  The variational quantum eigensolver (VQE) stands as a prominent
quantum-classical hybrid algorithm for near-term quantum computers to obtain
the ground states of molecular Hamiltonians in quantum chemistry. However, due
to the non-commutativity of the Pauli operators in the Hamiltonian, the number
of measurements required on quantum computers increases significantly as the
system size grows, which may hinder practical applications of VQE. In this
work, we present a protocol termed joint Bell measurement VQE (JBM-VQE) to
reduce the number of measurements and speed up the VQE algorithm. Our method
employs joint Bell measurements, enabling the simultaneous measurement of the
absolute values of all expectation values of Pauli operators present in the
Hamiltonian. In the course of the optimization, JBM-VQE estimates the absolute
values of the expectation values of the Pauli operators for each iteration by
the joint Bell measurement, while the signs of them are measured less
frequently by the conventional method to measure the expectation values. Our
approach is based on the empirical observation that the signs do not often
change during optimization. We illustrate the speed-up of JBM-VQE compared to
conventional VQE by numerical simulations for finding the ground states of
molecular Hamiltonians of small molecules, and the speed-up of JBM-VQE at the
early stage of the optimization becomes increasingly pronounced in larger
systems. Our approach based on the joint Bell measurement is not limited to VQE
and can be utilized in various quantum algorithms whose cost functions are
expectation values of many Pauli operators.
