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Register a new userQuantum Algorithm for the Vlasov Equation
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The Vlasov-Maxwell system of equations, which describes classical plasma physics, is extremely challenging to solve, even by numerical simulation on powerful computers. By linearizing and assuming a Maxwellian background distribution function, we convert the Vlasov-Maxwell system into a Hamiltonian simulation problem. Then for the limiting case of electrostatic Landau damping, we design and verify a quantum algorithm, appropriate for a future error-corrected universal quantum computer. While the classical simulation has costs that scale as $mathcal{O}(N_v t)$ for a velocity grid with $N_v$ grid points and simulation time $t$, our quantum algorithm scales as $mathcal{O}(text{polylog}(N_v) t/delta)$ where $delta$ is the measurement error, and weaker scalings have been dropped. Extensions, including electromagnetics and higher dimensions, are discussed. A quantum computer could efficiently handle a high-resolution, six-dimensional phase-space grid, but the $1/delta$ cost factor to extract an accurate result remains a difficulty. This paper provides insight into the possibility of someday achieving efficient plasma simulation on a quantum computer.
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Numerical methods that approximate the solution of the Vlasov-Poisson equation by a low-rank representation have been considered recently. These methods can be extremely effective from a computational point of view, but contrary to most Eulerian Vlasov solvers, they do not conserve mass and momentum, neither globally nor in respecting the corresponding local conservation laws. This can be a significant limitation for intermediate and long time integration. In this paper we propose a numerical algorithm that overcomes some of these difficulties and demonstrate its utility by presenting numerical simulations.
The quantum search algorithm is a technique for searching N possibilities in only sqrt(N) steps. Although the algorithm itself is widely known, not so well known is the series of steps that first led to it, these are quite different from any of the generally known forms of the algorithm. This paper describes these steps, which start by discretizing Schrodingers equation. This paper also provides a self-contained introduction to the quantum search algorithm from a new perspective.
A quantum algorithm that solves the time-dependent Dirac equation on a digital quantum computer is developed and analyzed. The time evolution is performed by an operator splitting decomposition technique that allows for a mapping of the Dirac operator to a quantum walk supplemented by unitary rotation steps in spinor space. Every step of the splitting method is decomposed into sets of quantum gates. It is demonstrated that the algorithm has an exponential speedup over the implementation of the same numerical scheme on a classical computer, as long as certain conditions are satisfied. Finally, an explicit decomposition of this algorithm into elementary gates from a universal set is carried out to determine the resource requirements. It is shown that a proof-of-principle calculation may be possible with actual quantum technologies.
The Poisson equation occurs in many areas of science and engineering. Here we focus on its numerical solution for an equation in d dimensions. In particular we present a quantum algorithm and a scalable quantum circuit design which approximates the solution of the Poisson equation on a grid with error varepsilon. We assume we are given a supersposition of function evaluations of the right hand side of the Poisson equation. The algorithm produces a quantum state encoding the solution. The number of quantum operations and the number of qubits used by the circuit is almost linear in d and polylog in varepsilon^{-1}. We present quantum circuit modules together with performance guarantees which can be also used for other problems.
Solving differential equations is one of the most promising applications of quantum computing. Recently we proposed an efficient quantum algorithm for solving one-dimensional Poisson equation avoiding the need to perform quantum arithmetic or Hamiltonian simulation. In this letter, we further develop this algorithm to make it closer to the real application on the noisy intermediate-scale quantum (NISQ) devices. To this end, we first develop a new way of performing the sine transformation, and based on it the algorithm is optimized by reducing the depth of the circuit from n2 to n. Then, we analyze the effect of common noise existing in the real quantum devices on our algorithm using the IBM Qiskit toolkit. We find that the phase damping noise has little effect on our algorithm, while the bit flip noise has the greatest impact. In addition, threshold errors of the quantum gates are obtained to make the fidelity of the circuit output being greater than 90%. The results of noise analysis will provide a good guidance for the subsequent work of error mitigation and error correction for our algorithm. The noise-analysis method developed in this work can be used for other algorithms to be executed on the NISQ devices.
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