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arXiv:2106.08682 · Quantum Information Processing

A comparison of various classical optimizers for a variational quantum linear solver

Aidan Pellow-Jarman, Ilya Sinayskiy, Anban W. Pillay, Francesco Petruccione

Published Jun 1, 2021• 105 citations
arXivHTMLPDF
DOI
10.1007/s11128-021-03140-x

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This paper presents a quantum-enhanced optimization approach for solving optimal power flow (OPF) by integrating the interior point method (IPM) with a coherent variational quantum linear solver (CVQLS). The objective is to explore the applicability of quantum computing to power systems optimization and address the associated challenges. A comparative analysis of state-of-the-art quantum linear solvers—Harrow-Hassidim-Lloyd (HHL), variational quantum linear solver (VQLS), and CVQLS—revealed that CVQLS is most suitable for OPF due to its stability with ill-conditioned matrices, such as the Hessian in IPM. To ensure high-quality solutions, prevent suboptimal convergence, and avoid the barren plateau problem, we propose a quantum circuit parameter initialization technique along with a method to guide the IPM along the central path. Moreover, we design an ansatz tailored for OPF, optimizing the expressibility and trainability of the quantum circuit to ensure efficient convergence and robustness in solving quantum OPF. Various optimizers are also tested for quantum circuit parameter optimization to select the best one. We evaluate our approaches on multiple systems to show their effectiveness in providing reliable OPF solutions. Simulations for the 2-bus system are conducted on a commercial IBMQ quantum device, while simulations for the other larger cases are performed using the IBM quantum simulator. While promising, CVQLS is limited by current quantum hardware, especially for larger systems. We use a quantum noise simulator to test scalability.
We present a variational quantum linear solver (VQLS) for the Poisson equation built on an exact block encoding of the discrete Laplacian, and demonstrate its performance on physically motivated benchmarks. Unlike LCU-based VQLS where the number of distinct circuits required per cost-function evaluation is O(L2)\mathcal{O}(L^2)O(L2), where LLL is the number of terms in the LCU decomposition of the discrete Laplacian operator, this approach requires only a single circuit for cost evaluation. We further empirically demonstrate that the choice of classical optimizer materially affects where the variational optimization ceases to make progress. The solver is benchmarked on three problems: a Poisson equation with sinusoidal forcing and a steady-state heat conduction problem with a localized Gaussian source, both with Dirichlet boundaries, and the pressure-Poisson equation of a two-dimensional lid-driven cavity flow, in which the solver is invoked once per time step under Neumann boundary conditions.
Voltage source converters (VSCs) play a critical role in modern power systems, where their simulation requires solving linear system problems and remains computationally expensive. Quantum linear system solvers, particularly the variational quantum linear solver (VQLS) tailored for noisy intermediate-scale quantum devices, offer a competitive paradigm with the potential for developing linear solvers with logarithmically scaled computational complexity. We propose Acc-VQLS, a domain-specific quantum algorithm and circuit co-optimization method that accelerates traditional VQLS for VSC simulation. We adopt constant vector-based variational quantum circuits for fast convergence with efficient initial state preparation and employ multi-core processing to parallelize the cost function evaluation and gradient computation. To demonstrate the potential applicability of Acc-VQLS in VSC operating scenarios, we employ the commonly used fault behavior model for VQLS-based VSC simulation. Quantitative experimental evaluations and comparisons across the tested benchmarks show that the proposed Acc-VQLS reduces iterations required for cost function convergence by an average of 82.5%, and our multi-core parallel computing strategy achieves a 53.9 × speedup on the AMD EPYC 9554 CPU. By incorporating a short-circuit fault in the VSC simulation, the proposed Acc-VQLS with error compensation achieves less than 10 − 9 in both the mean absolute error and the root mean square error.