On the Impossibility of General Parallel Fast-forwarding of Hamiltonian Simulation

• URL: http://arxiv.org/abs/2305.12444v1
• Date: Sun, 21 May 2023 12:30:00 GMT
• Title: On the Impossibility of General Parallel Fast-forwarding of Hamiltonian Simulation
• Authors: Nai-Hui Chia, Kai-Min Chung, Yao-Ching Hsieh, Han-Hsuan Lin, Yao-Ting Lin, Yu-Ching Shen
• Abstract summary: Hamiltonian simulation is one of the most important problems in the field of quantum computing. Existing simulation algorithms require running time at least linear in the evolution time $T$. It is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism.
• Score: 4.925967492198012
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Hamiltonian simulation is one of the most important problems in the field of quantum computing. There have been extended efforts on designing algorithms for faster simulation, and the evolution time $T$ for the simulation turns out to largely affect algorithm runtime. While there are some specific types of Hamiltonians that can be fast-forwarded, i.e., simulated within time $o(T)$, for large enough classes of Hamiltonians (e.g., all local/sparse Hamiltonians), existing simulation algorithms require running time at least linear in the evolution time $T$. On the other hand, while there exist lower bounds of $\Omega(T)$ circuit size for some large classes of Hamiltonian, these lower bounds do not rule out the possibilities of Hamiltonian simulation with large but "low-depth" circuits by running things in parallel. Therefore, it is intriguing whether we can achieve fast Hamiltonian simulation with the power of parallelism. In this work, we give a negative result for the above open problem, showing that sparse Hamiltonians and (geometrically) local Hamiltonians cannot be parallelly fast-forwarded. In the oracle model, we prove that there are time-independent sparse Hamiltonians that cannot be simulated via an oracle circuit of depth $o(T)$. In the plain model, relying on the random oracle heuristic, we show that there exist time-independent local Hamiltonians and time-dependent geometrically local Hamiltonians that cannot be simulated via an oracle circuit of depth $o(T/n^c)$, where the Hamiltonians act on $n$-qubits, and $c$ is a constant.

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