Time-optimal decompositions in SU(2)

A connected Lie group G is generated by its two 1-parametric subgroups exp(tX), exp(tY ) if and only if the Lie algebra of G is generated by {X, Y }. We consider decompositions of elements of G into a product of such exponentials with times t > 0 and study the problem of minimizing the total time of the decompositions for a fixed element of G. We solve this problem for the group SU2 and describe the structure of the time-optimal decompositions. 0. Introduction. In the axiomatics of quantum mechanics a state of a quantum system is a unit vector in a complex hermitian vector space and its evolution is given by unitary transformations. In quantum computing the space of states is finite-dimensional and the quantum computation is an element of SUN , which is a compact connected real Lie group. In order to carry out a quantum computation in physical reality, we need to be able to transform an initial state ψ of a quantum processor into a state gψ, where g is the element of SUN representing the quantum computation. In the classical (non-quantum) case, a computation can be viewed as a boolean function in several variables. A processor can not implement an arbitrary function directly, but instead decomposes the computation into elementary steps, which is reflected in the process of programming. An efficient algorithm for a classical computation is a program that minimizes the number of elementary steps, which results in a shorter run time. Likewise, an implementation of a quantum computation g is a factorization of g into certain elementary factors, called the quantum gates (in analogy with the classical logical gates used for the decomposition of an arbitrary boolean function of several variables in a disjunctive normal form, for example). Some quantum gates are the direct analogues of the classical logical gates and are discrete, while others depend on a continuous time parameter. Time evolution of a quantum system is governed by a Hamiltonian H and is given by the exponential exp(itH) ∈ SUN . Here X = iH belongs to the Lie algebra suN . We are going to assume that all available quantum gates are continuous, since the discrete gates are physically realized as continuous gates applied for a specific finite time. This leads to the following quantum control problem: given a set S ⊂ suN of quantum controls (gates), decompose an element g ∈ SUN into a product g = exp(t1C1)× . . .× exp(tnCn), (0.1) where Ci ∈ S, ti ∈ R. This quantum control problem admits a solution for every g precisely when the set S generates the Lie algebra suN (see Theorem 1.1 below). This 1 School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, K1S 5B6, Canada. E-mail: [email protected] 2010 Mathematics Subject Classification: 81P68, 22E70, 57R27. 1 result is proved using topological methods and does not provide an effective procedure for finding such decompositions. Minimal sets of quantum gates for quantum computing were discussed in [2] and [6], where it is shown that a set of two generic Hamiltonians is sufficient for controllability. A review of quantum control methods in physical chemistry is given in [7]. The problem of finding explicit factorizations of type (0.1) goes back to Euler [3], who studied it for the group SO3 of rotations in R . The Lie algebra so3 consists of skew-symmetric 3× 3 matrices and the exponentials exp(tCi) corresponding to C1 =   0 0 0 0 0 −1 0 1 0 