Variational quantum Hamiltonian engineering

Estimating the expectation value of a Hamiltonian and performing a Hamiltonian simulation are two fundamental tasks in quantum computation, with their efficiency heavily dependent on the Pauli norm of the Hamiltonian, which sums the absolute values of its Pauli coefficients. In this work, we propose a variational quantum algorithm called variational quantum Hamiltonian engineering (VQHE) to minimize the Pauli norm of a Hamiltonian, thereby reducing the overhead for expectation value estimation and Hamiltonian simulation. We first develop a theoretical framework that transforms the Pauli norm minimization problem into a vector <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:msub><a:mi>l</a:mi><a:mn>1</a:mn></a:msub></a:math>-norm minimization problem. We then design an appropriate cost function and employ parametrized quantum circuits to variationally minimize this cost function. Numerical experiments demonstrate the effectiveness of VQHE in reducing the Pauli norm for both the Ising Hamiltonian and molecular Hamiltonians. Furthermore, we show that VQHE is compatible with grouping strategies, enabling further reductions in measurement complexity for expectation value estimation. Our results highlight the potential of VQHE to enhance the efficiency of quantum algorithms in near-term quantum devices, offering a promising approach for practical quantum computing applications.