Practical Verification of Quantum Properties in Quantum Approximate Optimization Runs

In order to assess whether quantum resources can provide an advantage over classical computation, it is necessary to characterize and benchmark the non-classical properties of quantum algorithms in a practical manner. This paper presents a method using measurements in no more than 3 out of the possible 3N bases to reconstruct single-qubit reduced density matrices and measure coherent superpositions, with potential verification of entanglement across all N qubits involved. We introduce a family of generalized Bell-type observables, establishing an upper bound for expectation values in fully separable states through a generalization of the Cauchy-Schwarz inequality, which bears independent interest. We demonstrate that a subset of these observables can act as entanglement witnesses for QAOA-MaxCut states, providing an entanglement potency metric. Furthermore, a subset of these observables also certifies, albeit weakly, the entanglement in GHZ states. The construction of such witnesses is directly linked to the cost Hamiltonian rather than the standard technique involving the projector of the state to be certified, offering insights for developing similar constructs for other variational algorithms in the NISQ era. Proof-of-concept experiments on the Rigetti Aspen-9 chip validate our approach with ansätze containing up to 24 qubits.