  The Hamiltonian of a quantum system is represented in terms of operators
corresponding to the kinetic and potential energies of the system. The
expectation value of a Hamiltonian and Hamiltonian simulation are two of the
most fundamental tasks in quantum computation. The overheads for realizing the
two tasks are determined by the Pauli norm of Hamiltonian, which sums over all
the absolute values of Pauli coefficients. In this work, we propose a
variational quantum algorithm (VQA) called variational quantum Hamiltonian
engineering (VQHE) to minimize the Pauli norm of Hamiltonian, such that the
overhead for executing expectation value estimation and Hamiltonian simulation
can be reduced. First, we develop a theory to encode the Pauli norm
optimization problem into the vector L1-norm minimization problem. Then we
devise an appropriate cost function and utilize the parameterized quantum
circuits (PQC) to minimize the cost function. We also conduct numerical
experiments to reduce the Pauli norm of the Ising Hamiltonian and molecules'
Hamiltonian to show the efficiency of the proposed VQHE.
