P $\neq$ NP的物理结果和DMRG退火猜想
Physical consequences of P$\neq$NP and the DMRG-annealing conjecture
摘要：
计算复杂性理论包含了大量的定理和定理，这些定理和定理是关于图灵机解决某些类型的问题所需的时间，作为输入大小的函数。自然界{ \em不一定}是图灵机，因此这些定理并不直接适用于它。但是物理过程的{ \em经典模拟}是在图灵机上运行的程序，因此，它们服从图灵机。在这项工作中，计算复杂性理论应用于经典模拟系统进行绝热量子计算（AQC），基于退火扩展的密度矩阵重整化群（DMRG）。我们推测，这些经典模拟所需的计算时间是完全控制的{ \em最大纠缠}过程中发现。因此，可以提供关于纠缠随系统尺寸的增长的下限。在某些情况下，量子相变可以预测发生在某些非均匀系统。具体地，从复杂性类{ \bf P}和{ \bf NP}不同的假设中得出物理结论。作为一个副产品，提出了一种替代措施的纠缠，通过切比雪夫不等式，允许建立严格的界限所需的计算时间。
英文摘要：
Computational complexity theory contains a corpus of theorems and conjectures regarding the time a Turing machine will need to solve certain types of problems as a function of the input size. Nature {\em need not} be a Turing machine and, thus, these theorems do not apply directly to it. But {\em classical simulations} of physical processes are programs running on Turing machines and, as such, are subject to them. In this work, computational complexity theory is applied to classical simulations of systems performing an adiabatic quantum computation (AQC), based on an annealed extension of the density matrix renormalization group (DMRG). We conjecture that the computational time required for those classical simulations is controlled solely by the {\em maximal entanglement} found during the process. Thus, lower bounds on the growth of entanglement with the system size can be provided. In some cases, quantum phase transitions can be predicted to take place in certain inhomogeneous systems. Concretely, physical conclusions are drawn from the assumption that the complexity classes {\bf P} and {\bf NP} differ. As a by-product, an alternative measure of entanglement is proposed which, via Chebyshev's inequality, allows to establish strict bounds on the required computational time.
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