In this paper, we introduce some new quantum numerical techniques of midpoint and trapezoidal type essentially by using the decomposition technique. We also check the order of convergence of our suggested iterative methods. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy.
In this paper, we introduce some new quantum numerical techniques of midpoint and trapezoidal type essentially by using the decomposition technique. We also check the order of convergence of our suggested iterative methods. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy.
Abstract Quantum methods are proposed for solving differential equations that are based on a gradual improvement of the solution via an iterative process, and are targeted at applications in fluid dynamics. First, the Jacobi iteration is implemented on a quantum register that utilizes a linear combination of unitaries (LCU) approach to store the trajectory information. Second, quantum methods are extended to Gauss‐Seidel iterative methods. Additionally, a quantum‐suitable resolvent decomposition is proposed based on the Woodbury identity. From a technical perspective, tools are developed and utilized for the block encoding of specific matrices as well as their multiplication. The approach is benchmarked on paradigmatic fluid dynamics problems. These results stress that instead of inverting large matrices, one can program quantum computers to perform multigrid‐type computations and leverage corresponding advances in scientific computing.
This chapter covers quantum algorithms for numerically solving differential equations and the areas of application where such capabilities might be useful, such as computational fluid dynamics, semiconductor chip design, and many engineering workflows. We focus mainly on algorithms for linear differential equations (covering both partial and ordinary linear differential equations), but we also mention the additional nuances that arise for nonlinear differential equations. We discuss important caveats related to both the data input and output aspects of an end-to-end differential equation solver, and we place these quantum methods in the context of existing classical methods currently in use for these problems.
Geometric integrators of the Schrödinger equation conserve exactly many invariants of the exact solution. Among these integrators, the split-operator algorithm is explicit and easy to implement but, unfortunately, is restricted to systems whose Hamiltonian is separable into kinetic and potential terms. Here, we describe several implicit geometric integrators applicable to both separable and nonseparable Hamiltonians and, in particular, to the nonadiabatic molecular Hamiltonian in the adiabatic representation. These integrators combine the dynamic Fourier method with the recursive symmetric composition of the trapezoidal rule or implicit midpoint method, which results in an arbitrary order of accuracy in the time step. Moreover, these integrators are exactly unitary, symplectic, symmetric, time-reversible, and stable and, in contrast to the split-operator algorithm, conserve energy exactly, regardless of the accuracy of the solution. The order of convergence and conservation of geometric properties are proven analytically and demonstrated numerically on a two-surface NaI model in the adiabatic representation. Although each step of the higher order integrators is more costly, these algorithms become the most efficient ones if higher accuracy is desired; a thousand-fold speedup compared to the second-order trapezoidal rule (the Crank-Nicolson method) was observed for a wavefunction convergence error of 10−10. In a companion paper [J. Roulet, S. Choi, and J. Vaníček, J. Chem. Phys. 150, 204113 (2019)], we discuss analogous, arbitrary-order compositions of the split-operator algorithm and apply both types of geometric integrators to a higher-dimensional system in the diabatic representation.
We evaluate the accuracy of the quantum inverse (Q-Inv) algorithm, in which the multiplication of Ĥ – k to the reference wave function is replaced by the Fourier transformed multiplication of e –iλ Ĥ, as a function of the integration parameters and the iteration power k for various systems, including H 2, LiH, BeH 2 and the notorious H 4 molecule at square geometry. We further consider the possibility of employing the Gaussian-quadrature rule as an alternate integration method and compared it to the results employing trapezoidal integration. The Q-Inv algorithm is compared to the inverse iteration method using the Ĥ –1 inverse (I-Iter) and the exact inverse by lower-upper decomposition. Energy values are evaluated as the expectation values of the Hamiltonian. Results suggest that the Q-Inv method provides lower energy results than the I-Iter method up to a certain k, after which the energy increases due to errors in the numerical integration that are dependent on the integration interval. A combined Gaussian-quadrature and trapezoidal integration method proved to be more effective at reaching convergence while decreasing the number of operations. For systems like H 4, in which the Q-Inv cannot reach the expected error threshold, we propose a combination of Q-Inv and I-Iter methods to further decrease the error with k at lower computational cost. Finally, we summarize the recommended procedure when treating unknown systems.
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables n, which means our algorithm's complexity is less than O(n3) of the corresponding classical ones when the set of equations has many variables and not so high orders. Different with most popular way of representing the results by state vector, we get the results in computation basis which is readable directly.
The construction of quantum time of arrival operator conjugate to the system Hamiltonian leads to a particular linear homogeneous Goursat problem. In this work, we demonstrate how to approximate the solution of the mentioned differential equation both semi-analytically and numerically with the goal of calculating the largest eigenvalue of the associated confined time of arrival operator. In the analytical approximation, we used the partial sum expansion of the solution and showed that the resulting largest eigenvalue converges as the number of terms increases. The result shows that for the parameters considered in this paper, the approximation is sufficient up to the fourth order correction term. In the numerical approximation, we develop a non-iterative formula to obtain the numerical solution of the Goursat problem. The performance of the non-iterative method is compared with the known numerical techniques in literature. Numerical results show that the non-iterative algorithm is more accurate and faster compared to the other techniques considered in this paper. Specifically, the proposed algorithm was able to approximate the largest eigenvalue of the confined time of arrival operator up to the third correction term for a particular nonlinear system.