Some Remarks on Quantum Brachistochrone

We study some aspects of the Quantum Brachistochrone Problem. Physical realizability of the faster pseudo Hermitian version of the problem is also discussed. This analysis, applied to simple quantum gates, supports an informational interpretation of the problem that is quasi Hermitian invariant.Comment: 9 pages, 3 figure

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

Quantum Adiabatic Brachistochrone

We formulate a time-optimal approach to adiabatic quantum computation (AQC). A corresponding natural Riemannian metric is also derived, through which AQC can be understood as the problem of finding a geodesic on the manifold of control parameters. This geometrization of AQC is demonstrated through two examples, where we show that it leads to improved performance of AQC, and sheds light on the roles of entanglement and curvature of the control manifold in algorithmic performance.Comment: 4 pages, 2 figure

Complementarity between quantum entanglement, geometrical and dynamical appearances in NN spin-1/21/2 system under all-range Ising model

With the growth of geometric science, including the methods of exploring the world of information by means of modern geometry, there has always been a mysterious and fascinating ambiguous link between geometric, topological and dynamical characteristics with quantum entanglement. Since geometry studies the interrelations between elements such as distance and curvature, it provides the information sciences with powerful structures that yield practically useful and understandable descriptions of integrable quantum systems. We explore here these structures in a physical system of $N$ interaction spin-$1/2$ under all-range Ising model. By performing the system dynamics, we determine the Fubini-Study metric defining the relevant quantum state space. Applying Gaussian curvature within the scope of the Gauss-Bonnet theorem, we proved that the dynamics happens on a closed two-dimensional manifold having both a dumbbell-shape structure and a spherical topology. The geometric and topological phases appearing during the system evolution processes are sufficiently discussed. Subsequently, we resolve the quantum brachistochrone problem by achieving the time-optimal evolution. By restricting the whole system to a two spin-$1/2$ system, we investigate the relevant entanglement from two viewpoints; The first is of geometric nature and explores how the entanglement level affects derived geometric structures such as the Fubini-Study metric, the Gaussian curvature, and the geometric phase. The second is of dynamic nature and addresses the entanglement effect on the evolution speed and the related Fubini-Study distance. Further, depending on the degree of entanglement, we resolve the quantum brachistochrone problem