量子系统中的局部极小值
Local minima in quantum systems
AI总结：
本研究针对量子系统局部极小值求解问题，证明经典计算机求解该问题存在计算难度，而量子计算机可通过模拟自然冷却的热梯度下降算法高效求解，揭示了量子计算在该问题上的优势。
摘要：
众所周知，寻找量子多体系统的基态对经典计算机和量子计算机而言都是难题。因此，当自然界在低温热浴中冷却量子系统时，往往无法高效地找到基态，反而会得到一个能量的局部极小值。本研究探讨了热扰动下量子系统的局部极小值求解问题。尽管局部极小值远比基态容易找到，但我们证明，对经典计算机而言，求解局部极小值在计算上是困难的，即便任务仅为输出任意一个局部极小值处的单比特可观测量也是如此。与之相反，我们证明量子计算机总能通过一种模拟自然界冷却过程的热梯度下降算法，高效地找到局部极小值。为证明求解局部极小值的经典难度，我们考虑了一族二维哈密顿量（Hamiltonians），任何可由多项式时间量子算法求解的问题都可归约为求解这些哈密顿量的基态问题。我们证明，对于这类哈密顿量，所有局部极小值都是全局极小值。因此，在量子计算能力强于经典计算的假设下，求解局部极小值对经典计算而言是困难的，而对量子计算而言则是容易的。
英文摘要：
Finding ground states of quantum many-body systems is known to be hard for both classical and quantum computers. As a result, when Nature cools a quantum system in a low-temperature thermal bath, the ground state cannot always be found efficiently. Instead, Nature finds a local minimum of the energy. In this work, we study the problem of finding local minima in quantum systems under thermal perturbations. While local minima are much easier to find than ground states, we show that finding a local minimum is computationally hard for classical computers, even when the task is to output a single-qubit observable at any local minimum. In contrast, we prove that a quantum computer can always find a local minimum efficiently using a thermal gradient descent algorithm that mimics the cooling process in Nature. To establish the classical hardness of finding local minima, we consider a family of two-dimensional Hamiltonians such that any problem solvable by polynomial-time quantum algorithms can be reduced to finding ground states of these Hamiltonians. We prove that for such Hamiltonians, all local minima are global minima. Therefore, assuming quantum computation is more powerful than classical computation, finding local minima is classically hard and quantumly easy.
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