitary matrix is shown to be a product of certain {em basic} unitary matrices and the product is unique up to order. A basic unitary matrix itself is a product of {em minimal basic} matrices. A unitary $ntimes n$ matrix can be expressed as a product of at most $n$ basic matrices. A basic matrix is defined in terms of an idempotent matrix; the idempotent matrix used in the definition of a minimal basic unitary matrix is a {em density} matrix. This gives builders for unitary matrices. Quantum logic gates are represented by unitary matrices thus giving unique building material for quantum logic gates.