Time-dependent unbounded Hamiltonian simulation with vector norm scaling

The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations.

THE CONVERGENCE ON ALGEBRAIC LATTICE NORMED SPACES

The multiplicative convergence on Riesz algebras introduced and investigated with respect to norm and order convergences. If X is a Riesz space and E is a Riesz algebra then the vector norm μ:X→E_+ can be considered. Then (X,μ,E) is called algebraic lattice normed spaces. A net (x_α )_(α∈A) in an (X,μ,E) is said to be multiplicative μ-convergent to x∈X if μ(x_α-x)∙u□(→┴o ) 0 holds for all u∈E_+. In this paper, the general properties of this convergence are studied.

Guaranteeing String Stability of Multiple Interconnected Vehicles Using Heterogeneous Controllers

In this paper the issue of string stability for acceleration-controlled vehicles interconnected in a chain is studied. String stability is concerned with having bounded displacements between vehicles in such a way that displacements should not grow unboundedly with respect to the perturbation. Different definitions can be given to string stability: one that relates to the amplification of a local disturbance acting on one vehicle towards the whole vehicle chain, more strict definition that is related to the boundedness of vector norm of displacements with respect to the bounded vector norm of disturbance inputs acting on all vehicles; and, most practical definition that considers the boundedness of signal norm of each individual displacement with respect to the bounded signal norm of disturbance inputs acting on all vehicles, independently from the number of vehicles. It has been proven that these definitions are all impossible to be achieved using any linear homogeneous unidirectional distributed controllers with constant spacing policy. This paper proposes linear heterogeneous controllers where each vehicle behaves differently from others in a vehicle chain. We prove that three different definitions of string stability can be attained using the proposed heterogenous controller. We propose sufficient conditions to guarantee string stability and boundedness of acceleration of each vehicle. Finally, simulation results are given to illustrate the effectiveness of proposed heterogenous control synthesis.

Time-Varying Vector Norm and Lower and Upper Bounds on the Solutions of Uniformly Asymptotically Stable Linear Systems

Based on the eigenvalue idea and the time-varying weighted vector norm in the state space R n we construct here the lower and upper bounds of the solutions of uniformly asymptotically stable linear systems. We generalize the known results for the linear time-invariant systems to the linear time-varying ones.

One-bit compressed sensing with partial Gaussian circulant matrices

In this paper we consider memoryless one-bit compressed sensing with randomly subsampled Gaussian circulant matrices. We show that in a small sparsity regime and for small enough accuracy $\delta$, $m\simeq \delta ^{-4} s\log (N/s\delta )$ measurements suffice to reconstruct the direction of any $s$-sparse vector up to accuracy $\delta$ via an efficient program. We derive this result by proving that partial Gaussian circulant matrices satisfy an $\ell _1/\ell _2$ restricted isometry property property. Under a slightly worse dependence on $\delta$, we establish stability with respect to approximate sparsity, as well as full vector recovery results, i.e., estimation of both vector norm and direction.

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.