The following tables list several unitary quantum logic gates, together with their common name, how they are represented, and some of their properties.
Controlled or conjugate transpose (adjoint) versions of some of these gates may not be listed.
, most of the times this gate is not indicated in circuit diagrams, but it is useful when describing mathematical results.
A quantum state is uniquely defined up to a phase.
Because of the Born rule, a phase factor has no effect on a measurement outcome:
when the global phase gate is applied to a single qubit in a quantum register, the entire register's global phase is changed.
These gates can be extended to any number of qubits or qudits.
This table includes commonly used Clifford gates for qubits.
Note that if a Clifford gate A is not in the Pauli group,
Implementation: The phase shift is a family of single-qubit gates that map the basis states
is unchanged after applying this gate, however it modifies the phase of the quantum state.
This is equivalent to tracing a horizontal circle (a line of latitude), or a rotation along the z-axis on the Bloch sphere by
to a rotation about a generic phase of both basis states of a 2-level quantum system (a qubit) can be done with a series circuit:
Arbitrary single-qubit phase shift gates
are natively available for transmon quantum processors through timing of microwave control pulses.
: The controlled-Z (or CZ) gate is the special case where
are the analog rotation matrices in three Cartesian axes of SO(3),[c] along the x, y or z-axes of the Bloch sphere projection.
As Pauli matrices are related to the generator of rotations, these rotation operators can be written as matrix exponentials with Pauli matrices in the argument.
unitary matrix in SU(2) can be written as a product (i.e. series circuit) of three rotation gates or less.
Note that for two-level systems such as qubits and spinors, these rotations have a period of 4π.
A rotation of 2π (360 degrees) returns the same statevector with a different phase.
It's possible to work out the adjoint action of rotations on the Pauli vector, namely rotation effectively by double the angle a to apply Rodrigues' rotation formula: Taking the dot product of any unit vector with the above formula generates the expression of any single qubit gate when sandwiched within adjoint rotation gates.
gate represents a rotation of π/2 about the x axis at the Bloch sphere
Similar rotation operator gates exist for SU(3) using Gell-Mann matrices.
Implementation: Implementation: The qubit-qubit Ising coupling or Heisenberg interaction gates Rxx, Ryy and Rzz are 2-qubit gates that are implemented natively in some trapped-ion quantum computers, using for example the Mølmer–Sørensen gate procedure.
[17][18] Note that these gates can be expressed in sinusoidal form also, for example
The √SWAP gate arises naturally in systems that exploit exchange interaction.
[21][1] For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap[22] or iSWAP.
SWAPα arises naturally in spintronic quantum computers.
controlled square root NOT canonical decomposition Controlled-fermionic SWAP