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

TL;DR

This work addresses efficient ground-state preparation for Hamiltonians with large spectral radii by bridging the gap between Linear Combination of Unitaries (LCU) and Quantum Signal Processing (QSP) methods under fast-forwarded Hamiltonian evolution. It introduces a novel multi-level QSP-based algorithm that leverages staged, easily implementable filter functions to progressively reduce the effective spectral radius, achieving a logarithmic dependence on the spectral radius and gap under ideal fast-forwarding. The approach matches or surpasses LCU performance in ideal FF settings, while dramatically reducing ancillary overhead by omitting the PREPARE oracle and using a compression gadget for coherence. These results enhance the practicality of ground-state preparation and open paths to efficiently compute low-energy subspaces and related matrix functions.

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 $\mathcal{O}(\log^2(\|H\| Δ^{-1}))$ 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 $\mathcal{O}(\|H\| Δ^{-1})$. 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 $\mathcal{O}(\log(\|H\| Δ^{-1}))$. Additionally, our multi-level QSP method requires only $\mathcal{O}(\log(\|H\| Δ^{-1}))$ coefficients for implementing single qubit rotations. This eliminates the need for constructing the PREPARE oracle in LCU, which prepares a state encoding $\mathcal{O}(\|H\| Δ^{-1})$ coefficients regardless of whether the Hamiltonian can be fast-forwarded.

Figures (6)

• Figure 1: Comparing the filter functions used in the standard QSP-based method and at the final level of the multi-level QSP-based method. We assume the spectral radius of the Hamiltonian is $\left\lVert H\right\rVert = 20$ and the spectral gap is $\Delta = 1$. The shaded area stands for the region containing only ground-state energy.
• Figure 2: An example illustrating methods for ground state preparation. Panel (a) visualizes the multi-level QSP-based method, while panel (b) displays the process using standard QSP-based or LCU-based methods. For simplicity, we consider a Hamiltonian with equally spaced eigen values whose spectral radius is $20$. The height of each bar stands for the magnitude of each coefficient with respect to the eigen basis before and after applying the filter function.
• Figure 3: Multi-level QSP-based quantum circuits. (a) QETU circuit which is used in implementing filter functions. (b) A multi-level circuit based on intermediate measurements and post-selection. (c) A coherent implementation of the multi-level circuit based on compression gadget. Here, $m = \lceil \log_2(L + 2) \rceil$ extra ancilla qubits are introduced to track the success of each level.
• Figure 4: Numerical demonstration of the filter function. The dashed vertical lines stand for the boundary values $\lambda = \pi / 4$ and $\pi / 2$.
• Figure 5: Building block of the QETU circuit (see \ref{['fig:qetu']} for details).

Theorems & Definitions (13)

• Definition 1: Fast-forwarded Hamiltonian evolution
• Lemma 2: Long-time Hamiltonian evolution synthesis using fast-forwarded assumption
• proof
• Theorem 3: Fourier approximation to the LCU-based filter function
• proof
• Theorem 4: LCU-based ground state preparation using fast-forwarded Hamiltonian evolution
• Theorem 5: QSP-based ground state preparation
• Theorem 6: Multi-level QSP-based ground state preparation using fast-forwarded Hamiltonian evolution
• Theorem 7: Multi-level QSP-based ground state preparation using fast-forwarded Hamiltonian evolution and amplitude amplification
• Theorem 8: LCU-based ground state preparation using $\alpha$-soft fast-forwarded Hamiltonian evolution