Enhanced Krylov Methods for Molecular Hamiltonians: Reduced Memory Cost and Complexity Scaling via Tensor...

Area of Science:

• Quantum Chemistry
• Computational Physics
• Materials Science

Background:

• Accurate simulation of molecular Hamiltonians is crucial for understanding chemical reactions and material properties.
• Matrix-product states (MPS) offer a powerful framework for representing quantum states, but their application can be computationally demanding.
• Existing methods for applying Hamiltonians to MPS often face challenges with memory and computational scaling.

Purpose of the Study:

• To introduce a novel, memory-efficient, and low-scaling algorithm for applying ab initio molecular Hamiltonians to MPS.
• To leverage the tensor-hypercontraction (THC) format for computational gains.
• To enhance the performance of Krylov subspace methods for quantum simulations.

Main Methods:

• Developed an algorithm representing the molecular Hamiltonian as a sum of products of four MPOs (matrix-product operators), each with a bond dimension of 2.
• Applied the MPOs iteratively to MPS, followed by summation and recompression.
• Integrated this approach with Krylov subspace methods for finding eigenstates and simulating time evolution.

Main Results:

• Achieved memory cost equivalent to the bare MPS.
• Demonstrated reduced computational cost scaling compared to conventional MPO constructions.
• Validated theoretical findings with numerical experiments, showcasing significant advantages.
• Confirmed high parallelizability for large-scale HPC simulations.

Conclusions:

• The proposed algorithm offers a significant improvement in efficiency and scalability for quantum chemistry simulations.
• This method enables accurate simulations of quantum time evolution and finding low-lying eigenstates.
• The approach is well-suited for tackling complex problems on modern high-performance computing architectures.