Hierarchical Multigrid Ansatz for Variational Quantum Algorithms

TL;DR

This work introduces a hierarchical Multigrid Ansatz for variational quantum algorithms, specifically implementing a Multigrid VQE that solves progressively larger instances by reusing optimized parameters from coarser levels. By seeding refinements with zero-initialized parameters to interpolate states between levels, the approach achieves improved solution quality for the discrete Dirichlet Laplacian and for NP-hard combinatorial problems such as MaxCut and MaxE_k-Sat, compared to standard hardware-efficient VQAs. The numerical results demonstrate faster convergence, reduced variance in optimizer calls, and greater robustness across shot counts, suggesting Multigrid VQE as a promising alternative to QAOA in certain optimization settings. The method leverages a polylogarithmic refinement depth and quadratic two-qubit gate costs, highlighting practical scalability on NISQ devices and applicability to problems with hierarchical structure.

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.

Figures (9)

• Figure 1: A process diagram for the Multigrid VQE. When one level of the Hamiltonian is optimized, refine the problem and use the previous level's angles to optimize. This process proceeds until refinement produces the original Hamiltonian.
• Figure 2: A refinement layer of a Multigrid Hierarchy of Variational Ansätze. The circuit $U_{n-1}(\theta_1,\theta_2,\ldots,\theta_m)$ is the previous refinement layer with angles from the previous optimization round. The solution to the previous round's coarser problem on four qubits is extended to a refined problem on five qubits by putting a new qubit in the $\ket{+}$ state with an H gate and entangling it to all previous qubits with $\texttt{CZ}$ (connected black dots) and $\texttt{RY}$ gates. In our implementation, the new angles $\theta_{m+1},\theta_{m+2},\theta_{m+3},$ and $\theta_{m+4}$ are each initialized to zero in order to create a constant interpolation with respect to computational basis states.
• Figure 3: An Efficient SU(2) ansatz with two repetitions and a final rotation layer. This circuit exhibits the reverse linear entanglement pattern, which is the default in Qiskit. Ladders of reverse linear entangling $CX$ gates separate blocks of parameterized $Y$ and $Z$ rotations to form a simple yet robust ansatz for heuristic VQA applications.
• Figure 4: The smallest eigenvectors of $\nabla^2_D$ (for dimensions $m=2^2, 2^3, 2^4, 2^5$, respectively) form a refinement structure well-suited to multigrid methods. The $X$ axes represent computational basis states expressed as decimal integers, and the $Y$ axes show the amplitudes of the eigenvector in the respective entries. Ignoring a complex global phase, we assume entries in $\mathbb{R}_+$.
• Figure 5: An increment circuit for the shift operation $P$ of Sato et al. sato2021variational. This circuit sends big-endian basis states $\ket{b}$ to their cyclically shifted counterparts $\ket{\texttt{rem}(b+1,2^n)}$, and can be decomposed into linearly many CNOTs gidney2015largegates.