The Z gate is a commonly used quantum gate that acts as a phase shift on a qubit. It applies a phase factor of -1 to the state |1⟩ while leaving the state |0⟩ unchanged. In matrix form, the Z gate can be represented as:
Z = |0⟩⟨0| - |1⟩⟨1| = [[1, 0], [0, -1]]
To apply the Z gate to qubits in positions 0, 2, and 4, we need to consider a multi-qubit system and specify the target qubits to which the gate is applied. Assuming we have a system of five qubits indexed from 0 to 4, and we want to apply the Z gate to qubits 0, 2, and 4, the resulting operator for this multi-qubit system would be:
Z⊗Z⊗Z⊗I⊗Z
where I represents the identity gate acting on the third qubit, which is not targeted for the Z gate.
In matrix form, this operator can be represented as a 32x32 matrix, where each entry corresponds to the transformation of a specific input state to the output state.
Please note that this representation assumes a specific ordering of the qubits, typically the computational basis order, where qubit 0 is the rightmost qubit. The specific implementation of quantum gates and their actions on qubits can vary depending on the quantum computing platform or software you are using.