2024/06/04 by Yulong Dong, Lin Lin, Dong, Yulong +1 · 1 citation
Physics and Astronomy · Computer Science · #Spectroscopy and Quantum Chemical Studies #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography
paper · pdf · doi:10.48550/arxiv.2406.02086
The preparation of the ground state of a Hamiltonian H with a large spectral radius has applications in many areas such as electronic structure theory and quantum field theory. Given an initial state with a constant overlap with the ground state, and assuming that the Hamiltonian H can be efficiently simulated with an ideal fast-forwarding protocol, we first demonstrate that employing a linear combination of unitaries (LCU) approach can prepare the ground state at a cost of O(log2(‖H‖ Δ-1)) queries to controlled Hamiltonian evolution. Here ‖H‖ is the spectral radius of H and Δ the spectral gap. However, traditional Quantum Signal Processing (QSP)-based methods fail to capitalize on this efficient protocol, and its cost scales as O(‖H‖ Δ-1). To bridge this gap, we develop a multi-level QSP-based algorithm that exploits the fast-forwarding feature. This novel algorithm not only matches the efficiency of the LCU approach when an ideal fast-forwarding protocol is available, but also exceeds it with a reduced cost that scales as O(log(‖H‖ Δ-1)). Additionally, our multi-level QSP method requires only O(log(‖H‖ Δ-1)) coefficients for implementing single qubit rotations. This eliminates the need for constructing the PREPARE oracle in LCU, which prepares a state encoding O(‖H‖ Δ-1) coefficients regardless of whether the Hamiltonian can be fast-forwarded.