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Hamiltonian engineering is an important approach for quantum information processing, when appropriate materials do not exist in nature or are unstable. So far there is no stable material for the Kitaev spin Hamiltonian with anisotropic interactions on a honeycomb lattice, which plays a crucial role in the realization of both Abelian and non-Abelian anyons. Here, we show two methods to dynamically realize the Kitaev spin Hamiltonian from the conventional Heisenberg spin Hamiltonian using pulse-control techniques based on the Baker-Campbell-Hausdorff (BCH) formula. In the first method, the Heisenberg interaction is changed into Ising interactions in the first process of the pulse sequence. In the next process of the first method, we transform them to a desirable anisotropic Kitaev spin Hamiltonian. In the second more efficient method, we show that if we carefully design two-dimensional pulses that vary depending on the qubit location, we can obtain the desired Hamiltonian in only one step of applying the BCH formula. As an example, we apply our methods to spin qubits based on quantum dots, in which the effects of both the spin-orbit interaction and the hyperfine interaction are estimated.

We present an approach that allows performing computations related to the Baker-Campbell-Haussdorff (BCH) formula and its generalizations in an arbitrary Hall basis, using labeled rooted trees. In particular, we provide explicit formulas (given in terms of the structure of certain labeled rooted trees) of the continuous BCH formula. We develop a rewriting algorithm (based on labeled rooted trees) in the dual Poincare-Birkhoff-Witt (PBW) basis associated to an arbitrary Hall set, that allows handling Lie series, exponentials of Lie series, and related series written in the PBW basis. At the end of the paper we show that our approach is actually based on an explicit description of an epimorphism ν of Hopf algebras from the commutative Hopf algebra of labeled rooted trees to the shuffle Hopf algebra and its kernel ker ν.

We estimate Grobner-Shirshov bases for the Lie algebra An given arbitrary orders of generators (nodes of a Dynkin graph). Previously, the Grobner-Shirshov basis was computed in [1] for the particular case where nodes of the Dynkin graph are ordered successively.

In this paper, we consider the counting process for a class of Markovian arrival processes (MAPs). We assume that the representing matrices in such MAPs are expanded in terms of matrix representations of the standard generators in the Lie algebra of the special linear group. The primary purpose of this paper is to construct an explicit solution of the time-dependent distribution and factorial moments of the number of arrival events in (0,t] of the counting process for this class of MAPs. Our construction relies on the Baker--Hausdorff lemma and the specific structure of the representing matrices. To investigate the efficiency of CPU usage with the explicit solution, we have conducted numerical experiments on computing the time-dependent distribution of the counting process through the explicit solution and uniformization-based method. We show that the CPU times required to compute the time-dependent distribution of the number of arrival events in (0,t] through the explicit solution have little sensitivity to t, while the consumption of CPU times with the uniformization-based method becomes greater as t increases. For illustrative purposes, we present a system performance analysis of a queueing system for possible use in automatic call distribution (ACD) systems. As an application of the explicit solution, we use it to express the waiting time distribution of the queueing system. Some numerical examples are also given with comparisons to computer simulations.

Two-forms in Minkowski space-time may be considered as generators of Lorentz transformations. Here, the covariant and general expression for the composition law (Baker-Campbell-Hausdorff formula) of two Lorentz transformations in terms of their generators is obtained. Every subalgebra of the Lorentz algebra of such generators, up to one, may be generated by a sole pair of generators. When the subalgebra is known, the above BCH formula for the two two-forms simplifies. Its simplified expressions for all such subalgebras are also given., 16 pages, no figures

We find local closed-form expression for the Baker—Campbell—Hausdorff formula in the Lie algebra so(3), and interpret the formula geometrically in terms of rotation vectors in ℝ3.

We describe realizations of the colour analogue of the Heisenberg Lie algebra by power series in non-commuting indeterminates satisfying Heisenberg's canonical commutation relations of quantum mechanics. The obtained formulas are used to construct new operator representations of the colour Heisenberg Lie algebra. These representations are shown to be closely connected with some combinatorial identities and functional difference-differential interpolation formulae involving Euler numbers.

A recursive algorithm is derived, allowing the expansion in λ of z = exp [x + λy] for noncommuting x and y, written as ordered product of exponentials. Such an expansion is the inverse of the usual Baker-Campbell-Hausdorff formula. The explicit form of the terms, up to third order in λ is also given. The same method provides the explicit expansion to any order for the matrix elements of z.

In the theory of supersymmetricSU(2) Yang-Mills fields described on the 8th dimensional superspace, the local gauge transformations constitute a group whose Lie algebra has as coefficients belonging to the Weyl-spinorial Grassmann algebra. We present here a Baker-Campbell-Hausdorff formula for the chiralSU(2) supergroup and using this formula we give the finite form of each element of this group in terms of the local fields emering in the infinitesimal real superscalar generator.

The dynamics of a classical system, specified by a HamiltonianH 0(q,p) and a number of constraint functionsϕ k (q,p) in Euclidean phase space (q,p), is described by means of a new HamiltonianH(q,p), which is an invariant of the (closed) Poisson-bracket Lie algebra (H 0,ϕ k ). Fixed values ofϕ k (not necessarily zero) are given by initial conditions, and they are conserved along the trajectories determined by the Hamilton equations. The quantization is performed by the standard Heisenberg commutation relations in the embedding phase space, while the constraint functions are put in correspondence with constraint operators $$\hat \phi _k$$ which generate the Lie algebra of quantum commutators. A subset of commuting constraint operators may be chosen to have certain values in the initial state; and as soon as the Hamiltonian operatorĤ is an invariant of the Lie algebra, these conditions are maintained permanently. Simple examples are presented. Systems with both Bose and Fermi degrees of freedom (and constraints) can be treated universally.

The convergence of the Zassenhaus formula is proven under an appropriate condition as well as for other exponential operators such as the Baker-Campbell-Hausdorff formula.

This paper considers perturbationsH=H 0+eV of the Hamiltonian operatorH 0 of a free scalar Boson field.V is a polynomial in the annihilation creation operators. Terms of any order are allowed inV, but point interactions, such as ∫:0(x)4(x)4:dx, are not considered. Unnormalized solutions for the Schrodinger equation are found. For e→0, these solutions have a partial asymptotic expansion in powers of e. The set of all possible pertubation termsV forms a Lie algebra. General properties of this Lie algebra are investigated.

Six different proofs of this formula are given in [1-6]. We believe that our proof below is more elementary than any one of these six. Let A be the algebra of formal power series in three independent variables t, x, y with rational coefficients, where we postulate that t commutes with x and y bht x and y do not commute. We denote by D the operator of formal differentiation with respect to the variable t. With respect to the bracket operation [a, b] = a b b a , A is a Lie algebra over (2We say that a e A is a Lie element if it belongs to the Lie subalgebra of A generated by x and y. For a e A we denote by L a and R a the left and right multiplications by a in A, respectively. We put Ad a = L , R . The power series a e A with zero constant term form an ideal A + of A. The maps exp: A + ~ I +A + and log: 1 +A+--*A +, defined by usual power series, are inverse to each other. We define h e A by h = log (e t ~ e t Y) and put h = ~,__> 1 h, (x, y) t". Since L x R:, = R~ L x we have

The author obtains a compact form of expression of the BakerCampbell-Hausdorff formula. In it, the dependence of the boson superfields, which take values on the Grassmann hull of the Lie algebra, on the generators of the two-dimensional Grassmann algebra is factorized in the form of a single exponential.