自然双能级系统中单向量子计算的No-go定理
No-go Theorem for One-way Quantum Computing on Naturally Occurring Two-level Systems
摘要：
单向量子计算通过对多体纠缠态（称为资源态）进行单粒子测量来实现量子计算的全部功能。由于单粒子测量相对容易实现，因此资源状态的准备成为一项至关重要的任务。一个吸引人的方法是简单地将强关联的量子多体系统冷却到它的基态。除了要求系统的基态对于单向量子计算是普适的之外，我们还希望哈密顿量具有由固定的能隙保护的非简并基态，仅涉及两体相互作用，并且是无阻挫的，以便在计算过程中的测量将剩余的粒子留在基空间中。近年来，人们在寻找自旋晶格系统中自然出现的基态资源态方面做了大量的工作。在自旋为5/2和自旋为3/2的系统中，该方法被证明是成功的。然而，在自旋为1/2的情况下，是否存在这样一种自然资源状态仍然是一个悬而未决的问题，即，量子比特系统在这里，我们给出了一个否定的答案，证明了一个真正的纠缠量子比特态不可能是任何两体无阻挫哈密顿量的非简并基态。更重要的是，我们证明了每个自旋为1/2的无阻挫哈密顿量与两体相互作用总是有一个基态，是一个产品的单或两个量子比特的状态，一个更强的结果，是有趣的独立的上下文的单向量子计算。
英文摘要：
One-way quantum computing achieves the full power of quantum computation by performing single particle measurements on some many-body entangled state, known as the resource state. As single particle measurements are relatively easy to implement, the preparation of the resource state becomes a crucial task. An appealing approach is simply to cool a strongly correlated quantum many-body system to its ground state. In addition to requiring the ground state of the system to be universal for one-way quantum computing, we also want the Hamiltonian to have non-degenerate ground state protected by a fixed energy gap, to involve only two-body interactions, and to be frustration-free so that measurements in the course of the computation leave the remaining particles in the ground space. Recently, significant efforts have been made to the search of resource states that appear naturally as ground states in spin lattice systems. The approach is proved to be successful in spin-5/2 and spin-3/2 systems. Yet, it remains an open question whether there could be such a natural resource state in a spin-1/2, i.e., qubit system. Here, we give a negative answer to this question by proving that it is impossible for a genuinely entangled qubit states to be a non-degenerate ground state of any two-body frustration-free Hamiltonian. What is more, we prove that every spin-1/2 frustration-free Hamiltonian with two-body interaction always has a ground state that is a product of single- or two-qubit states, a stronger result that is interesting independent of the context of one-way quantum computing.
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