Quantum gates are reversible operations on qubits. A gate changes amplitudes, phases, or correlations without reading the qubit state. A quantum circuit combines gates, then measures one or more qubits to produce classical data.
The gate names look small. Their combinations create every algorithm covered by current gate-model software.
A qubit before gates
A qubit state uses two complex amplitudes:
|ψ⟩ = α|0⟩ + β|1⟩
The measurement probabilities are |α|² for 0 and |β|² for 1. Their sum equals one. A gate changes the amplitudes while preserving total probability.
A global phase has no measurement effect. A relative phase does affect later interference. This distinction explains why phase gates matter even when immediate measurement probabilities stay unchanged.
The X gate
The X gate is the quantum version of a bit flip:
X|0⟩ = |1⟩X|1⟩ = |0⟩
On the Bloch sphere, X rotates a state by 180 degrees around the x axis. Two X gates return the original state because every quantum gate used in a circuit has an inverse.
The Y and Z gates
The Y gate combines a bit flip with a phase change. The Z gate leaves |0⟩ unchanged and adds a minus sign to |1⟩:
Z|0⟩ = |0⟩Z|1⟩ = -|1⟩
A Z gate often appears invisible when followed by immediate measurement in the computational basis. A later H gate converts phase information into measurable probability differences.
X, Y, and Z are Pauli gates. Pauli operators also describe error syndromes, observables, and many Hamiltonians used in quantum algorithms.
The H gate
The Hadamard gate creates equal superposition from a basis state:
H|0⟩ = (|0⟩ + |1⟩) / √2H|1⟩ = (|0⟩ - |1⟩) / √2
The minus sign in the second expression is relative phase. Applying H twice returns the input. A common pattern is H, apply a phase-dependent operation, then H again so interference moves phase information into measurement probabilities.
Superposition does not mean a measurement returns both values. Each shot returns one classical result. Repeated shots estimate the distribution.
S and T phase gates
The S gate adds a 90-degree phase to |1⟩. The T gate adds a 45-degree phase. Both are diagonal gates, so computational-basis probabilities stay unchanged immediately after application.
S and T matter because phase differences combine with later gates. T gates also matter in fault-tolerant resource estimates because non-Clifford operations require special error-correction resources in many architectures.
The CNOT gate
CNOT, or controlled-X, uses two qubits. The first qubit is the control. The second is the target.
- Control
0: target stays unchanged. - Control
1: target receives X.
Start with |00⟩. Apply H to qubit 0, then CNOT with qubit 0 as control and qubit 1 as target:
|00⟩ → (|00⟩ + |10⟩) / √2 → (|00⟩ + |11⟩) / √2
Measurement returns 00 or 11, never 01 or 10 in an ideal circuit. The pair is entangled. CNOT plus arbitrary single-qubit rotations forms a universal gate set.
Controlled-Z and SWAP
Controlled-Z adds a phase only when both qubits equal 1. Unlike CNOT, it does not exchange computational-basis values. A change of basis turns one controlled gate into another, which lets compilers select hardware-native operations.
SWAP exchanges two qubit states. A standard decomposition uses three CNOT gates. Hardware with limited connectivity often pays extra SWAP cost when a circuit needs interactions between distant qubits.
Rotation gates
Rotation gates vary continuously by angle:
Rx(θ)rotates around the x axis.Ry(θ)rotates around the y axis.Rz(θ)rotates around the z axis.
Variational algorithms such as VQE and QAOA optimize these angles with a classical loop. The circuit supplies expectation values. The optimizer changes parameters and repeats the run.
Measurement is not a gate
Measurement converts a quantum state into classical data and usually destroys the measured state. A circuit sometimes measures in the computational basis or rotates first to measure another observable.
A measurement shot gives one bit string. Thousands of shots build counts. Expectation values estimate averages such as the mean of Z measurements. Read our getting started guide for a complete Bell-state example.
How to read a circuit
Read left to right. Track each wire as a qubit. Mark single-qubit rotations, controlled operations, measurement bases, and barriers. Count two-qubit gates separately because they often produce more errors than single-qubit gates.
Then ask four questions:
- Which amplitudes changed?
- Which relative phases changed?
- Which qubits became correlated?
- Which measurement basis reveals the result?
Those four questions explain most beginner circuits before advanced linear algebra enters.