Classically optimized Hamiltonian simulation (2205.11427v5)

Abstract: Hamiltonian simulation is a promising application for quantum computers to achieve a quantum advantage. We present classical algorithms based on tensor network methods to optimize quantum circuits for this task. We show that, compared to Trotter product formulas, the classically optimized circuits can be orders of magnitude more accurate and significantly extend the total simulation time.

Overview of Classically Optimized Hamiltonian Simulation

The presented paper explores the methodologies aimed at reducing quantum hardware demands for Hamiltonian simulation, a prominent application envisioned to leverage quantum computing in providing advantages over classical approaches. The authors propose several classical algorithms grounded in tensor network strategies to optimize quantum circuits tasked with simulating Hamiltonian time evolution. By leveraging these classical optimizations, the organized circuits exhibit marked improvements in accuracy compared to conventional Trotter product formulas and effectively manage extended simulation times without necessitating deep quantum circuits.

Key Contributions

1. Classical Optimization of Quantum Circuits: The paper details classical algorithms that optimize quantum circuits using tensor network methods, allowing for shallow yet efficient circuits. This is achieved by representing quantum circuits via matrix product operators (MPOs) and applying tensor network contraction techniques for optimization.
2. Comparative Performance Analysis: The classical optimization of circuits shows a substantial improvement over Trotter methods in terms of approximation error and simulation time. Especially for circuits with depth greater than one, the improvement in accuracy is several orders of magnitude.
3. Scalability and Versatility: The optimization techniques are applicable to various Hamiltonian types and circuit structures, including both open and periodic boundary condition systems, local and nonlocal interactions, and brickwall or sequential circuit structures. The algorithms are hardware-agnostic and compatible with diverse native quantum gates.
4. Error Scaling Insights: Through numerical experiments for systems up to 60 qubits, the error of the optimized circuits scales linearly with the number of qubits, a crucial insight indicating potential for efficient computation beyond the limit of classical simulability.

Numerical Results

The paper thoroughly investigates the approximation errors for classically optimized circuits and compares them to various order Trotter product formulas. For circuits with multiple layers, the classically optimized circuits achieve lower approximation errors and better scalability in terms of computation time per error threshold. Moreover, detailed error scaling analysis presents how optimized circuits markedly outperform traditional Trotter approaches in both prefactors and exponents.

Implications and Future Directions

• Practical Enhancements: By reducing the requirement for deep quantum circuits and minimizing quantum hardware demands, these optimizations offer practical improvements for currently challenging Hamiltonian simulations in quantum chemistry and materials science.
• Algorithmic Integrations: The classical optimization methods may supplement variational quantum algorithms, offering classical pre-optimizations that reduce quantum computational costs during hybrid algorithms.
• Exploration of Extended Time Evolution: For handling longer times, extended slicing and optimization techniques can be explored, potentially even enabling advances in quantum simulation scalability.
• Adaptations for Error Mitigation: As computational fidelity improves with reduction in quantum gate errors, the classically optimized circuits are likely to show increasing advantages over traditional approaches, motivating further research in error correction and mitigation strategies.
• Expanded Algorithm Portfolio: Incorporating alternative approximations like Jacobi-Anger expansions or higher-order Taylor series may enhance the efficiency of classical optimizations, reducing the slice number necessary for high accuracy.

In summary, the paper introduces significant advancements in classical circuit optimizations for Hamiltonian simulation, offering a promising pathway towards more efficient, scalable quantum simulations with lower hardware constraints. The insights gained lay foundational stones for future theoretical and practical explorations in the evolving sphere of quantum computing applications.