Ball置换的计算复杂性
The Computational Complexity of Ball Permutations
摘要：
受二维量子理论的启发，我们定义了几种基于置换可区分粒子（我们称之为球）的计算模型，并描述了它们的计算复杂性。在量子环境中，我们发现该模型的计算能力依赖于初始输入态。更确切地说，在标准基输入状态下，我们展示了如何使用模型DQC 1（用一个干净的量子位可解决的问题类）在加性误差内近似该模型的振幅，证明在这种情况下该模型比通用量子计算弱。然而，对于特定的输入状态的选择，该模型被证明是普遍的BQP在编码的意义上。我们使用的对称群的表示理论，部分分类的计算复杂性，这个模型的任意输入状态。有趣的是，我们发现了一些输入状态，产生了DQC 1和BQP之间的模型中间。此外，我们认为这个模型的限制版本的基础上，在1+1维的可积散射问题。我们表明，它是普遍的后选择下，如果我们允许中间破坏性测量和特定的输入状态。因此，存在任何经典的程序，从这个模型的输出分布在乘性误差内的采样意味着多项式层次的崩溃到它的第三级。最后，我们定义了一个经典版本的这个模型，其中一个可以概率置换球。我们发现这产生了一个复杂性类，这是中间L和BPP。此外，我们发现这个模型的一个非确定性版本是NP完全的。
英文摘要：
Inspired by connections to two dimensional quantum theory, we define several models of computation based on permuting distinguishable particles (which we call balls), and characterize their computational complexity. In the quantum setting, we find that the computational power of this model depends on the initial input states. More precisely, with a standard basis input state, we show how to approximate the amplitudes of this model within additive error using the model DQC1 (the class of problems solvable with one clean qubit), providing evidence that the model in this case is weaker than universal quantum computing. However, for specific choices of input states, the model is shown to be universal for BQP in an encoded sense. We use representation theory of the symmetric group to partially classify the computational complexity of this model for arbitrary input states. Interestingly, we find some input states which yield a model intermediate between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an integrable scattering problem in 1+1 dimensions. We show it is universal under postselection, if we allow intermediate destructive measurements and specific input states. Therefore, the existence of any classical procedure to sample from the output distribution of this model within multiplicative error implies collapse of polynomial hierarchy to its third level. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP. Moreover, we find a nondeterministic version of this model is NP-complete.
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