An efficient algorithm for approximate shadow Hamiltonian simulation
We introduce an efficient algorithm based on shadow Hamiltonian simulation for the approximate real-time simulation of observable dynamics under time-independent interacting Hamiltonians. Shadow Hamiltonian simulation operates within the operator algebra generated by observables through commutators with the Hamiltonian. However, exact quantum state encoding in this framework is typically inefficient for interacting systems due to the exponential growth of the algebra. Our algorithm overcomes this bottleneck by systematically identifying the algebraic elements most relevant to the target observables. This targeted approach is a controlled approximation that yields a highly efficient quantum state encoding that substantially reduces the required qubit register size. Consequently, our method not only provides an efficient quantum algorithm but also pushes the boundaries of classical simulation of quantum dynamics.
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