Correlation-Informed Permutation of Qubits for Reducing Ansatz Depth in the Variational Quantum Eigensolver

The variational quantum eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers. However, the circuit depth is expected to grow significantly with the problem size. Increased depth can both degrade the accuracy of the results and reduce trainability. In this work, we propose an approach to reduce ansatz circuit depth. Our approach, called "PermVQE," adds an additional optimization loop to the VQE that permutes qubits in order to solve for the qubit Hamiltonian that maximally localizes correlations in the ground state. The choice of permutations is based on mutual information, which is a measure of interaction between electrons and/or holes in spin-orbitals. Encoding strongly entangled spin-orbitals into proximal qubits on a quantum chip naturally reduces the circuit depth needed to prepare the ground state. For representative molecular systems, LiH, H-2, (H-2)(2), H-4(not equal), H-3(+), and N-2, we demonstrate that placing entangled qubits in close proximity leads to shallower depth circuits required to reach a given eigenvalue-eigenvector accuracy. The approach is designed for hardware-efficient ansatz of any qubit connectivity, and examples are demonstrated for linear and two-dimensional grid architectures. The main ideas can also be applied to simulate molecules with other ansatz as well as variational quantum algorithms beyond the VQE. In particular, we demonstrate the beneficial effect of qubit permutations to build fermionic-adaptive derivative assembled pseudo-Trotter ansatz on a linear qubit connectivity architecture with nearly a twofold reduction of the number of controlled NOT gates.