Time-optimal Unitary Operations in Ising Chains II: Unequal Couplings and Fixed Fidelity

We analytically determine the minimal time and the optimal control laws required for the realization, up to an assigned fidelity and with a fixed energy available, of entangling quantum gates ($\mathrm{CNOT}$) between indirectly coupled qubits of a trilinear Ising chain. The control is coherent and open loop, and it is represented by a local and continuous magnetic field acting on the intermediate qubit. The time cost of this local quantum operation is not restricted to be zero. When the matching with the target gate is perfect (fidelity equal to one) we provide exact solutions for the case of equal Ising coupling. For the more general case when some error is tolerated (fidelity smaller than one) we give perturbative solutions for unequal couplings. Comparison with previous numerical solutions for the minimal time to generate the same gates with the same Ising Hamiltonian but with instantaneous local controls shows that the latter are not time-optimal.Comment: 11 pages, no figure

Brachistochrone of Entanglement for Spin Chains

We analytically investigate the role of entanglement in time-optimal state evolution as an appli- cation of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly cou- pled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initial pure quantum states of a tripartite system. The integrals of the motion of the brachistochrone are determined by fixing the minimal time at which the residual entanglement is maximized. Entan- glement plays a role for W and GHZ initial quantum states, and for the bi-separable initial state in which the indirectly coupled qubits have a nonzero value of the 2-tangle.Comment: 9 pages, 4 figure

A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations

We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the dependence of the coefficients is nonlinear and nonlocal in time with respect to the unknowns. We extend the numerical scheme proposed and studied recently by the authors for a single FPK equation of this type. We analyse the convergence of the scheme and we study its applicability in two examples. The first one concerns a population model involving two interacting species and the second one concerns two populations Mean Field Games

Time-optimal CNOT between indirectly coupled qubits in a linear Ising chain

We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $\mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3 is $T=\sqrt{3/2} J^{-1}$, i.e. faster than a previous estimate based on a similar Hamiltonian and the assumption of local unitaries with zero time cost. Furthermore, performing a simple Walsh-Hadamard rotation in the Hlibert space of qubit 3 shows that the time-optimal synthesis of the $\mathrm{CNOT}^{\pm}(1, 3)$ (which acts as the identity when the control qubit 1 is in the state $\ket{0}$, while if the control qubit is in the state $\ket{1}$ the target qubit 3 is flipped as $\ket{\pm}\rightarrow \ket{\mp}$) also requires the same time $T$.Comment: 9 pages; minor modification

Non-adiabatic Fast Control of Mixed States based on Lewis-Riesenfeld Invariant

We apply the inversely-engineered control method based on Lewis-Riesenfeld invariants to control mixed states of a two-level quantum system. We show that the inversely-engineered control passages of mixed states - and pure states as special cases - can be made significantly faster than the conventional adiabatic control passages, which renders the method applicable to quantum computation. We devise a new type of inversely-engineered control passages, to be coined the antedated control passages, which further speed up the control significantly. We also demonstrate that by carefully tuning the control parameters, the inversely-engineered control passages can be optimized in terms of speed and energy cost.Comment: 9 pages, 9 figures, version to appear in J. Phys. Soc. Jp

Coprophagous features in carnivorous Nepenthes plants: a task for ureases

Most terrestrial carnivorous plants are specialized on insect prey digestion to obtain additional nutrients. Few species of the genus Nepenthes developed mutualistic relationships with mammals for nitrogen supplementation. Whether dietary changes require certain enzymatic composition to utilize new sources of nutrients has rarely been tested. Here, we investigated the role of urease for Nepenthes hemsleyana that gains nitrogen from the bat Kerivoula hardwickii while it roosts inside the pitchers. We hypothesized that N. hemsleyana is able to use urea from the bats’ excrements. In fact, we demonstrate that 15N-enriched urea provided to Nepenthes pitchers is metabolized and its nitrogen is distributed within the plant. As ureases are necessary to degrade urea, these hydrolytic enzymes should be involved. We proved the presence and enzymatic activity of a urease for Nepenthes plant tissues. The corresponding urease cDNA from N. hemsleyana was isolated and functionally expressed. A comprehensive phylogenetic analysis for eukaryotic ureases, including Nepenthes and five other carnivorous plants’ taxa, identified them as canonical ureases and reflects the plant phylogeny. Hence, this study reveals ureases as an emblematic example for an efficient, low-cost but high adaptive plasticity in plants while developing a further specialized lifestyle from carnivory to coprophagy

Cosmology as Geodesic Motion

For gravity coupled to N scalar fields with arbitrary potential V, it is shown that all flat (homogeneous and isotropic) cosmologies correspond to geodesics in an (N+1)-dimensional `augmented' target space of Lorentzian signature (1,N), timelike if V>0, null if V=0 and spacelike if V<0. Accelerating cosmologies correspond to timelike geodesics that lie within an `acceleration subcone' of the `lightcone'. Non-flat (k=-1,+1) cosmologies are shown to evolve as projections of geodesic motion in a space of dimension (N+2), of signature (1,N+1) for k=-1 and signature (2,N) for k=+1. This formalism is illustrated by cosmological solutions of models with an exponential potential, which are comprehensively analysed; the late-time behviour for other potentials of current interest is deduced by comparison.Comment: 26 pages, 2 figures, journal version with additional reference

Fundamental Constants and the Problem of Time

We point out that for a large class of parametrized theories, there is a constant in the constrained Hamiltonian which drops out of the classical equations of motion in configuration space. Examples include the mass of a relativistic particle in free fall, the tension of the Nambu string, and Newton's constant for the case of pure gravity uncoupled to matter or other fields. In the general case, the classically irrelevant constant is proportional to the ratio of the kinetic and potential terms in the Hamiltonian. It is shown that this ratio can be reinterpreted as an {\it unconstrained} Hamiltonian, which generates the usual classical equations of motion. At the quantum level, this immediately suggests a resolution of the "problem of time" in quantum gravity. We then make contact with a recently proposed transfer matrix formulation of quantum gravity and discuss the semiclassical limit. In this formulation, it is argued that a physical state can obey a (generalized) Poincar\'e algebra of constraints, and still be an approximate eigenstate of 3-geometry. Solutions of the quantum evolution equations for certain minisuperspace examples are presented. An implication of our proposal is the existence of a small, inherent uncertainty in the phenomenological value of Planck's constant.Comment: 46 pages + 5 figures, LaTex, NBI-HE-94-3