Time complexity and gate complexity

We formulate and investigate the simplest version of time-optimal quantum computation theory (t-QCT), where the computation time is defined by the physical one and the Hamiltonian contains only one- and two-qubit interactions. This version of t-QCT is also considered as optimality by sub-Riemannian geodesic length. The work has two aims: one is to develop a t-QCT itself based on physically natural concept of time, and the other is to pursue the possibility of using t-QCT as a tool to estimate the complexity in conventional gate-optimal quantum computation theory (g-QCT). In particular, we investigate to what extent is true the statement: time complexity is polynomial in the number of qubits if and only if so is gate complexity. In the analysis, we relate t-QCT and optimal control theory (OCT) through fidelity-optimal computation theory (f-QCT); f-QCT is equivalent to t-QCT in the limit of unit optimal fidelity, while it is formally similar to OCT. We then develop an efficient numerical scheme for f-QCT by modifying Krotov's method in OCT, which has monotonic convergence property. We implemented the scheme and obtained solutions of f-QCT and of t-QCT for the quantum Fourier transform and a unitary operator that does not have an apparent symmetry. The former has a polynomial gate complexity and the latter is expected to have exponential one because a series of generic unitary operators has a exponential gate complexity. The time complexity for the former is found to be linear in the number of qubits, which is understood naturally by the existence of an upper bound. The time complexity for the latter is exponential. Thus the both targets are examples satisfyng the statement above. The typical characteristics of the optimal Hamiltonians are symmetry under time-reversal and constancy of one-qubit operation, which are mathematically shown to hold in fairly general situations.Comment: 11 pages, 6 figure

Entanglement vs. the quantum-to-classical transition

We analyze the quantum-to-classical transition (QCT) for coupled bipartite quantum systems for which the position of one of the two subsystems is continuously monitored. We obtain the surprising result that the QCT can emerge concomitantly with the presence of highly entangled states in the bipartite system. Furthermore the changing degree of entanglement is associated with the back-action of the measurement on the system and is itself an indicator of the QCT. Our analysis elucidates the role of entanglement in von Neumann's paradigm of quantum measurements comprised of a system and a monitored measurement apparatus

Conditions for the Quantum to Classical Transition: Trajectories vs. Phase Space Distributions

We contrast two sets of conditions that govern the transition in which classical dynamics emerges from the evolution of a quantum system. The first was derived by considering the trajectories seen by an observer (dubbed the ``strong'' transition) [Bhattacharya, et al., Phys. Rev. Lett. 85: 4852 (2000)], and the second by considering phase-space densities (the ``weak'' transition) [Greenbaum, et al., Chaos 15, 033302 (2005)]. On the face of it these conditions appear rather different. We show, however, that in the semiclassical regime, in which the action of the system is large compared to $\hbar$, and the measurement noise is small, they both offer an essentially equivalent local picture. Within this regime, the weak conditions dominate while in the opposite regime where the action is not much larger than Planck's constant, the strong conditions dominate.Comment: 8 pages, 2 eps figure

Machine Learning for Observables: Reactant to Product State Distributions for Atom-Diatom Collisions

Machine learning-based models to predict product state distributions from a distribution of reactant conditions for atom-diatom collisions are presented and quantitatively tested. The models are based on function-, kernel- and grid-based representations of the reactant and product state distributions. While all three methods predict final state distributions from explicit quasi-classical trajectory simulations with R$^2$ > 0.998, the grid-based approach performs best. Although a function-based approach is found to be more than two times better in computational performance, the kernel- and grid-based approaches are preferred in terms of prediction accuracy, practicability and generality. The function-based approach also suffers from lacking a general set of model functions. Applications of the grid-based approach to nonequilibrium, multi-temperature initial state distributions are presented, a situation common to energy distributions in hypersonic flows. The role of such models in Direct Simulation Monte Carlo and computational fluid dynamics simulations is also discussed

On the influence of collisional rate coefficients on the water vapour excitation

Water is a key molecule in many astrophysical studies. Its high dipole moment makes this molecule to be subthermally populated under the typical conditions of most astrophysical objects. This motivated the calculation of various sets of collisional rate coefficients (CRC) for H$_2$O (with He or H$_2$) which are necessary to model its rotational excitation and line emission. We performed accurate non--local non--LTE radiative transfer calculations using different sets of CRC in order to predict the line intensities from transitions that involve the lowest energy levels of H$_2$O (E $<$ 900 K). The results obtained from the different CRC sets are then compared using line intensity ratio statistics. For the whole range of physical conditions considered in this work, we obtain that the intensities based on the quantum and QCT CRC are in good agreement. However, at relatively low H$_2$ volume density ($n$(H$_2$) $<$ 10$^7$ cm$^{-3}$) and low water abundance ($\chi$(H$_2$O) $<$ 10$^{-6}$), these physical conditions being relevant to describe most molecular clouds, we find differences in the predicted line intensities of up to a factor of $\sim$ 3 for the bulk of the lines. Most of the recent studies interpreting early Herschel Space Observatory spectra used the QCT CRC. Our results show that although the global conclusions from those studies will not be drastically changed, each case has to be considered individually, since depending on the physical conditions, the use of the QCT CRC may lead to a mis--estimate of the water vapour abundance of up to a factor of $\sim$ 3