作为非线性量子模拟的宇宙：中心自旋模型的大 $n$ 极限
The universe as a nonlinear quantum simulation: Large $n$ limit of the central spin model
AI总结：
本文研究了映射到大 $n$ 极限中心自旋模型的非线性量子比特演化，证明了其严格对偶于非线性量子比特，支持非线性量子计算与模拟，并揭示了线性和非线性量子力学演化间无明确界限。
摘要：
我们研究了基于映射到大 $n$ 极限下的 $n$-比特中心自旋模型（CSM）的非线性比特演化模型，其中平均场理论是精确的。通过扩展 Erdös 和 Schlein 的一个定理，我们确立了当 $n \rightarrow \infty$ 时，CSM 严格对偶于一个非线性比特。这种对偶性支持在诸如凝聚体等系统中的非线性量子计算，在这些系统中，大量辅助比特对称地耦合到一个“中心”比特上。它还使得非线性量子模拟的门模型实现具有严格的误差界限。该模型的两个变体（有无辅助比特间的耦合）映射到了具有不同非线性和对称性的有效模型。从量子基础的角度来看，这里讨论的对偶性可能也很有趣。长期以来，人们一直对量子力学是否可能具有某种微小的、未被观察到的非线性感兴趣。如果不是，禁止它的原理是什么？对偶性意味着，按照线性和非线性量子力学演化的宇宙之间没有明确的区别：在大爆炸时刻制备于纯态 $| φ\rangle $ 并对称耦合到制备于相同状态的辅助比特的单比特“宇宙”，只要存在指数级多的辅助比特 $n \gg {\rm exp}(O(t))$，对于任何有限时间 $t>0$ 都将表现为非线性演化。
英文摘要：
We investigate models of nonlinear qubit evolution based on mappings to an $n$-qubit central spin model (CSM) in the large $n$ limit, where mean field theory is exact. Extending a theorem of Erdös and Schlein, we establish that the CSM is rigorously dual to a nonlinear qubit when $n \rightarrow \infty$. The duality supports a type of nonlinear quantum computation in systems, such as a condensate, where a large number of ancilla couple symmetrically to a "central" qubit. It also enables a gate-model implementation of nonlinear quantum simulation with a rigorous error bound. Two variants of the model, with and without coupling between ancilla, map to effective models with different nonlinearity and symmetry. The duality discussed here might also be interesting from a quantum foundations perspective. There has long been interest in whether quantum mechanics might possess some type of small, unobserved nonlinearity. If not, what is the principle prohibiting it? The duality implies that there is not a sharp distinction between universes evolving according to linear and nonlinear quantum mechanics: A one-qubit "universe" prepared in a pure state $| φ\rangle $ at the time of the big bang and symmetrically coupled to ancilla prepared in the same state, would appear to evolve nonlinearly for any finite time $t>0$ as long as there are exponentially many ancilla $n \gg {\rm exp}(O(t))$.
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