  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.
