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 χ1 be a SWAP gate applied on qubits 1, n. Then χ n 1vχ n 1 = k1 a k2 is a KGD for a = spanR{χ1(| j〉〈N − j−1|− |N − j−1〉〈 j|)χ1} if and only if v = (χ1k1χ1)(χ1aχ1)(χ1k2χ1) is a CSD. Any fixed-time closed-system evolution of the n-qubit state space may be modelled mathematically by multiplication of a state vector |ψ〉 by some N ×N unitary matrix v, where N = 2n throughout. By choice of global phase, we may multiply v by det(v)1/N , so that without loss of generality v ∈ SU(N), the Lie group [Kna98, H01] of determinant one unitary matrices. Matrix decompositions are algorithms for factoring matrices. In the context of qubit dynamics, such a decomposition would split an evolution v into component subevolutions. A matrix decomposition may be proven by explicitly specifying an algorithm that computes it. Alternately, several theorems in Lie theory posit factorizations of a group across certain subgroups. These may in essence be viewed as meta-decomposition theorems; they often allow many degrees of freedom in the choice of subgroups and the group being factored. Without describing the appropriate hypotheses, we mention examples such as the global Cartan decomposition G = exp(p)K, the Iwasawa decomposition G = NAK, and its generalization the Langlands decomposition G = UMAK. For G = Gl(n,C) the Lie group of all invertible complex matrices, wellknown algorithms often exist for the outputs of these theorems. For example, the global Cartan decomposition outputs the usual polar decomposition which writes a matrix as a product of a Hermitian and unitary matrix, while the Iwasawa decomposition reduces to the QR decomposition. The G = KAK metadecomposition theorem has seen several overt and hidden applications in quantum computing. A sample output is SU(21)= {Ry(θ)}{Rz(α)}{Ry(θ)}, where Ry(θ)= cos 2 |0〉〈0|+sin 2 |0〉〈1|−sin 2 |0〉〈1|+ cos 2 |1〉〈1| and Rz(α) = e−iα/2|0〉〈0|+eiα/2|1〉〈1|. This is the factorization of any Bloch-sphere rotation into rotations about orthogonal axes. For compact groups such as SU(N), the definitive statement of the G = KAK theorem is found in Helgason [H01, thm8.6,§VII.8]. The Khaneja Glaser decomposition (KGD) is constructed by an explicit invocation of the theorem, with G = SU(2n). In fact, one formulation [KG01, Cor.3] requires two applications of the theorem. In this note, we discuss the statement of Corollary 2 ibid. Also, we note that the Cosine Sine decomposition (CSD) [GvL96, pg.77] [PW94] of numerical linear algebra is the output of the theorem for one standard choice of inputs for G = SU(2n). Per the statement of the abstract, these two matrix decompositions are closely related. Indeed, one results from the other after swapping the labels on the first and last qubit. We derive this result in context, using the G = KAK language. As such, this note is another instance of a Lie theoretic decomposition specializing to matrix analysis. The theorem has three inputs, each dependent on the last. The first is the Lie group G with Lie algebra g. Should G ⊆ Gl(n,C) as a closed subgroup, i.e. G is linear, then g is the set (in fact vector space) of matrix logarithms of elements of G. The algebra operation in this case is given by [X ,Y ] = XY −YX for X ,Y ∈ g. The ∗Mathematical and Computational Sciences Division, National Institute of Standards and Technology, Gaithersburg, MD, 20899, [email protected]