Researchers have developed two quantum algorithms that establish the optimal query complexity for preparing ground states of a Hamiltonian. These algorithms are capable of preparing a quantum state within a trace distance ε of the ground state, assuming an energy threshold in the spectral gap is known. This gap, Δ, is defined as the minimum energy separating the ground state from excited states, and state preparation relies on an oracle that generates a state with an overlap γ with the ground state.

The proposed algorithms use a number of calls to a Hamiltonian oracle U_H that scales as O((α/Δ)(γ⁻¹+log(1/ε))) in expectation and O((α/(γΔ))log(1/ε)) in the worst case, where α is a measure of the Hamiltonian encoding complexity. These bounds have been shown to be optimal, matching theoretical lower limits. For calls to the state preparation oracle U_I, the bounds are O(1/γ) in expectation and O(γ⁻¹log(1/ε)) in the worst case, also being optimal in (N+1)-dimensional systems under certain conditions.

The core method of the algorithms involves using a constant-accuracy spectral filter to construct a purifier. This purifier is sequentially composed during an amplitude amplification process, allowing the preparation of a state with constant overlap with the ground state. The expected-query algorithm repeats the preparation followed by a high-accuracy spectral filter until success, while the worst-case algorithm employs filters of increasing accuracy, limiting the total number of queries.