Geometric representations of solutions provides intuitive physical insights. To which end studying dynamics of Quantum systems via
su(n) Lie algebra proves to be convenient way of obtaining geometric solution. In this paper link is established between two formalisms that made use of Lie algebra to describe equation of motion for quantum system. In both approaches the Hamiltonian and the density matrix are expressed as a linear combination of the Lie group. To exemplify the approach we consider a very well studied two level system coupled by a laser pulse. Beyond establishing link between these two formalism we obtained two constants of motion by assuming time dependent detuning whose time profile is assumed to be same as the laser pulse. Consequently we have shown how one can have two disjoint subspaces whose evolution vector is independent of each other.
Abstract Simulating the time evolution of a Hamiltonian system on a classical computer is hard—The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge—Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution, is therefore essential for algorithm design and circuit implementation. Most current techniques are fallible because of truncation errors or the stagnation at local solutions. This work offers an innovative avenue by tackling the Cartan decomposition with the notion of Lax dynamics. Within the integration errors that is controllable, this approach gives rise to a genuine unitary synthesis that not only is numerically feasible, but also can be utilized to gauge the quality of results produced by other means, and extend the knowledge to a wide range of applications. This paper aims at establishing the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics.
Four-level systems in quantum optics, and for representing two qubits in quantum computing, are difficult to solve for general time-dependent Hamiltonians. A systematic procedure is presented which combines analytical handling of the algebraic operator aspects with simple solutions of classical, first-order differential equations. In particular, by exploiting $\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{su}(2)$ and $\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{su}(2)\ensuremath{\bigoplus}\mathrm{u}(1)$ subalgebras of the full SU(4) dynamical group of the system, the nontrivial part of the final calculation is reduced to a single Riccati (first-order, quadratically nonlinear) equation, itself simply solved. Examples are provided of two-qubit problems from the recent literature, including implementation of two-qubit gates with Josephson junctions.
The quantum dynamics of arbitrary
N-level systems, including dissipative systems, are modeled exactly here by the dynamics of classical coupled oscillators. A one-to-one correspondence is established between the quantum states and the positions of the oscillators. Quantum coherence, expectation values, and measurement probabilities for system observables can therefore be realized from the corresponding classical states. Although the well-known equivalence [SU(2), SO(3) homomorphism] of two-level quantum dynamics to a rotation in real, physical space cannot be generalized to arbitrary
N-level systems, the representation of quantum dynamics by a system of coupled harmonic oscillators in one physical dimension
is general for any
N. The time evolution of an
N-level system [generated by a complex element of the SU(
N) group], is first represented as the rotation of a real state vector in (unphysical) hyperspace, as previously known for density matrix states and also extended here to include Schr\"odinger states. The resulting rotor in
n Euclidean dimensions [the rotation group SO(
n)] is then mapped directly to
n oscillators in one physical dimension, which significantly reduces the level of abstraction required to visualize quantum dynamics compared to vector models or generalized Bloch spheres in higher dimensions. The number of such oscillators needed to represent
N-level systems scales as
N2 for the density matrix formalism but increases only linearly with
N for Schr\"odinger states. Values for the classical coupling constants are readily derived from the system Hamiltonian, allowing construction of classical mechanical systems that provide insight into the dynamics of abstract quantum systems (new dynamical invariants) as well as a metric for characterizing the interface between quantum and classical mechanics. A distinctive attribute of the quantum-classical connection as presented here is the necessity for both positive and negative couplings and, in the case of dissipative systems, antisymmetric couplings.
Dynamical symmetries, time-dependent operators that almost commute with the Hamiltonian, extend the role of ordinary symmetries. Motivated by progress in quantum technologies, we illustrate a practical algebraic approach to computing such time-dependent operators. Explicitly we expand them as a linear combination of time-independent operators with time-dependent coefficients. There are possible applications to the dynamics of systems of coupled coherent two-state systems, such as qubits, pumped by optical excitation and other addressing inputs. Thereby, the interaction of the system with the excitation is bilinear in the coherence between the two states and in the strength of the time-dependent excitation. The total Hamiltonian is a sum of such bilinear terms and of terms linear in the populations. The terms in the Hamiltonian form a basis for Lie algebra, which can be represented as coupled individual two-state systems, each using the population and the coherence between two states. Using the factorization approach of Wei and Norman, we construct a unitary quantum mechanical evolution operator that is a factored contribution of individual two-state systems. By that one can accurately propagate both the wave function and the density matrix with special relevance to quantum computing based on qubit architecture. Explicit examples are derived for the electronic dynamics in coupled semi-conducting nanoparticles that can be used as hardware for quantum technologies.