Hamiltonian simulation for low-energy states with optimal time dependence

08/12/2024, 9:40am-10:00am

Alexander Zlokapa, Rolando Somma

We consider the task of simulating time evolution under a Hamiltonian H within its low-energy subspace. Assuming access to a block-encoding of H′=(H−E)/λ for some E∈R, the goal is to implement an ϵ-approximation to e−itH when the initial state is confined to the subspace corresponding to eigenvalues [−1,−1+Δ/λ] of H′. We present a quantum algorithm that uses O(tλΓ−−−√+λ/Γ−−−√log(1/ϵ)) queries to the block-encoding for any Γ such that Δ≤Γ≤λ. When log(1/ϵ)=o(tλ) and Δ/λ=o(1), this result improves over generic methods with query complexity Ω(tλ). Our quantum algorithm leverages spectral gap amplification and the quantum singular value transform. Using standard access models for H, we show that the ability to efficiently block-encode H′ is equivalent to H being what we refer to as a “gap-amplifiable” Hamiltonian. This includes physically relevant examples such as frustration-free systems, and it encompasses all previously considered settings of low-energy simulation algorithms. We also provide lower bounds for low-energy simulation. In the worst case, we show that the low-energy condition cannot be used to improve the runtime of Hamiltonian simulation. For gap-amplifiable Hamiltonians, we prove that our algorithm is tight in the query model with respect to t, Δ, and λ. In the practically relevant regime where log(1/ϵ)=o(tΔ) and Δ/λ=o(1), we also prove a matching lower bound in gate complexity (up to log factors). To establish the query lower bounds, we consider PARITY∘OR and degree bounds on trigonometric polynomials. To establish the lower bound on gate complexity, we use a circuit-to-Hamiltonian reduction acting on a low-energy state.