The intensive pursuit for quantum advantage in terms of computational complexity has further led to a modernized crucial question of when and how will quantum computers outperform classical computers. The next milestone is undoubtedly the realization of quantum acceleration in practical problems. Here we provide a clear evidence and arguments that the primary target is likely to be condensed matter physics. Our primary contributions are summarized as follows: 1) Proposal of systematic error/runtime analysis on state-of-the-art classical algorithm based on tensor networks; 2) Dedicated and high-resolution analysis on quantum resource performed at the level of executable logical instructions; 3) Clarification of quantum-classical crosspoint for ground-state simulation to be within runtime of hours using only a few hundreds of thousand physical qubits for 2d Heisenberg and 2d Fermi-Hubbard models, assuming that logical qubits are encoded via the surface code with the physical error rate of p = 10⁻³. To our knowledge, we argue that condensed matter problems offer the earliest platform for demonstration of practical quantum advantage that is order-of-magnitude more feasible than ever known candidates, in terms of both qubit counts and total runtime.

对计算复杂性方面量子优势的深入追求进一步引发了一个现代化的关键问题：量子计算机何时以及如何超越经典计算机。下一个里程碑无疑是在实际问题中实现量子加速。在这里，我们提供了明确的证据和论据，表明主要目标很可能是凝聚态物理。我们的主要贡献总结如下： 1）提出基于张量网络的最先进经典算法的系统误差/运行时分析； 2）在可执行逻辑指令级别上对量子资源进行专门的高分辨率分析； 3) 对 2d 海森堡和 2d 费米-哈伯德模型仅使用数十万个物理量子位，在几个小时的运行时间内澄清基态模拟的量子经典交叉点，假设逻辑量子位通过表面代码进行编码物理错误率为p  = 10 ⁻³。据我们所知，我们认为凝聚态问题为展示实际量子优势提供了最早的平台，就量子位计数和总运行时间而言，该优势比已知的候选者更可行。