Entanglement-Dependent Error Bounds for Hamiltonian Simulation
Prateek P. Kulkarni
TL;DR
This work establishes that entanglement entropy governs the efficiency of Hamiltonian simulation with product formulas. By combining Lieb-Robinson locality, a novel commutator-entropy bound, and tensor-network reasoning, the authors derive explicit entanglement-dependent error bounds: for first-order Trotterization the error scales as $\mathcal{O}\big( t^2 S_{\max} \operatorname{polylog}(n)/r \big)$ (with refinements involving $S^*$ for growth), and higher-order Suzuki formulas gain a factor $2^{pS^*/2}$. They show a fundamental separation: volume-law systems require $\tilde\Omega(n)$ more Trotter steps than area-law systems to achieve the same precision, with the separation tight up to polylog factors. The results yield concrete improvements for 1D and 2D lattices (e.g., $\tilde\Omega(n^2)$ and $\tilde\Omega(n^{3/2})$ savings, respectively) and offer practical guidance for quantum chemistry and condensed-matter simulations. Numerically, the theory is validated on 1D models, confirming the state-dependence of errors and the dramatic advantage of area-law dynamics. Overall, the paper elevates entanglement structure to a central metric for resource estimation in quantum simulation, with immediate implications for near-term devices and fault-tolerant planning.
Abstract
We establish tight connections between entanglement entropy and the approximation error in Trotter-Suzuki product formulas for Hamiltonian simulation. Product formulas remain the workhorse of quantum simulation on near-term devices, yet standard error analyses yield worst-case bounds that can vastly overestimate the resources required for structured problems. For systems governed by geometrically local Hamiltonians with maximum entanglement entropy $S_\text{max}$ across all bipartitions, we prove that the first-order Trotter error scales as $\mathcal{O}(t^2 S_\text{max} \operatorname{polylog}(n)/r)$ rather than the worst-case $\mathcal{O}(t^2 n/r)$, where $n$ is the system size and $r$ is the number of Trotter steps. This yields improvements of $\tildeΩ(n^2)$ for one-dimensional area-law systems and $\tildeΩ(n^{3/2})$ for two-dimensional systems. We extend these bounds to higher-order Suzuki formulas, where the improvement factor involves $2^{pS^*/2}$ for the $p$-th order formula. We further establish a separation result demonstrating that volume-law entangled systems fundamentally require $\tildeΩ(n)$ more Trotter steps than area-law systems to achieve the same precision. This separation is tight up to logarithmic factors. Our analysis combines Lieb-Robinson bounds for locality, tensor network representations for entanglement structure, and novel commutator-entropy inequalities that bound the expectation value of nested commutators by the Schmidt rank of the state. These results have immediate applications to quantum chemistry, condensed matter simulation, and resource estimation for fault-tolerant quantum computing.