Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916118951624704 |
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| author | Zhuk, Sergiy Robertson, Niall Bravyi, Sergey |
| author_facet | Zhuk, Sergiy Robertson, Niall Bravyi, Sergey |
| contents | Multi-product formulas (MPF) are linear combinations of Trotter circuits offering high-quality simulation of Hamiltonian time evolution with fewer Trotter steps. Here we report two contributions aimed at making multi-product formulas more viable for near-term quantum simulations. First, we extend the theory of Trotter error with commutator scaling developed by Childs, Su, Tran et al. to multi-product formulas. Our result implies that multi-product formulas can achieve a quadratic reduction of Trotter error in 1-norm (nuclear norm) on arbitrary time intervals compared with the regular product formulas without increasing the required circuit depth or qubit connectivity. The number of circuit repetitions grows only by a constant factor. Second, we introduce dynamic multi-product formulas with time-dependent coefficients chosen to minimize a certain efficiently computable proxy for the Trotter error. We use a minimax estimation method to make dynamic multi-product formulas robust to uncertainty from algorithmic errors, sampling and hardware noise. We call this method Minimax MPF and we provide a rigorous bound on its error. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12569 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation Zhuk, Sergiy Robertson, Niall Bravyi, Sergey Quantum Physics Numerical Analysis Mathematical Physics Optimization and Control Multi-product formulas (MPF) are linear combinations of Trotter circuits offering high-quality simulation of Hamiltonian time evolution with fewer Trotter steps. Here we report two contributions aimed at making multi-product formulas more viable for near-term quantum simulations. First, we extend the theory of Trotter error with commutator scaling developed by Childs, Su, Tran et al. to multi-product formulas. Our result implies that multi-product formulas can achieve a quadratic reduction of Trotter error in 1-norm (nuclear norm) on arbitrary time intervals compared with the regular product formulas without increasing the required circuit depth or qubit connectivity. The number of circuit repetitions grows only by a constant factor. Second, we introduce dynamic multi-product formulas with time-dependent coefficients chosen to minimize a certain efficiently computable proxy for the Trotter error. We use a minimax estimation method to make dynamic multi-product formulas robust to uncertainty from algorithmic errors, sampling and hardware noise. We call this method Minimax MPF and we provide a rigorous bound on its error. |
| title | Trotter error bounds and dynamic multi-product formulas for Hamiltonian simulation |
| topic | Quantum Physics Numerical Analysis Mathematical Physics Optimization and Control |
| url | https://arxiv.org/abs/2306.12569 |