An algebraic approach for quantum computation
Journal of Computing Sciences in Colleges
A new algorithm for producing quantum circuits using KAK decompositions
Quantum Information & Computation
Synthesis of quantum circuits for d-level systems by using cosine-sine decomposition
Quantum Information & Computation
On the CNOT-cost of TOFFOLI gates
Quantum Information & Computation
| Hi-index | 0.00 |
Recently, Vatan and Williams utilize a matrix decomposition of SU(2n) introduced byKhaneja and Glaser to produce CNOT-efficient circuits for arbitrary three-qubit unitaryevolutions. In this note, we place the Khaneja Glaser Decomposition (KGD) in contextas 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 computedusing mature, stable algorithms. In the course of our proof that the new decomposition istype AIII, we further establish the following. Khaneja and Glaser allow for a particulardegree 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 vχ1n = k1 a k2 is a KGD fora = span R{χ1n(|j〉〈N-j-1|-|N-j-1〉〈j|)χ1n} if and only if v = (χ1nk1χ1n)(χ1naχ1n)(χ1nk2χ1n) is a CSD.