Hardware-efficient entangled measurements for variational quantum algorithms

Variational algorithms have received significant attention in recent years due to their potential to solve practical problems in noisy intermediate-scale quantum (NISQ) devices. A fundamental step of these algorithms is the evaluation of the expected value of Hamiltonians, and hence, efficient schemes to perform this task are required. The standard approach employs local measurements of Pauli operators and requires a large number of circuits. An alternative is to make use of entangled measurements, which significantly reduces the number of circuits but involves entangling gates between non-physically connected qubits, introducing intermediate entangling operations that increase the depth of the circuits. As a solution to this problem we propose hardware-efficient entangled measurements (HEEM), that is, measurements that only permit entanglement between physically connected qubits. We show that this strategy enhances the evaluation of molecular Hamiltonians in NISQ devices, reducing the number of circuits required without increasing their depth. We provide quantitative metrics of how this approach offers better results than only local measurements and arbitrarily entangled ones. We estimate with classical simulators and quantum hardware the ground state energy of the H2O molecule by the variational quantum eigensolver using HEEM