Trotterization is substantially efficient for low-energy states

Abstract: Trotterization is one of the central approaches for simulating quantum many-body dynamics on quantum computers or tensor networks. In addition to its simple implementation, recent studies have revealed that its error and cost can be reduced if the initial state is closed in the low-energy subspace. However, the improvement by the low-energy property rapidly vanishes as the Trotter order grows in the previous studies, and thus, it is mysterious whether there exists genuine advantage of low-energy initial states. In this Letter, we resolve this problem by proving the optimal error bound and cost of Trotterization for low-energy initial states. For generic local Hamiltonians composed of positive-semidefinite terms, we show that the Trotter error is at most linear in the initial state energy $\Delta$ and polylogarithmic in the system size $N$. As a result, the computational cost becomes substantially small for low-energy states with $\Delta \in o(Ng)$ compared to the one for arbitrary initial states, where $g$ denotes the energy per site and $Ng$ means the whole-system energy. Our error bound and cost of Trotterization achieve the theoretically-best scaling in the initial state energy $\Delta$. In addition, they can be partially extended to weakly-correlated initial states having low-energy expectation values, which are not necessarily closed in the low-energy subspace. Our results will pave the way for fast and accurate simulation of low-energy states, which are one central targets in condensed matter physics and quantum chemistry.

Trotterization's Efficiency for Low-Energy Quantum State Simulation

The paper titled "Trotterization is substantially efficient for low-energy states" presents a comprehensive analysis of the Trotterization approach in the context of simulating quantum many-body dynamics. This technique, which is central to implementing Hamiltonian simulations on quantum computers, stands out due to its straightforward circuit construction and favorable computational cost scalings.

Key Findings

The study delves into whether Trotterization offers performance improvements specifically for low-energy initial states, compared to more generic initial states. The primary contribution of this work is a rigorous proof that low-energy states do enjoy a substantial efficiency boost. Specifically, when simulating local Hamiltonians composed of positive-semidefinite terms, the authors demonstrate that the Trotter error depends linearly on the energy of the initial state and polylogarithmically on the system size (N). This results in a significant reduction in computational cost—as measured by the Trotter number—when dealing with low-energy states. The paper achieves this by employing advanced bounds on nested commutators and meticulously accounting for errors in low-energy projections.

Numerical Results

The paper provides explicit calculations and numerical demonstrations using models like the Affleck-Kennedy-Lieb-Tasaki (AKLT) and Majumdar-Ghosh (MG) Hamiltonians. These models support the theoretical findings that Trotter errors grow much more slowly in system size (N) for low-energy states than for generic initial states.

Theoretical Implications

The implications of these findings are profound for quantum computing, particularly in condensed matter physics and quantum chemistry, where low-energy states are frequently the focus. Theoretically, these results present a more optimized pathway for using quantum computers to simulate systems at low energies, providing a potentially exponential computational advantage for specific classes of initial states.

Practical Considerations

Practically, this study forecasts a future where simulating low-energy states could be implemented with substantially fewer resources on both current and future quantum hardware. Importantly, the findings set a pathway for developing more nuanced quantum algorithms that consider the nature of the initial states in question.

Future Directions

Looking ahead, the exploration of Trotterization's efficiency could extend to other settings such as time-dependent systems or more complex interaction models. The insights on initial-state-dependent simulations hold potential for broad applicability, suggesting new directions for quantum algorithmic development that optimize computational resources by leveraging the physical characteristics of the system being simulated.

In conclusion, this paper provides detailed theoretical and numerical evidence supporting the use of Trotterization for low-energy states, promising substantial computational savings and advancing the understanding of quantum simulation efficiencies.