X, Y and Z are the three fundamental single-qubit gates, each a 180-degree rotation of the Bloch vector about its own axis. This simulator lets you set an arbitrary starting state, apply any sequence of the three gates, and watch the vector trace its rotation path in real time.
• Independent theta/phi sliders for setting an arbitrary starting Bloch vector. • X, Y and Z gate buttons that apply instantly and log to a running sequence. • A dashed trail on the Bloch sphere showing every state visited so far. • Undo (remove the last gate) and full-reset controls. • Live P(0)/P(1) probability readout and the full 2x2 matrix for each gate.
• Start at |0>, apply X, and confirm the vector jumps straight to |1> — the quantum NOT. • Start at |+> (theta=90, phi=0) and apply Z; watch the vector reflect to |-> even though P(0) and P(1) both stay at 50%. • Apply the same gate twice in a row (X then X, or Z then Z) and confirm the vector returns exactly to where it started.
X changes what you would measure (a true bit flip), while Z changes only the hidden relative phase (a phase flip) and Y does both simultaneously. This distinction matters immediately once qubits interact with anything else: phase-flip errors are invisible to a direct measurement in the same way phase itself is, which is exactly the vulnerability that quantum error correction has to guard against separately from bit-flip errors.
Each is a 180-degree rotation of the Bloch vector about its own named axis: X rotates about the X axis, Y about the Y axis, and Z about the Z axis. X flips |0> and |1> into each other (the quantum NOT), Z flips the relative phase of |1> while leaving measurement probabilities unchanged, and Y does a combination of both with an extra phase.
Z leaves |0> alone and multiplies the amplitude of |1> by -1. That is a pure phase change — the probability of measuring |1> is unaffected because probability depends on the squared magnitude of the amplitude, not its sign. It only becomes visible once another gate recombines the amplitudes.
You return to the original state. X, Y and Z all square to the identity matrix, since each is a 180-degree rotation and two 180-degree rotations about the same axis add up to a full 360-degree turn.
Not on their own — X, Y and Z together with the identity form a specific finite set of operations. Reaching arbitrary rotations requires additional gates such as the Hadamard and phase gates shown in the Quantum Circuit lab, which combine with the Pauli gates to generate any single-qubit operation.