A Theory of Digital Quantum Simulations in the Low-Energy Subspace
CoRR(2023)
摘要
Digital quantum simulation has broad applications in approximating unitary
evolutions of Hamiltonians. In practice, many simulation tasks for quantum
systems focus on quantum states in the low-energy subspace instead of the
entire Hilbert space. In this paper, we systematically investigate the
complexity of digital quantum simulation based on product formulas in the
low-energy subspace. We show that the simulation error depends on the effective
low-energy norm of the Hamiltonian for a variety of digital quantum simulation
algorithms and quantum systems, allowing improvements over the previous
complexities for full unitary simulations even for imperfect state
preparations. In particular, for simulating spin models in the low-energy
subspace, we prove that randomized product formulas such as qDRIFT and random
permutation require smaller step complexities. This improvement also persists
in symmetry-protected digital quantum simulations. We prove a similar
improvement in simulating the dynamics of power-law quantum interactions. We
also provide a query lower bound for general digital quantum simulations in the
low-energy subspace.
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