Reference

Quantum Gate Library

13 gates and measurement. Each card shows the matrix representation, Bloch sphere geometry, truth table (where applicable), and a link to try it in the simulator.

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Single-qubit

Multi-qubit

X Gate

Single-qubit \u00B7 Basic

bit flip

Flips |0⟩ ↔ |1⟩ for the target qubit.

(0110)

Observable effect

The measured bit result flips.

Bloch sphere

Rotates 180° around the X axis. Swaps north and south poles.

Learn more

Z Gate

Single-qubit \u00B7 Phase

phase

Adds a phase flip to |1⟩ without flipping the bit.

(10 0 - 1)

Observable effect

Histogram may look unchanged right now.

Hidden effect

Phase shift — a later Hadamard reveals it.

Bloch sphere

Rotates 180° around the Z axis. Flips equatorial phase without moving the poles.

Try it in the simulator Learn more

Hadamard

Single-qubit \u00B7 Basic

superposition

Creates equal superposition from a basis state.

\frac{1}{2} (11 1 - 1)

Observable effect

Measurement odds spread more evenly.

Hidden effect

Relative phase may affect later interference.

Bloch sphere

Rotates 180° around the axis halfway between X and Z. Maps north pole to equator.

Try it in the simulator Learn more

CNOT

2-qubit \u00B7 Entanglement

entangle

Flips the target when the control is |1⟩.

1000010000010010

Observable effect

Joint two-qubit probability structure changes.

Hidden effect

Creates entanglement when control is in superposition.

Bloch sphere

Conditional rotation of target qubit. Creates entanglement from superposed control.

Truth table

|00⟩\u2192|00⟩

|01⟩\u2192|01⟩

|10⟩\u2192|11⟩

|11⟩\u2192|10⟩

Try it in the simulator Learn more

CCX

Toffoli

3-qubit \u00B7 Multi-qubit

AND

Flips target only when both controls are |1⟩.

I_{8} with ∣110 ⟩ \leftrightarrow ∣111 ⟩

Observable effect

Classical AND logic applied to amplitudes.

Hidden effect

Still acts on superpositions — not purely classical.

Bloch sphere

Conditional X on target when both controls are |1⟩.

Truth table

|000⟩\u2192|000⟩

|110⟩\u2192|111⟩

|111⟩\u2192|110⟩

Try it in the simulator Learn more

Swap

2-qubit \u00B7 Multi-qubit

swap

Exchanges the quantum states of two qubits.

1000001001000001

Observable effect

Qubit roles swap in the output.

Bloch sphere

Exchanges the Bloch vectors of two qubits.

Truth table

|00⟩\u2192|00⟩

|01⟩\u2192|10⟩

|10⟩\u2192|01⟩

|11⟩\u2192|11⟩

Learn more

Phase

Single-qubit \u00B7 Phase

+π/2

Adds a quarter-turn of phase to the |1⟩ branch.

(10 0 i)

Observable effect

Measurement odds stay the same.

Hidden effect

Quarter-turn phase — visible only after another mixing gate.

Bloch sphere

Rotates 90° around the Z axis. Quarter-turn of equatorial phase.

Learn more

RX Gate

Single-qubit \u00B7 Rotation

X-rot

Rotates around the X axis by angle θ.

(cos \frac{θ}{2} - i sin \frac{θ}{2} - i sin \frac{θ}{2} cos \frac{θ}{2})

Observable effect

Gradually mixes |0⟩ and |1⟩ amplitudes.

Hidden effect

At θ=π this becomes the X gate.

Bloch sphere

Rotates by θ around the X axis. At θ=π, becomes X gate.

Learn more

RY Gate

Single-qubit \u00B7 Rotation

Y-rot

Rotates around the Y axis by angle θ.

(cos \frac{θ}{2} sin \frac{θ}{2} - sin \frac{θ}{2} cos \frac{θ}{2})

Observable effect

Creates real-valued superpositions.

Hidden effect

At θ=π this becomes the Y gate (up to phase).

Bloch sphere

Rotates by θ around the Y axis. Stays in the XZ plane.

Learn more

RZ Gate

Single-qubit \u00B7 Rotation

Z-rot

Rotates around the Z axis by angle θ.

(e^{- i θ /2} 0 0 e^{i θ /2})

Observable effect

Adjusts relative phase between |0⟩ and |1⟩.

Hidden effect

Phase-only change — visible after mixing gates.

Bloch sphere

Rotates by θ around the Z axis. At θ=π, becomes Z gate.

Learn more

U3 Gate

Single-qubit \u00B7 Rotation

universal

Most general single-qubit gate — any rotation via θ, φ, λ.

(cos \frac{θ}{2} e^{i ϕ} sin \frac{θ}{2} - e^{iλ} sin \frac{θ}{2} e^{i (ϕ + λ)} cos \frac{θ}{2})

Observable effect

Can change both probabilities and phase simultaneously.

Hidden effect

Subsumes all single-qubit gates as special cases.

Bloch sphere

Arbitrary rotation. Three parameters cover any point on the sphere.

Learn more

Controlled-Z

2-qubit \u00B7 Entanglement

ctrl-Z

Applies Z to target when control is |1⟩.

diag (1, 1, 1, - 1)

Observable effect

Joint two-qubit phase structure changes.

Hidden effect

Creates entanglement when control is in superposition.

Bloch sphere

Conditional phase flip. Symmetric between qubits — no control/target distinction.

Learn more

Controlled-Phase

2-qubit \u00B7 Entanglement

ctrl-phase

Applies a configurable phase rotation to the |11⟩ state.

diag (1, 1, 1, e^{i θ})

Observable effect

Joint phase structure changes by angle θ.

Hidden effect

At θ=π this becomes the CZ gate.

Bloch sphere

Conditional phase rotation by θ. At θ=π, becomes CZ.

Learn more

Measure

Single-qubit \u00B7 Measurement

collapse

Collapses quantum state to a classical result.

∣ ψ ⟩ \to ∣ k ⟩ with p_{k} = ∣ ⟨ k ∣ ψ ⟩ ∣^{2}

Observable effect

One outcome is sampled from the distribution.

Hidden effect

The full superposition information is discarded.

Bloch sphere

Projects the Bloch vector to the nearest pole. The sphere collapses to |0⟩ or |1⟩.

Learn more

Notation

Matrices are in the computational basis {|0\u27E9, |1\u27E9} for single-qubit gates and {|00\u27E9, |01\u27E9, |10\u27E9, |11\u27E9} for two-qubit gates. Rotation angles are in radians. Parameterized gates (RX, RY, RZ, U3, CP) accept user-defined parameters through the inline editor. Reference: Nielsen & Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010).

Notation