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.

翻译：哈密顿量模拟是量子计算领域最重要的问题之一。为设计更快速的模拟算法，学界已做出大量努力，而模拟的演化时间$T$在很大程度上影响算法运行时间。尽管某些特定类型的哈密顿量可以实现加速（即能在$o(T)$时间内模拟），但对于足够大类的哈密顿量（例如所有局域/稀疏哈密顿量），现有模拟算法所需的运行时间至少与演化时间$T$呈线性关系。另一方面，虽然对于某些大类哈密顿量存在电路规模的$\Omega(T)$下界，但这些下界并未排除通过并行计算实现"低深度"大电路进行哈密顿量模拟的可能性。因此，能否借助并行能力实现快速哈密顿量模拟是一个令人困惑的问题。在本工作中，我们对上述开放问题给出了否定性结论，证明稀疏哈密顿量和（几何）局域哈密顿量无法实现并行加速。在预言机模型下，我们证明存在与时间无关的稀疏哈密顿量，其无法通过深度为$o(T)$的预言机电路模拟。在普通模型中，基于随机预言机启发法，我们证明存在与时间无关的局域哈密顿量和与时间相关的几何局域哈密顿量，若此类哈密顿量作用于$n$量子比特且$c$为常数，则无法通过深度为$o(T/n^c)$的预言机电路模拟。