Experimental Verification of Group Non-Membership in Optical Circuits

The class quantum Merlin–Arthur（QMA）, as the quantum analog of nondeterministic polynomial time, contains the decision problems whose YES instance can be verified efficiently with a quantum computer. The problem of deciding the group non-membership（GNM） of a group element is conjectured to be a member of QMA. Previous works on the verification of GNM, which still lacks experimental demonstration, required a quantum circuit with O（n~5） group oracle calls. Here, we provide an efficient way to verify GNM problems, in which each quantum circuit only contains O（1） group of oracle calls, and the number of qubits in each circuit is reduced by half. Based on this protocol, we then experimentally demonstrate the new verification process with a four-element group in an all-optical circuit. The new protocol is validated experimentally by observing a significant completeness-soundness gap between the probabilities of accepting elements in and outside the subgroup. This work efficiently simplifies the verification of GNM and is helpful in constructing more quantum protocols based on the near-term quantum devices.