Hierarchical Multigrid Ansatz for Variational Quantum Algorithms

TL;DR

This paper introduces a multigrid-inspired ansatz for variational quantum algorithms, demonstrating improved performance over standard methods in solving eigenvalue problems and combinatorial optimization tasks.

Contribution

It proposes a novel hierarchical multigrid ansatz for VQAs, leveraging smaller problem solutions to enhance larger problem optimization.

Findings

Outperforms standard hardware-efficient ansatz in solution quality.

Effective for Laplacian eigensolver and combinatorial optimization problems.

Establishes multigrid ansatz as a promising alternative to QAOA.

Abstract

Quantum computing is an emerging topic in engineering that promises to enhance supercomputing using fundamental physics. In the near term, the best candidate algorithms for achieving this advantage are variational quantum algorithms (VQAs). We design and numerically evaluate a novel ansatz for VQAs, focusing in particular on the variational quantum eigensolver (VQE). As our ansatz is inspired by classical multigrid hierarchy methods, we call it "multigrid" ansatz. The multigrid ansatz creates a parameterized quantum circuit for a quantum problem on $n$ qubits by successively building and optimizing circuits for smaller qubit counts $j < n$, reusing optimized parameter values as initial solutions to next level hierarchy at $j+1$. We show through numerical simulation that the multigrid ansatz outperforms the standard hardware-efficient ansatz in terms of solution quality for the Laplacian eigensolver as well as for a large class of combinatorial optimization problems with specific examples for MaxCut and Maximum $k$-Satisfiability. Our studies establish the multi-grid ansatz as a viable candidate for many VQAs and in particular present a promising alternative to the QAOA approach for combinatorial optimization problems.