Evaluating quantum state verification for d-level graph states

Abstract

Quantum State Verification (QSV) attempts to figure out if, at a certain confidence level, an unknown quantum state is sufficiently near to a fixed target state. The spectral gap of the verification operator is inversely proportional to the number of identical, independent copies, N, needed for this operation. A significant class of quantum states that often feature in quantum computing are stabilizer states. These states’ structural characteristics make it easier to express and simulate quantum states mathematically. They are specified by a collection of stabilizer operators that are derived from Pauli matrices in qubit systems. These characteristics also make stabilizer states particularly suitable for local measurement-based verification processes. Previous research by Dangniam et al. has demonstrated that the spectral gap for qubit stabilizer states cannot exceed 2/3 when the verification operator is restricted to local measurements. Building upon their methodology, we extend these results to graph states in d-level systems, where d is a prime number. Specifically, we establish that the spectral gap for these systems is upper-bounded by d/(d + 1). This problem is reduced to the case of bipartite maximally entangled d-level systems, for which the optimal spectral gap is known from the work of Li et al. Our derivation leverages the property that d-level graph states exhibit maximal entanglement across all possible bipartitions of the graph