对数空间中的全对全量子动力学的经典与量子模拟
Classical and Quantum Simulation of All-to-All Quantum Dynamics in Logarithmic Space
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中文总结 AI 辅助
该研究针对全对全自旋模型的无序平均淬火动力学，提出一种仅需对数空间和多项式时间的量子算法，并证明经典算法若匹配此资源将蕴含BQL⊆SC，从而为同时时间-空间量子优势提供候选。
中文摘要 AI 辅助
我们研究了对于全对全自旋哈密顿量（如量子Hopfield、Sherrington-Kirkpatrick（SK）或Dicke型模型）估计无序平均淬火动力学问题的经典与量子复杂度。在此设置中，无序平均恢复了置换不变性，因此$N$个自旋的状态可以存储在$\mathcal O(\log N)$个量子比特中。虽然已知对数空间量子计算可以在多项式时间或对数空间内经典模拟，但单一算法能否同时实现两者仍是开放问题，这提出了同时量子优势的可能性。为此，我们给出了一种针对无无序的置换不变$k$-局域动力学的量子算法，使用多项式时间和$\mathcal O(\log N)$的量子与经典空间。我们证明，任何匹配这些资源的经典算法都将蕴含$\mathsf{BQL}\subseteq \mathsf{SC}$。然而，对于短时间，我们提供了此类算法，包括针对SK模型的算法。对于Hopfield模型，我们讨论了如何将无序稀释到辅助量子比特上，从而将问题简化为扩大系统上的无无序情况，进而将经典和量子结果扩展到这些模型。我们给出了一些证据表明它们的长时间动力学是非平凡的。因此，它们成为量子模拟中同时时间-空间优势的候选者。
英文摘要
We study both the classical and the quantum complexity of the problem of estimating disorder-averaged quench dynamics for all-to-all spin Hamiltonians such as quantum Hopfield, Sherrington--Kirkpatrick (SK) or Dicke-type models. Disorder averaging restores permutation invariance in this setting, so states of $N$ spins can be stored in $\mathcal O(\log N)$ qubits. While logarithmic-space quantum computations are known to be simulatable classically in polynomial time or in polylogarithmic space, whether a single algorithm can achieve both is open, raising the possibility of simultaneous quantum advantage. To this end, we give a quantum algorithm for permutation-invariant $k$-local dynamics without disorder using polynomial time and $\mathcal O(\log N)$ quantum and classical space. We show that any classical algorithm matching these resources would imply $\mathsf{BQL}\subseteq \mathsf{SC}$. For short times, however, we provide such algorithms, including for SK models. For Hopfield models, we discuss how dilating the disorder to ancilla qubits reduces the problem to the disorder-free case on an enlarged system, thus extending both classical and quantum results to them. We give some evidence that their long-time dynamics are non-trivial. They thus emerge as a candidate for a simultaneous time--space advantage in quantum simulation.
发表机构
- Max-Planck-Institut für Quantenoptik（马克斯·普朗克量子光学研究所）
- Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）
机构由 AI 辅助整理，请以论文原文为准。
