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 AffleckKennedy-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 measurementbased 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 measurementbased 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.