Multi-level quantum signal processing with applications to ground state preparation using fast-forwarded Hamiltonian evolution

TL;DR

This paper introduces a multi-level quantum signal processing algorithm that efficiently prepares ground states of Hamiltonians by leveraging fast-forwarded Hamiltonian evolution, surpassing traditional methods in query complexity.

Contribution

Develops a novel multi-level QSP algorithm that exploits fast-forwarding to reduce the cost of ground state preparation, eliminating the need for PREPARE oracle construction.

Findings

Achieves a query complexity of O(log(||H||/Δ)) with fast-forwarding.

Matches LCU approach efficiency when ideal fast-forwarding is available.

Reduces the implementation cost of single qubit rotations.

Abstract

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 $\mathcal{O}(\log^2(\|H\| \Delta^{-1}))$ queries to controlled Hamiltonian evolution. Here $\|H\|$ is the spectral radius of $H$ and $\Delta$ the spectral gap. However, traditional Quantum Signal Processing (QSP)-based methods fail to capitalize on this efficient protocol, and its cost scales as $\mathcal{O}(\|H\| \Delta^{-1})$. 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 $\mathcal{O}(\log(\|H\| \Delta^{-1}))$. Additionally, our multi-level QSP method requires only $\mathcal{O}(\log(\|H\| \Delta^{-1}))$ coefficients for implementing single qubit rotations. This eliminates the need for constructing the PREPARE oracle in LCU, which prepares a state encoding $\mathcal{O}(\|H\| \Delta^{-1})$ coefficients regardless of whether the Hamiltonian can be fast-forwarded.