Note on the Khaneja Glaser Decomposition

Recently, Vatan and Williams utilize a matrix decomposition of SU(2^n) introduced by Khaneja and Glaser to produce CNOT-efficient circuits for arbitrary three-qubit unitary evolutions. In this note, we place the Khaneja Glaser Decomposition (KGD) in context as a SU(2^n)=KAK decomposition by proving that its Cartan involution is type AIII, given n is greater than 2. The standard type AIII involution produces the Cosine-Sine Decomposition (CSD), a well-known decomposition in numerical linear algebra which may be computed using mature, stable algorithms. In the course of our proof that the KGD is type AIII, we further establish the following. Khaneja and Glaser allow for a particular degree of freedom, namely the choice of a commutative algebra a, in their construction. Let SWAP denote a swap between qubits 1 and n. Then for appropriate choice of a, the KGD by (SWAP v SWAP)=k1 a k2 may be recovered from a CSD by v = (SWAP k1 SWAP) (SWAP a SWAP) (SWAP k2 SWAP).