Exact & formalMulti-qubit gate
CCZ (doubly-controlled Z) gate
The phase-only sibling of the Toffoli gate that applies a -1 phase exactly when all three qubits are |1⟩, symmetric in all three qubits and the natural three-qubit generalization of CZ.
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The defining behavior was checked exactly: a mathematical identity, a full statevector or stabilizer simulation, or an exhaustive basis-state truth table.
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The phase-only sibling of the Toffoli gate that applies a -1 phase exactly when all three qubits are |1⟩, symmetric in all three qubits and the natural three-qubit generalization of CZ.
A controlled version of the Hadamard gate that puts the target into superposition only when the control qubit is |1⟩, a canonical example of promoting a single-qubit gate to a controlled operation.
A parametrized entangling gate that applies e^{iλ} exactly when both qubits are |1⟩, generalizing CZ and forming the entangling primitive of the quantum Fourier transform.
A parametrized controlled gate that applies an RZ(θ) rotation to the target exactly when the control is |1⟩, the entangling generalization of RZ used when a Z-rotation itself needs to be conditioned on another qubit.
The canonical two-qubit controlled operation used to create correlation and entanglement.
A controlled version of the Pauli-Y gate that flips the target bit and attaches a ±i phase exactly when the control qubit is |1⟩, completing the CX/CY/CZ family of controlled Paulis.
A diagonal two-qubit entangling gate that applies a -1 phase exactly when both qubits are |1⟩, and is symmetric between the two qubits it acts on.
A two-qubit permutation gate defined as two CNOTs with reversed control and target applied back to back — a genuinely different two-CNOT gate from SWAP (which uses three), and notable for not being self-inverse.
The maximally entangling two-qubit gate native to IBM's newer superconducting processors, built from an 'echoed' pair of opposite-sign cross-resonance pulses around a control-qubit X flip, and locally equivalent to CX.
A three-qubit controlled-swap gate that conditionally exchanges two target qubits, useful for reversible routing logic and as a component of the SWAP test for state overlap.
The smallest useful reference for putting a qubit into an equal superposition.
The single-qubit no-op gate that leaves every state exactly unchanged, used as a placeholder in circuit diagrams, a timing/idle slot on real hardware, and the base case for gate-composition identities.
A two-qubit gate that swaps |01⟩ and |10⟩ while attaching a factor of i, arising naturally from XY-type physical couplings and native to several superconducting and photonic platforms.
The Pauli gate that combines a bit flip with a phase rotation around the y axis.
A phase-flip gate whose behavior is invisible to a basis measurement until interference is introduced.
The generalized phase gate that fixes the |0⟩ amplitude at 1 and applies e^{iλ} to |1⟩, unifying S, T, and Z as special cases of one continuous parameter.
A parametrized single-qubit rotation about the Bloch-sphere X axis, one of the two generators (with RZ) used to synthesize arbitrary single-qubit unitaries.
A parametrized two-qubit gate implementing evolution under the Ising XX coupling, the RZZ gate's basis-rotated sibling used in Heisenberg-model simulation and trapped-ion Mølmer–Sørensen-style entangling operations.
A parametrized single-qubit rotation about the Bloch-sphere Y axis whose matrix entries are always real, making it the standard tool for encoding real-valued amplitudes.
A parametrized single-qubit rotation about the Z axis that adds a relative phase between |0⟩ and |1⟩ without changing measurement probabilities in the computational basis.
A parametrized two-qubit gate implementing evolution under the Ising ZZ coupling, the entangling primitive most directly native to superconducting and trapped-ion hardware built from always-on or tunable ZZ interactions.
The Hermitian conjugate (and inverse) of the S gate, applying a -π/2 phase to |1⟩ and undoing whatever an S gate did earlier in a circuit.
A quarter-turn phase gate that exposes the difference between global and relative phase.
A routing primitive that exchanges two qubit states and makes hardware connectivity explicit.
A Clifford square root of the Pauli-X gate that is the native physical single-qubit gate (an X_{π/2} pulse) on most superconducting hardware.
The Hermitian conjugate (and inverse) of the non-Clifford T gate, applying a -π/4 phase to |1⟩ and appearing throughout fault-tolerant circuits as the T-count partner of T.
The three-qubit doubly-controlled NOT gate that implements reversible classical AND/NAND logic and is universal for classical reversible computation.
A non-Clifford phase gate that marks the boundary between a compact Clifford set and universal synthesis.
The three-parameter gate that can reach any single-qubit unitary up to global phase, used as the universal target when a compiler synthesizes one-qubit operations from Euler angles.
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