Quantum simulation for time-dependent Hamiltonians -- with applications to non-autonomous ordinary and partial differential equations
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| Format: | Preprint |
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2023
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| _version_ | 1866909585125670912 |
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| author | Cao, Yu Jin, Shi Liu, Nana |
| author_facet | Cao, Yu Jin, Shi Liu, Nana |
| contents | Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case it corresponds to having a time-dependent Hamiltonian. However, the quantum simulation of these systems often needs to appeal to rather complicated procedures involving the Dyson series, considerations of time-ordering, requirement of time steps to be discrete and/or requiring multiple measurements and postselection. These procedures are generally much more complicated than the quantum simulation of time-independent Hamiltonians. Here we propose an alternative formalism that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e., quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrodingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2312_02817 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum simulation for time-dependent Hamiltonians -- with applications to non-autonomous ordinary and partial differential equations Cao, Yu Jin, Shi Liu, Nana Quantum Physics Mathematical Physics Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case it corresponds to having a time-dependent Hamiltonian. However, the quantum simulation of these systems often needs to appeal to rather complicated procedures involving the Dyson series, considerations of time-ordering, requirement of time steps to be discrete and/or requiring multiple measurements and postselection. These procedures are generally much more complicated than the quantum simulation of time-independent Hamiltonians. Here we propose an alternative formalism that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e., quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrodingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients. |
| title | Quantum simulation for time-dependent Hamiltonians -- with applications to non-autonomous ordinary and partial differential equations |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2312.02817 |