Construction of Superposition States of Energy Eigenstates via Classically Emulated Digital Quantum Simulation: The Hydrogen Molecule as an Example

TL;DR

Addresses the challenge of preparing energy eigenstates and superpositions for molecular Hamiltonians using classically emulated digital quantum simulation. Introduces twirling operations to extract eigenstates and a controlled unitary U10 to build superpositions such as |E0> ± |E1> and |E0> ± i|E1>, validated by measured observable matrix elements. Demonstrates the method on a simple two-qubit hydrogen molecule Hamiltonian and a single-qubit model, using both algebraic and simulation-based versions of U10 and verifying results with consistency checks. Concludes that the method generalizes to arbitrary Pauli-expressible Hamiltonians and offers a practical route to energy-superposition state preparation, with open-source code provided.

Abstract

We construct superposition states of energy eigenstates of the hydrogen molecule using classically emulated digital quantum simulation. We generate the ground state and excited states of the system via the twirling operation method, and construct superposition states of the ground state and an excited state of the system by applying a controlled excitation unitary transformation on the ground state with one ancillary qubit as the control. To verify the correctness of the resulting superposition state, we calculate matrix elements of several physical observables.