Hierarchical Multigrid Ansatz for Variational Quantum Algorithms

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.

11 pages, 9 figures

References in corpus (19)

• A variational eigenvalue solver on a quantum processor
• Variational Quantum Algorithms
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• A Quantum Approximate Optimization Algorithm
• Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
• The Variational Quantum Eigensolver: a review of methods and best practices
• An adaptive variational algorithm for exact molecular simulations on a quantum computer
• Quantum Computation by Adiabatic Evolution
• Variational quantum algorithms for nonlinear problems
• Obstacles to State Preparation and Variational Optimization from Symmetry Protection
• Warm-starting quantum optimization
• Variational Quantum Linear Solver
• Quantum Algorithm Implementations for Beginners
• A Quantum Approximate Optimization Algorithm Applied to a Bounded Occurrence Constraint Problem
• Layer VQE: A Variational Approach for Combinatorial Optimization on Noisy Quantum Computers
• On the practical usefulness of the Hardware Efficient Ansatz
• Basic Elements for Simulations of Standard Model Physics with Quantum Annealers: Multigrid and Clock States
• Numerical Evidence for Exponential Speed-up of QAOA over Unstructured Search for Approximate Constrained Optimization
• The Quantum Alternating Operator Ansatz for Satisfiability Problems

Cited by in corpus (1)

• MLQAOA: Graph Learning Accelerated Hybrid Quantum-Classical Multilevel QAOA