The Tool Desk
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What the Pauli X and Z gates do
A one-qubit gate transforms the state of the selected qubit. The two Pauli gates have different effects on computational-basis states:
| Gate | Matrix | Basis-state action | Common description |
|---|---|---|---|
| Pauli X | [[0, 1], [1, 0]] |
|0⟩ → |1⟩; |1⟩ → |0⟩ |
Bit flip |
| Pauli Z | [[1, 0], [0, −1]] |
|0⟩ → |0⟩; |1⟩ → −|1⟩ |
Phase flip |
For a general one-qubit state α|0⟩ + β|1⟩, X gives α|1⟩ + β|0⟩, while Z gives α|0⟩ − β|1⟩. Thus, Z changes the relative phase between the two components; it does not swap their basis labels. The gate definitions and circuit methods are documented in the Qiskit XGate reference and Qiskit ZGate reference.
Apply X or Z in a Qiskit circuit
Create a QuantumCircuit with enough qubits, then call the method for the gate you want on the selected qubit index. For the circuit-construction approach shown in IBM Quantum Learning’s bits, gates, and circuits lesson, the calls are:
qc.x(q)applies Pauli X to qubitq.qc.z(q)applies Pauli Z to qubitq.
Example: apply both gates
from qiskit import QuantumCircuit
qc = QuantumCircuit(1)
qc.x(0) # apply X to qubit 0
qc.z(0) # then apply Z to qubit 0
print(qc.draw())Instructions are added in circuit order. Starting from the default all-zero state, the X first changes the qubit from |0⟩ to |1⟩; the following Z changes that component’s sign, giving −|1⟩. Drawing the circuit helps check which operation is applied to which wire.
Example: target different qubits
qc = QuantumCircuit(2)
qc.x(0)
qc.z(1)Each instruction acts on the specified qubit and leaves the other subsystem untouched. When interpreting a displayed multi-qubit bitstring, follow Qiskit’s qubit and display-order conventions rather than assuming the leftmost displayed bit is qubit 0.
Check the result and interpret the gate order
In IBM Quantum Learning’s workflow, the circuit’s state can be inspected with Statevector(qc). For a circuit initialized in |0⟩, applying X produces |1⟩. Applying Z directly to |0⟩ leaves the state unchanged. To observe Z’s phase effect, the qubit must have a component in |1⟩, as in a superposition or after an X operation.
On the same qubit, the operators satisfy XZ = −ZX. The minus sign is a global phase when comparing the resulting isolated state, so it does not change that state’s measurement probabilities. But order and phase must not be discarded when building controlled operations or comparing exact unitaries.
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Pauli gates versus π rotations
In Qiskit’s gate conventions, an X rotation by π is RX(π) = −iX, and a Z rotation by π is RZ(π) = −iZ. These differ from the Pauli gates by a global phase. That phase does not affect measurement probabilities for an isolated state, but it matters if you are comparing exact unitary matrices or incorporating the operation into a larger controlled construction.
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