分解量子态 [英] Factoring a quantum state

问题描述

我正在寻找采用任意量子态的算法，该量子态由加权的经典态之和组成，这些态由位组成，像这样：

I'm looking for algorithms that take an arbitrary quantum state made up of a sum of weighted classical states made up of bits, like this:

|0000>/2 - |0011>/2 + |0100>/2 - |0111>/2

并使用张量积将其分解为更紧凑的形式，例如：

and factor it into a more compact form using tensor products, like this:

|0> x (|0> + |1>) x (|00> - |11>) / 2

我想将该算法用作可视化/简化（模拟）量子电路状态的一种方式。

I want to use the algorithm as a way of visualizing/simplifying the state of a (simulated) quantum circuit.

对于单个量子位，我知道我可以将所有翻转位，并检查每对在状态之间是否具有相同的x：y关系。在上面的示例中，翻转第二个位始终会为您提供权重为1：1的状态，因此第二个位分解为（1 | 0> + 1 | 1>）。

For individual qubits I know I can just pair all the states with the state where the bit is flipped and check that every pair has the same x:y relation between the states. In the example above, flipping the second bit always gives you a state with a 1:1 weighting, so the second bit factors out as (1|0> + 1|1>).

但是将这种方法扩展为检测纠缠位（例如示例中的第三和第四位）会使它至少花费Ω（n ^ c）时间（可能更多，我还没有完全考虑过），其中 n 是州的数量， c 是纠缠位数。由于 n 已经随着不理想的位数呈指数增长。

But extending that approach to detect entangled bits (like the third and fourth in the example) causes it to take at least Ω(n^c) time (probably more, I haven't thought it all the way through), where n is the number of states and c is the number of entangled bits. Since n is already growing exponentially with the number of bits that's... not ideal.

有没有更好的算法？表示更容易从/作为要素？更改基础有多有用？

Are there better algorithms? Representations easier to factor from/to? How useful is changing the basis? Links to papers would be great.

推荐答案

有效的算法似乎很难：

来自维基百科：

一般来说，决定一个状态是否可分离的问题是
，有时在量子信息
理论中称为可分离性问题。这被认为是一个困难的问题。已经证明
是NP困难的。

The problem of deciding whether a state is separable in general is sometimes called the separability problem in quantum information theory. It is considered to be a difficult problem. It has been shown to be NP-hard.

Gurvits，L.，《埃德蒙兹问题的经典确定性复杂性和量子纠缠》，第35卷ACM计算理论专题讨论会，ACM出版社，纽约，2003年。

Gurvits, L., Classical deterministic complexity of Edmonds’ problem and quantum entanglement, in Proceedings of the 35th ACM Symposium on Theory of Computing, ACM Press, New York, 2003.

Sevag Gharibian，《量子可分离性问题的强NP硬度》，《量子信息与计算》，第一卷。 ，第3号和第4号，第343-360页，2010年。arXiv：0810.4507

Sevag Gharibian, Strong NP-Hardness of the Quantum Separability Problem, Quantum Information and Computation, Vol. 10, No. 3&4, pp. 343-360, 2010. arXiv:0810.4507

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