In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given Hamiltonian $H$. It was shown in [Gharibian and Sikora, ICALP15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of the problem is also QCMA-complete. This provides one of the first examples where commuting local Hamiltonians exhibit complexity theoretic hardness equivalent to general local Hamiltonians.
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机译：在这项工作中，我们考虑了通勤当地哈密顿人的地面空间连通性问题。地面空间连通性问题询问是否可以通过应用2量子位unit的多项式长度序列，同时始终保持给定的哈密顿量$ H $处于低能量状态，来从一种（有效可制备）状态转变为另一种状态。在[Gharibian and Sikora，ICALP15]中显示，对于一般的本地哈密顿人来说，此问题是QCMA完全的，其中QCMA被定义为具有经典见证人和BQP验证者的QMA。在这里，我们表明问题的通勤版本也是QCMA完整的。这提供了第一个例子，其中通勤的局部哈密顿量显示出与一般局部哈密顿量相等的复杂性理论硬度。
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