Physics umpulse equation maker11/26/2022 ![]() Specifically, VQAs employ shallow circuit depths to optimize a cost function, expressed in terms of an Ansatz with tunable parameters, through iterative evaluations of expectation values 5. Over the last few years, variational quantum algorithms (VQAs) have emerged as a leading strategy to realize a quantum advantage on NISQ devices. Instead, the machines we have today are imperfect, noisy intermediate-scale quantum (NISQ) devices 4 with both coherent and incoherent errors limiting practical circuit depths. Moreover, implementing HHL and many other quantum algorithms would require the use of a fault-tolerant quantum computer, which may not be available in the near future 3. Unfortunately, attractive as it may sound, the HHL algorithm works only in an idealized setting, and a list of caveats must be addressed before it can be used to realize a quantum advantage 2. Compared to classical algorithms, the HHL algorithm can be shown to exhibit an exponential speedup. A cornerstone of many of these quantum algorithms is the seminal Harrow-Hassidim-Lloyd (HHL) algorithm 1 for solving linear systems, which can be utilized to solve PDEs by discretizing the PDE and mapping it to a system of linear equations. Recently, the advent of quantum computers and the invention of new quantum algorithms have provided a novel paradigm for solving PDEs. Finding new methods to solve PDEs more efficiently-including making use of new algorithms or new types of hardware-has been an active area of research. Partial differential equations (PDEs) are fundamental to solving important problems in disciplines ranging from heat and mass transfer, fluid dynamics and electromagnetics to quantitative finance and human behavior. We present a semi-implicit scheme for solving systems of evolution equations with non-linear terms, such as the reaction–diffusion and the incompressible Navier–Stokes equations, and demonstrate its validity by proof-of-concept results. Our proposed algorithm extends economically to higher-order time-stepping schemes, such as the Crank–Nicolson method. Through statevector simulations of the heat equation, we demonstrate how the time complexity of our algorithm scales with the Ansatz volume for gradient estimation and how the time-to-solution scales with the diffusion parameter. The use of encoded source states informed by preceding solution vectors results in faster convergence compared to random re-initialization. Here, we propose a variational quantum algorithm for solving a general evolution equation through implicit time-stepping of the Laplacian operator. Variational quantum algorithms offer a promising new paradigm for solving partial differential equations on near-term quantum computers. ![]()
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