The Wei-Norman technique allows to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials. In particular it enables to find a time-ordered product of exponentials by solving a set of nonlinear differential equations. The method has numerous theoretical and computational advantages, in particular in optimal control theory. We show that in the unitary case, i.e.\ when the solution of the linear system is given by a unitary evolution operator, the nonlinear system can be by an appropriate choice of ordering reduced to a hierarchy of matrix Riccati equations. Our findings have a particular significance in quantum control theory since pure quantum evolution is unitary.
The Wei-Norman technique allows to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials. In particular it enables to find a time-ordered product of exponentials by solving a set of nonlinear differential equations. The method has numerous theoretical and computational advantages, in particular in optimal control theory. We show that in the unitary case, i.e.\ when the solution of the linear system is given by a unitary evolution operator, the nonlinear system can be by an appropriate choice of ordering reduced to a hierarchy of matrix Riccati equations. Our findings have a particular significance in quantum control theory since pure quantum evolution is unitary.
Wei-Norman decomposition is a well known theory for solving matrix differential equations.The Schrdinger Equation with unitary evolution equations is a matrix differential equations,so Wei-Norman decomposition can be used for solving the multi-level quantum systems.Owing to Wei-Norman decomposition′s application being confined in a little time area,in order to obtain the Unitary evolution matrix at any time,a small division of time is proposed in this paper,and Wei-Norman decomposition is used in every small division to obtain the unitary evolution.Moreover,the proposed method can be used to a two-level quantum system.The relationship between the states of this quantum system and the control fields is given according to the simulation test results.
Using the Wei-Norman theory, we obtain a time-dependent complex Riccati equation (TDCRE) as the solution of the time evolution operator (TEO) of quantum systems described by time-dependent (TD) Hamiltonians that are linear combinations of the generators of the su(1,1), su(2), and so(2,1) Lie algebras. Using a recently developed solution for the time evolution of these quantum systems, we solve the TDCRE recursively as generalized continued fractions, which are optimal for numerical implementations, and establish the necessary and sufficient conditions for the unitarity of the TEO in the factorized representation. The inherited symmetries of quantum systems can be recognized by a simple inspection of the TDCRE, allowing effective quantum Hamiltonians to be associated with it, as we show for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a generic TD system of the Lie algebra su(2). As an application, but also as a consistency test, we compare our solution with the analytic one for the Bloch-Riccati equation considering the Rabi frequency driven by a complex hyperbolic secant pulse generating spin inversion, showing an excellent agreement.
The purpose of controlling a quantum system is to steer the system from an initial state to a target state. Such a control problem can be changed into the problem of generating the desired unitary evolution operators by using an appropriate decomposition. By means of the desired state's exponential factor product theorem in the right-invariant system, the desired unitary evolution operators of a quantum system can be obtained. This paper investigates the functions and features of the Magnus decomposition, the Wei-Norman decomposition, the Lie Group decomposition and the Cartan decomposition contrastively and respectively. Properties and realization performance of those decompositions are analyzed, too. Each type of decomposition applied to different situations is also given.
In this paper, the Dyson series corresponding to the time-varying Hamiltonian of a finite dimensional quantum mechanical system is expanded in terms of products of exponentials of a complete basis of commutator superoperators in the corresponding Liouville space. The Cayley-Hamilton theorem and the Wei-Norman formula allow to express explicitly the functional relation between the Dyson series and the product of exponentials via a set of first order differential equations. Since the method is structure preserving, it can be used for the exact unitary integration of the driven Liouville-von Neumann equation.
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity O(νLt/ε), where the constant ν depends on the problem parameters, L involves a time integral of upper bounds on the norms of evolution operators, and ε is the error. In particular, νL is linear in t for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of Ω(νLt/ε) for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.
Many important quantum systems are characterized by Hamiltonians expressible as a linear combination of time-independent generators of a finite Lie algebra, H ̂ ( t ) = ∑ l = 1 L η l ( t ) g ̂ l . The Wei-Norman framework provides a powerful analytical method for computing the exact time-evolution operator in a factorized form, U ̂ ( t ) = ∏ l = 1 L e Λ l ( t ) g ̂ l . By exploiting the underlying algebraic structure, the method yields a system of differential equations—the Wei-Norman equations—which determine the time-evolution operator coefficients Λ l ( t ) from the Hamiltonian ones η l ( t ) . Although robust and formally extendable, applying the method by hand becomes increasingly difficult or even intractable for high-order systems ( L ≥ 6 ). In this work, we introduce , a Python library that automates the application of this method. Given the commutators between generators, the library symbolically derives the Wei-Norman equations and efficiently computes nested commutators, similarity transformations, and the differential equations that yield Baker-Campbell-Hausdorff-like relations. We validate the library by recovering known results for quantum systems associated with the Lie algebras su ( 1 , 1 ) , su ( 2 ) , sl ( 2 ) , and so ( 2 , 1 ) , and further demonstrate its capabilities by deriving the Wei-Norman equations for a system of two coupled time-dependent harmonic oscillators involving 11 generators. Additionally, we specialize the library to handle the Lie group SU ( N ) using a generic Cartan-Weyl basis for its algebra, and demonstrate its versatility by deriving the Wei-Norman equations for su ( 2 ) , su ( 3 ) , and su ( 4 ) , including the explicit construction of a set of quantum gates for universal quantum computation.