Finding resource states of measurement-based quantum computing is harder than quantum computing

Measurement-based quantum computing enables universal quantum computing with only adaptive single-qubit measurements on certain many-qubit states, such as the graph state, the Affleck-Kennedy-Lieb-Tasaki (AKLT) state, and several tensor-network states. Finding new resource states of measurement-based quantum computing is a hard task, since for a given state there are exponentially many possible measurement patterns on the state. In this paper, we consider the problem of deciding, for a given state and a set of unitary operators, whether there exists a way of measurement-based quantum computing on the state that can realize all unitaries in the set, or not. We show that the decision problem is QCMA-hard, which means that finding new resource states of measurement-based quantum computing is harder than quantum computing itself (unless BQP is equal to QCMA). We also derive an upperbound of the decision problem: the problem is in a quantum version of the second level of the polynomial hierarchy.