A Deterministic Decoration Scheme for Measurement-Based Variational Quantum Eigensolvers

The Variational Quantum Eigensolver (VQE) is a leading algorithm for finding ground states of quantum Hamiltonians on noisy quantum devices, but requires deep circuits that are challenging for current hardware. Measurement-Based VQE (MB-VQE) offers an alternative approach using single-qubit measurements on entangled resource states. However, measurement-based quantum computing requires deterministic computation, which is guaranteed only when the measurement pattern satisfies the generalized flow (gflow) condition. This work analyzes existing MB-VQE methods and identifies gflow violations that lead to non-deterministic computation. We focus on stabilizer Hamiltonians—quantum systems whose ground states are stabilizer states—and investigate how MB-VQE performs when small perturbations are added, making the perturbed ground state unknown and requiring variational approximation methods. To address the determinism issue, we develop a modified MB-VQE approach building on the existing edge-decoration scheme that ensures deterministic computation by preserving gflow conditions throughout the measurement process. Numerical simulations on perturbed Toric code and linear cluster Hamiltonians demonstrate that our deterministic MB-VQE approach successfully approximates the perturbed ground states while maintaining computational reliability.