Efficient and practical Hamiltonian simulation from time-dependent product formulas

Area of Science:

• Quantum Computing
• Quantum Simulation
• Computational Physics

Background:

• Simulating quantum systems is crucial for understanding complex phenomena.
• Existing methods like Trotter formulas face scaling challenges with system size and evolution time.
• Hamiltonians with disparate energy scales pose particular difficulties for accurate quantum simulation.

Purpose of the Study:

• To develop novel quantum algorithms for simulating the time-evolution of quantum systems.
• To improve the efficiency and scalability of quantum simulations, particularly for Hamiltonians with mixed energy scales.
• To provide practical quantum algorithms that outperform standard approaches in specific regimes.

Main Methods:

• Utilizing product formulas for the decomposition of quantum evolution operators.
• Designing quantum algorithms with provably better gate complexity and circuit depth compared to naive Trotter methods.
• Conducting extensive numerical simulations to validate algorithm performance across various models.

Main Results:

• The proposed quantum algorithms demonstrate superior scaling for systems with Hamiltonians featuring distinct large and small energy components.
• Numerical simulations confirm practical performance competitive with, and in some cases exceeding, state-of-the-art methods.
• For the 1D transverse-field Ising model, a one-order-of-magnitude improvement in simulated system size and evolution time was observed using a fixed gate budget.

Conclusions:

• The developed product formula approach offers a practical and efficient method for quantum system time-evolution.
• These algorithms provide a significant advantage for simulating systems with Hamiltonians characterized by different energy scales.
• The findings suggest a promising direction for advancing quantum simulation capabilities on current and future quantum hardware.