In general, the control bits of a multi-control bit quantum gate are in an AND relationship. Is there a quantum gate that makes the control bits an OR relationship?
If you can implement standard AND-controlled $U^\dagger$ gate you can implement your OR-controlled $U$ gate without any additional controls, as follows
This is an example for the $S$ gate and 2 control qubits, but it applies universally. A
qiskit code snippet for checking.
from qiskit import QuantumCircuit from qiskit.circuit.library import SGate from qiskit.quantum_info import Operator import numpy as np qc = QuantumCircuit(3) qc.x(0) qc.x(1) qc.s(2) qc.append(SGate().control(2).inverse(), [0, 1, 2]) qc.x(0) qc.x(1) u_target = np.kron(np.diag([1, 0, 0, 0]), np.identity(2))+np.kron(np.diag([0, 1, 1, 1]), np.array([[1, 0], [0, 1j]])) u_qs = Operator(qc.reverse_bits()).data print(np.allclose(u_target, u_qs)) qc.draw(output='mpl')