H\"uckel Molecular Orbital Theory on a Quantum Computer: A Scalable System-Agnostic Variational Implementation with Compact Encoding

Abstract

H\"uckel molecular orbital (HMO) theory provides a semi-empirical treatment of the electronic structure in conjugated {\pi}-electronic systems. A scalable system-agnostic execution of HMO theory on a quantum computer is reported here based on a variational quantum deflation (VQD) algorithm for excited state quantum simulation. A compact encoding scheme is proposed here that provides an exponential advantage over direct mapping and allows quantum simulation of the HMO model for systems with up to 2^N conjugated centers in N qubits. The transformation of the H\"uckel Hamiltonian to qubit space is achieved by two different strategies: a machine-learning-assisted transformation and the Frobenius-inner-product-based transformation. These methods are tested on a series of linear, cyclic, and hetero-nuclear conjugated {\pi}-electronic systems. The molecular orbital energy levels and wavefunctions from the quantum simulation are in excellent agreement with the exact classical results. The higher excited states of large systems, however, are found to suffer from error accumulation in the VQD simulation. This is mitigated by formulating a variant of VQD that exploits the symmetry of the Hamiltonian. This strategy has been successfully demonstrated for the quantum simulation of C_{60} fullerene containing 680 Pauli strings encoded on six qubits. The methods developed in this work are system-agnostic and hence are easily adaptable to similar problems of different complexity in other fields of research.