Multi-level quantum signal processing with applications to ground state preparation using fast-forwarded Hamiltonian evolution
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909216053133312 |
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| author | Dong, Yulong Lin, Lin |
| author_facet | Dong, Yulong Lin, Lin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02086 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi-level quantum signal processing with applications to ground state preparation using fast-forwarded Hamiltonian evolution Dong, Yulong Lin, Lin Quantum Physics Numerical Analysis 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. |
| title | Multi-level quantum signal processing with applications to ground state preparation using fast-forwarded Hamiltonian evolution |
| topic | Quantum Physics Numerical Analysis |
| url | https://arxiv.org/abs/2406.02086 |