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 {\tt CNOT}-efficient circuits for arbitrary three-qubit unitary evolutions. In this note, we place the Khaneja Glaser Decomposition ({\tt KGD}) in context as a $SU(2^n)=KAK$ decomposition by proving that its Cartan involution is type {\bf AIII}, given $n \geq 3$. The standard type {\bf 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 new decomposition is type {\bf AIII}, we further establish the following. Khaneja and Glaser allow for a particular degree of freedom, namely the choice of a commutative algebra $\mathfrak{a}$, in their construction. Let $\chi_1^n$ be a {\tt SWAP} gate applied on qubits $1$, $n$. Then $\chi_1^n v \chi_1^n=k_1\; a \; k_2$ is a KGD for $\mathfrak{a}=\mbox{span}_{\mathbb{R}} \{ \chi_1^n ( \ket{j}\bra{N-j-1} -\ket{N-j-1}\bra{j}) \chi_1^n \}$ if and only if $v=(\chi_1^n k_1 \chi_1^n) (\chi_1^n a \chi_1^n)(\chi_1^n k_2 \chi_1^n)$ is a CSD.