Note on the Khaneja Glaser decomposition

Recently, Vatan and Williams utilize a matrix decomposition of SU(2ⁿ) 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(2ⁿ)=KAK decomposition by proving that its Cartan involution is type { AIII}, given n ≥ 3. 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 new decomposition 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 χ₁ⁿ be a { SWAP} gate applied on qubits $1$, n. Then χ₁ⁿ v χ₁ⁿ=k₁\; a \; k₂ is a KGD for a=spanR \ χ₁ⁿ ( jN-j-1 -N-j-1j) χ₁ⁿ \ if and only if v=(χ₁ⁿ k₁ χ₁ⁿ) (χ₁ⁿ a χ₁ⁿ)(χ₁ⁿ k₂ χ₁ⁿ) is a CSD.