Variational Quantum Eigensolvers for Sparse Hamiltonians.

Area of Science:

• Quantum Computing
• Computational Chemistry
• Quantum Algorithms

Background:

• Hybrid quantum-classical variational algorithms like VQE and QAOA are key for noisy, intermediate-scale quantum computers.
• Current VQE and QAOA methods are restricted to Pauli representations of Hamiltonians.
• Many systems have sparse Hamiltonians without efficient Pauli representations, limiting current algorithms.

Purpose of the Study:

• To extend the variational quantum eigensolver (VQE) to accommodate general sparse Hamiltonians.
• To enable VQE's application to systems where sparse matrix representations are more efficient than Pauli representations.

Main Methods:

• Developed a decomposition method for fermionic second-quantized Hamiltonians into sparse, self-inverse, Hermitian terms.
• Showed a general d-sparse Hamiltonian can be decomposed into O(d^2) such terms.
• Demonstrated that each term can be sampled using two ansatz state preparations and at most six oracle queries.

Main Results:

• Successfully extended VQE to handle general sparse Hamiltonians.
• The number of samples required for precision scales as ε−2, consistent with Pauli-based VQE.
• The proposed decomposition is efficient for both fermionic and general sparse Hamiltonians.

Conclusions:

• This work significantly broadens the applicability of VQE to a wider range of quantum systems.
• Enables VQE for problems where Hamiltonians are naturally described by sparse matrices.
• Paves the way for more efficient quantum computations in chemistry and optimization.