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In this article, we compare the methods implementing the real-time evolution operator generated by a unitary diagonal matrix where its entries obey a known underlying real function. When the size of the unitary diagonal matrix is small, a well-known method based on Walsh operators gives a good and precise implementation. In contrast, as the number of qubits grows, the precise one uses exponentially increasing resources, and we need an efficient implementation based on suitable approximate functions. Using piecewise polynomial approximation of the function, we summarize the methods with different polynomial degrees. Moreover, we obtain the overheads of gate count for different methods concerning the error bound and grid parameter (number of qubits). This enables us to analytically find a relatively good method as long as the underlying function, the error bound, and the grid parameter are given. This study contributes to the problem of encoding a known function in the phase factor, which plays a crucial role in many quantum algorithms/subroutines. In particular, we apply our methods to implement the real-time evolution operator for the potential part in the first-quantized Hamiltonian simulation and estimate the resources (gate count and ancillary qubits) regarding the error bound, which indicates that the error coming from the approximation of the potential function is not negligible compared to the error from the Trotter-Suzuki formula.

This article is concerned with a semilinear time-fractional diffusion equation with a superlinear convex semilinear term in a bounded domain $\Omega$ with the homogeneous Dirichlet, Neumann, Robin boundary conditions and non-negative and not identically vanishing initial value. The order of the fractional derivative in time is between $1$ and $0$, and the elliptic part is with time-independent coefficients. We prove (i) The solution with any initial value blow-up if the eigenvalue $\lambda_1$ of the elliptic operator with the minimum real part is non-positive. (ii) Otherwise, the solution blows up if a weighted $L^1$-norm of initial value is greater than some critical value give by $\lambda_1$. We provide upper estimates of the blow-up times. The key is a comparison principle for time-fractional ordinary differential equations., Comment: 14 pages. arXiv admin note: text overlap with arXiv:2302.12724

During the noisy intermediate-scale quantum (NISQ) era, it is important to optimize the quantum circuits in circuit depth and gate count, especially entanglement gates, including the CNOT gate. Among all the unitary operators, diagonal unitary matrices form a special class that plays a crucial role in many quantum algorithms/subroutines. Based on a natural gate set {CNOT, Rz}, quantum circuits for general diagonal unitary matrices were discussed in several previous works, and an optimal synthesis algorithm was proposed in terms of circuit depth. In this paper, we are interested in the implementation of diagonal unitary matrices with reflection symmetry, which has promising applications, including the realization of real-time evolution for first quantized Hamiltonians by quantum circuits. Owing to such a symmetric property, we show that the quantum circuit in the existing work can be further simplified and propose a constructive algorithm that optimizes the entanglement gate count. Compared to the previous synthesis methods for general diagonal unitary matrices, the quantum circuit by our proposed algorithm achieves nearly half the reduction in both the gate count and circuit depth., Comment: 21 pages

Lying between traditional parabolic and hyperbolic equations, time-fractional wave equations of order $\alpha\in(1,2)$ in time inherit both decaying and oscillating properties. In this article, we establish a long-time asymptotic estimate for homogeneous time-fractional wave equations, which readily implies the strict positivity/negativity of the solution for $t\gg1$ under some sign conditions on initial values. As a direct application, we prove the uniqueness for a related inverse source problem on determining the temporal component., Comment: 15 pages, 1 figure

This article deals with the initial-boundary value problem for a moderately coupled system of time-fractional diffusion equations. Defining the mild solution, we establish fundamental unique existence, limited smoothing property and long-time asymptotic behavior of the solution, which mostly inherit those of a single equation. Owing to the coupling effect, we also obtain the uniqueness for an inverse problem on determining all the fractional orders by the single point observation of a single component of the solution., Comment: 24 pages

We consider the well-posedness of the initial-boundary value problem for a time-fractional partial differential equation with the fractional order lying in (1,2]. First, we show some results on the unique existence of solution to time-fractional ordinary differential equations. Then the unique existence of weak and strong solutions to a time-fractional partial differential equation is derived by using the Galerkin method.

We consider the inverse problem of determining the initial states or the source term of a hyperbolic equation damped by some non-local time-fractional derivative. This framework is relevant to medical imaging such as thermoacoustic or photoacoustic tomography. We prove a stability estimate for each of these two problems, with the aid of a Carleman estimate specifically designed for the governing equation.

In this paper, we study the asymptotic estimate of solution for a mixed-order time-fractional diffusion equation in a bounded domain subject to the homogeneous Dirichlet boundary condition. Firstly, the unique existence and regularity estimates of solution to the initial-boundary value problem are considered. Then combined with some important properties, including a maximum principle for a time-fractional ordinary equation and a coercivity inequality for fractional derivatives, the energy method shows that the decay in time of the solution is dominated by the term $t^{-\alpha}$ as $t\to\infty$., Comment: 27 pages

In this article, we consider a linearized magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. We first prove two kinds of Carleman estimates. This is done by combining the Carleman estimates for the parabolic and the elliptic equations. Then we apply the Carleman estimates to prove Holder type stability results for some inverse source problems., Comment: 36 pages

In this article, we consider a magnetohydrodynamics system for incompressible flow in a three-dimensional bounded domain. Firstly, we give the stability results for our inverse coefficients problem. Secondly, we establish and prove two Carleman estimates for both direct problem and inverse problem. Finally, we complete the proof of stability result in terms of the above Carleman estimates., Comment: 30 pages

We prove that the stationary magnetic potential vector and the electrostatic potential entering the dynamic magnetic Schr\"odinger equation can be Lipschitz stably retrieved through finitely many local boundary measurements of the solution. The proof is by means of a specific global Carleman estimate for the Schr\"odinger equation, established in the first part of the paper.

In this paper, we discuss an initial-boundary value problem (IBVP) for the multi-term time-fractional diffusion equation with x-dependent coefficients. By means of the Mittag-Leffler functions and the eigenfunction expansion, we reduce the IBVP to an equivalent integral equation to show the unique existence and the analyticity of the solution for the equation. Especially, in the case where all the coefficients of the time-fractional derivatives are non-negative, by the Laplace and inversion Laplace transforms, it turns out that the decay rate of the solution for long time is dominated by the lowest order of the time-fractional derivatives. Finally, as an application of the analyticity of the solution, the uniqueness of an inverse problem in determining the fractional orders in the multi-term time-fractional diffusion equations from one interior point observation is established.

In this article, we prove Carleman estimates for the generalized time-fractional advection-diffusion equations by considering the fractional derivative as perturbation for the first order time-derivative. As a direct application of the Carleman estimates, we show a conditional stability of a lateral Cauchy problem for the time-fractional advection-diffusion equation, and we also investigate the stability of an inverse source problem.