Quantum optimization algorithm for solving elliptic boundary value problems on D-Wave quantum annealing device

Abstract

The quantum annealing devices, which encode the solution to a computational problem in the ground state of a quantum Hamiltonian, are implemented in D-Wave systems with more than 2,000 qubits. However, quantum annealing can solve only a classical combinatorial optimization problem such as an Ising model, or equivalently, a quadratic unconstrained binary optimization (QUBO) problem. In this paper, we formulate the QUBO model to solve elliptic problems with Dirichlet and Neumann boundary conditions using the finite element method. In this formulation, we develop the objective function of quadratic binary variables represented by qubits and the system finds the binary string combination minimizing the objective function globally. Based on the QUBO formulation, we introduce an iterative algorithm to solve the elliptic problems. We discuss the validation of the modeling on the D-Wave quantum annealing system.

Practical quantum approximate optimization algorithms on IBM quantum processors

“…6 Additionally, quantum noise-resilient algorithms, such as Quantum Amplitude Estimation (QAE) [7][8][9][10][11][12][13] and Grover's search, [14][15][16][17] were developed to run the traditional quantum algorithms on noisy quantum computers. On the other hand, Variational Quantum Algorithms (VQAs), such as Quantum Approximate Optimization Algorithm (QAOA), 18,19 Variation Quantum Eigensolver (VQE), 20 and Quantum Machine Learning, [21][22][23][24][25] have drawn attention as a promising candidate to achieve quantum advantage on NISQ devices. These variational quantum algorithms work in conjunction with classical computers.…”

Quantum multi-programming for maximum likelihood amplitude estimation

“…In this vein, various quantum error mitigation methods 2 are developed and applied [3][4][5] to Hamiltonian simulations as well as pulse level optimization. 6 Additionally, Variational Quantum Algorithms (VQAs), such as Quantum Approximate Optimization Algorithm (QAOA), 7,8 Variation Quantum Eigensolver (VQE), 9 and Quantum Neural Networks, [10][11][12][13] have drawn attention as a promising candidate to achieve quantum advantage on NISQ devices. A different approach for NISQ devices is improving the traditional quantum algorithms such as Quantum Amplitude Estimation (QAE) [14][15][16][17][18][19][20] and Grover's search [21][22][23][24] to work on NISQ devices efficiently.…”

Computing a Sparse Approximate Inverse on Quantum Annealing Machines

“…al. showed how QUBO problems can be simplified using matrix congruence [18], while its application in solving 1D Poisson problems was demonstrated in [19]. If the matrix is positive-definite, which is often the case in engineering problems, it is much more efficient to use a potential-energy formulation, rather than the least-squares formulation, to pose QUBO problems.…”