Two QCMA-complete problems
Paweł Wocjan, Dominik Janzing, T. Beth · Quantum Information and Computation · 2003
QMA and QCMA are possible quantum analogues of the complexity class NP. In QMA the proof is a quantum state and the verification is a quantum circuit. In contrast, in QCMA the proof is restricted to be a classical state. It is not known whether QMA strictly contains QCMA. Here we show that two known QMA-complete problems can be modified to QCMA-complete problems in a natural way: (1) Deciding whether a 3-local Hamiltonian has low energy states (with energy smaller than a given value) that can be prepared with at most k elementary gates is QCMA-complete, whereas it is QMA-complete when the restriction on the complexity of preparation is dropped. (2) Deciding whether a (classically described) quantum circuit does not act as the identity on all basis states is QCMA-complete. It is QMA-complete to decide whether it does not act on all states as the identity.