Quantum approximate optimization of finite-state bosonic systems

Area of Science:

• Quantum Computing
• Quantum Information Science
• Computational Physics

Background:

• Many natural problems are described by finite D-dimensional states.
• Mapping these qudit states to multiqubit systems for quantum hardware can lead to exponentially larger, infeasible subspaces.
• Current methods using objective function penalization become inefficient as the infeasible subspace grows.

Purpose of the Study:

• To propose a Hamiltonian-based Quantum Approximate Optimization Algorithm (QAOA) approach to exclude infeasible configuration spaces.
• To investigate different mapping techniques (binary, symmetric, unary) for qudit to multiqubit Hilbert space mapping.
• To apply the developed framework to quantum approximate thermalization and solving the Bose-Hubbard model.

Main Methods:

• Devised appropriate mixing Hamiltonians within the Hamiltonian-based QAOA framework.
• Employed binary, symmetric, and unary mapping techniques to map qudit Hilbert space to multiqubit space.
• Analyzed implementation cost using controlled-NOT gate count for different mapping strategies.
• Applied the method to quantum approximate thermalization and finding the ground state of the Bose-Hubbard model.

Main Results:

• The standard mixing Hamiltonian is optimal for symmetric mapping, minimizing controlled-NOT gate count.
• Binary and unary mapping techniques result in a p-fold increase in gate count for a p-layer QAOA.
• The proposed QAOA framework effectively rules out infeasible configuration spaces.
• Successfully found the ground state of the repulsive Bose-Hubbard model in both strong and weak interaction regimes.

Conclusions:

• The Hamiltonian-based QAOA with specifically designed mixing Hamiltonians offers an efficient method to handle infeasible subspaces in quantum computations.
• Symmetric mapping with standard mixing Hamiltonians presents the most cost-effective implementation strategy in terms of gate count.
• This approach provides a viable framework for complex quantum simulations, including thermalization and condensed matter problems.