Gate

Qubits

Family

Matrix

Note

Identity (I)

pauli

(1001)

Does nothing; a wait of one cycle.

Pauli-X

pauli

(0110)

Bit flip, π about X.

Pauli-Y

pauli

(0 i - i 0)

Bit and phase flip, π about Y.

Pauli-Z

pauli

(10 0 - 1)

Phase flip, π about Z.

Hadamard (H)

clifford

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

Superposition; swaps X and Z axes.

S (√Z)

phase

(10 0 i)

Phase π/2 on |1⟩.

S†

phase

(10 0 - i)

Phase −π/2 on |1⟩.

T (⁴√Z)

phase

(10 0 e^{iπ /4})

Phase π/4; non-Clifford, needed for universality.

T†

phase

(10 0 e^{- iπ /4})

Phase −π/4.

√X

clifford

\frac{1}{2} (1 + i 1 - i 1 - i 1 + i)

Half an X; native on IBM.

√X†

clifford

\frac{1}{2} (1 - i 1 + i 1 + i 1 - i)

Inverse of √X.

RX(θ)

rotation

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

Rotation about X. RX(π) = −iX.

RY(θ)

rotation

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

Rotation about Y; real amplitudes.

RZ(φ)

rotation

(e^{- i φ /2} 0 0 e^{i φ /2})

Rotation about Z; free on IBM hardware.

P(λ)

rotation

(10 0 e^{iλ})

Phase gate; P(π)=Z, P(π/2)=S, P(π/4)=T.

U(θ, φ, λ)

general

(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})

The most general single-qubit gate; every 1-qubit gate is a U.

CNOT (CX)

two-qubit

1000000100100100

Flips q1 when q0 is 1 (Qiskit order: q0 is the rightmost bit).

Controlled-Y

two-qubit

C (Y)

Y on the target when the control is 1.

Controlled-Z

two-qubit

diag (1, 1, 1, - 1)

−1 phase on |11⟩; symmetric; native on Heron.

Controlled-H

two-qubit

C (H)

Hadamard on the target when the control is 1.

Controlled-phase CP(λ)

two-qubit

diag (1, 1, 1, e^{iλ})

Phase e^{iλ} on |11⟩ only; the QFT building block.

Controlled-RX(θ)

two-qubit

C (R_{X} (θ))

RX on the target when the control is 1.

Controlled-RY(θ)

two-qubit

C (R_{Y} (θ))

RY on the target when the control is 1; W-state preparation.

Controlled-RZ(φ)

two-qubit

C (R_{Z} (φ))

RZ on the target when the control is 1.

SWAP

two-qubit

1000001001000001

Exchanges the qubits; three CNOTs.

iSWAP

two-qubit

1000 00 i 0 0 i 00 0001

SWAP with an i phase on the swapped states; native on some superconducting chips.

ECR

two-qubit

\frac{1}{2} 001 - i 00 - i 1 1 i 00 i 100

Echoed cross-resonance: IBM Eagle's native entangler, CNOT up to 1-qubit gates.

DCX

two-qubit

1000001000010100

Two CNOTs back to back with controls swapped.

RXX(θ)

ising

e^{- i θ X \otimes X /2}

XX interaction, trapped-ion native (Mølmer–Sørensen).

RYY(θ)

ising

e^{- i θ Y \otimes Y /2}

YY interaction.

RZZ(θ)

ising

diag (e^{- i θ /2}, e^{i θ /2}, e^{i θ /2}, e^{- i θ /2})

ZZ interaction: the QAOA cost term; native on Heron (fractional gates).

Every gate, interactive

The palette covers the gates you draw most. Quantum computing has many more: general rotations, controlled versions, Ising interactions, and the native gates of real chips. Try any of them here on any input state and watch the matrix multiply the vector.

Gate Lab: try any gate on any state

Choose a gate, set its angle, pick the input qubit state(s). Everything below is computed live in your browser: the numeric matrix, matrix × vector, the output ket, the probabilities and the Bloch spheres.

Gate

Superposition; swaps X and Z axes.

Input state

Matrix (symbolic and numeric)

Calculation: U · |ψ_in⟩ = |ψ_out⟩

Probabilities

|0⟩50.0%
|1⟩50.0%

Bloch sphere(s) of the output

Catalog

All 30 gates in the lab, grouped by family. Two-qubit matrices use Qiskit ordering: qubit 0 is the rightmost bit and is the control.

Gate	Qubits	Family	Matrix	Note
Identity (I)	1	pauli	$(1001)$	Does nothing; a wait of one cycle.
Pauli-X	1	pauli	$(0110)$	Bit flip, π about X.
Pauli-Y	1	pauli	$(0 i - i 0)$	Bit and phase flip, π about Y.
Pauli-Z	1	pauli	$(10 0 - 1)$	Phase flip, π about Z.
Hadamard (H)	1	clifford	$\frac{1}{2} (11 1 - 1)$	Superposition; swaps X and Z axes.
S (√Z)	1	phase	$(10 0 i)$	Phase π/2 on \|1⟩.
S†	1	phase	$(10 0 - i)$	Phase −π/2 on \|1⟩.
T (⁴√Z)	1	phase	$(10 0 e^{iπ /4})$	Phase π/4; non-Clifford, needed for universality.
T†	1	phase	$(10 0 e^{- iπ /4})$	Phase −π/4.
√X	1	clifford	$\frac{1}{2} (1 + i 1 - i 1 - i 1 + i)$	Half an X; native on IBM.
√X†	1	clifford	$\frac{1}{2} (1 - i 1 + i 1 + i 1 - i)$	Inverse of √X.
RX(θ)	1	rotation	$(cos \frac{θ}{2} - i sin \frac{θ}{2} - i sin \frac{θ}{2} cos \frac{θ}{2})$	Rotation about X. RX(π) = −iX.
RY(θ)	1	rotation	$(cos \frac{θ}{2} sin \frac{θ}{2} - sin \frac{θ}{2} cos \frac{θ}{2})$	Rotation about Y; real amplitudes.
RZ(φ)	1	rotation	$(e^{- i φ /2} 0 0 e^{i φ /2})$	Rotation about Z; free on IBM hardware.
P(λ)	1	rotation	$(10 0 e^{iλ})$	Phase gate; P(π)=Z, P(π/2)=S, P(π/4)=T.
U(θ, φ, λ)	1	general	$(cos \frac{θ}{2} e^{i φ} sin \frac{θ}{2} - e^{iλ} sin \frac{θ}{2} e^{i (φ + λ)} cos \frac{θ}{2})$	The most general single-qubit gate; every 1-qubit gate is a U.
CNOT (CX)	2	two-qubit	$1000000100100100$	Flips q1 when q0 is 1 (Qiskit order: q0 is the rightmost bit).
Controlled-Y	2	two-qubit	$C (Y)$	Y on the target when the control is 1.
Controlled-Z	2	two-qubit	$diag (1, 1, 1, - 1)$	−1 phase on \|11⟩; symmetric; native on Heron.
Controlled-H	2	two-qubit	$C (H)$	Hadamard on the target when the control is 1.
Controlled-phase CP(λ)	2	two-qubit	$diag (1, 1, 1, e^{iλ})$	Phase e^{iλ} on \|11⟩ only; the QFT building block.
Controlled-RX(θ)	2	two-qubit	$C (R_{X} (θ))$	RX on the target when the control is 1.
Controlled-RY(θ)	2	two-qubit	$C (R_{Y} (θ))$	RY on the target when the control is 1; W-state preparation.
Controlled-RZ(φ)	2	two-qubit	$C (R_{Z} (φ))$	RZ on the target when the control is 1.
SWAP	2	two-qubit	$1000001001000001$	Exchanges the qubits; three CNOTs.
iSWAP	2	two-qubit	$1000 00 i 0 0 i 00 0001$	SWAP with an i phase on the swapped states; native on some superconducting chips.
ECR	2	two-qubit	$\frac{1}{2} 001 - i 00 - i 1 1 i 00 i 100$	Echoed cross-resonance: IBM Eagle's native entangler, CNOT up to 1-qubit gates.
DCX	2	two-qubit	$1000001000010100$	Two CNOTs back to back with controls swapped.
RXX(θ)	2	ising	$e^{- i θ X \otimes X /2}$	XX interaction, trapped-ion native (Mølmer–Sørensen).
RYY(θ)	2	ising	$e^{- i θ Y \otimes Y /2}$	YY interaction.
RZZ(θ)	2	ising	$diag (e^{- i θ /2}, e^{i θ /2}, e^{i θ /2}, e^{- i θ /2})$	ZZ interaction: the QAOA cost term; native on Heron (fractional gates).

Why these gates are enough

Any unitary on n qubits can be built from single-qubit rotations plus one entangling gate such as CNOT (Solovay–Kitaev, Barenco et al. 1995). The Clifford gates {H, S, CNOT} alone are not universal and can be simulated classically (Gottesman–Knill); adding T makes the set universal. That is why T-count matters in fault tolerance and why √X + RZ + CX/ECR/CZ is all a real IBM chip needs.

U = e^{i α} R_{Z} (β) R_{Y} (γ) R_{Z} (δ)

Z–Y decomposition: every single-qubit unitary is three rotations and a global phase. The transpiler uses this to merge any run of 1-qubit gates into one U, then into RZ·√X·RZ·√X·RZ.

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All gates, with the math

Dynamic circuits: if/else, loops and conditions