In
gate-based quantum computing, various sets of
quantum logic gate
In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, lik ...
s are commonly used to express quantum operations. The following tables lists several unitary quantum logic gates, together with their common name, how they are represented, and some of their properties.
Controlled or
Hermitian conjugate
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
wher ...
versions of some of these gates may not be listed.
Identity gate and global phase
The identity gate is the
identity operation
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
, most of the times this gate is not indicated in circuit diagrams, but it is useful when describing mathematical results.
It has been described as being a "wait cycle", and a
NOP.
The global phase gate introduces a global phase
to the whole qubit quantum state. A quantum state is uniquely defined up to a phase. Because of the
Born rule
The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of findin ...
, a
phase factor
For any complex number written in polar form (such as ), the phase factor is the complex exponential factor (). As such, the term "phase factor" is related to the more general term phasor, which may have any magnitude (i.e. not necessarily on the ...
have no effect on a
measurement
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
outcome:
for any
.
Because
when the global phase gate is applied to a single qubit in a
quantum register In quantum computing, a quantum register is a system comprising multiple qubits. It is the quantum analogue of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.
Definition ...
, the entire register's global phase is changed.
Also,
These gates can be extended to any number of
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
s or
qudits.
Clifford qubit gates
This table includes commonly used
Clifford gates
In quantum computing and quantum information theory, the Clifford gates are the elements of the Clifford group, a set of mathematical transformations which normalize the ''n''-qubit Pauli group, i.e., map tensor products of Pauli matrices to te ...
for qubits.
Other Clifford gates, including higher dimensional ones are not included here but by definition can be generated using
and
.
Note that if a Clifford gate ''A'' is not in the Pauli group,
or controlled-''A'' are not in the Clifford gates.
The Clifford set is not a universal quantum gate set.
Non-Clifford qubit gates
Relative phase gates
The phase shift is a family of single-qubit gates that map the basis states
and
. The probability of measuring a
or
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
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
by
radians. A common example is the ''T'' gate where
(historically known as the
gate), the phase gate. Note that some Clifford gates are special cases of the phase shift gate:
The argument to the phase shift gate is in
U(1)
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \.
...
, and the gate performs a phase rotation in U(1) along the specified basis state (e.g.
rotates the phase about . Extending
to a rotation about a generic phase of both basis states of a 2-level quantum system (a
qubit
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
) can be done with a
series circuit
Two-terminal components and electrical networks can be connected in series or parallel. The resulting electrical network will have two terminals, and itself can participate in a series or parallel topology. Whether a two-terminal "object" is an ...
:
. When
this gate is the
rotation operator gate and if
it is a global phase.
The ''T'' gate's historic name of
gate comes from the identity
, where
.
Arbitrary single-qubit phase shift gates
are natively available for
transmon
In quantum computing, and more specifically in superconducting quantum computing, a transmon is a type of superconducting charge qubit that was designed to have reduced sensitivity to charge noise. The transmon was developed by Robert J. Schoelko ...
quantum processors through timing of microwave control pulses. It can be explained in terms of
change of frame.
As with any single qubit gate one can build a controlled version of the phase shift gate. With respect to the computational basis, the 2-qubit controlled phase shift gate is: shifts the phase with
only if it acts on the state
:
:
The controlled-''Z'' (or CZ) gate is the special case where
.
The controlled-''S'' gate is the case when the controlled-
when
and is a commonly used gate.''
''
Rotation operator gates
The rotation operator gates
and
are the analog
rotation matrices in three
Cartesian axes of
SO(3)
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition.
By definition, a rotation about the origin is a tr ...
, the axes on the
Bloch sphere
In quantum quantum mechanics, mechanics and Quantum computing, computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level system, two-level quantum mechanical system (qubit), named after the physicist Felix ...
projection.
As Pauli matrices are related to the
generator
Generator may refer to:
* Signal generator, electronic devices that generate repeating or non-repeating electronic signals
* Electric generator, a device that converts mechanical energy to electrical energy.
* Generator (circuit theory), an eleme ...
of rotations, these rotation operators can be written as
matrix exponentials with Pauli matrices in the argument. Any
unitary matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is ...
in
SU(2)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
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
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
, these rotations have a period of . A rotation of (360 degrees) returns the same statevector with a different
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
.
We also have
and
for all
The rotation matrices are related to the Pauli matrices in the following way :
It's possible to work out the adjoint action of rotations on the
Pauli vector, namely rotation effectively by double the angle to apply
Rodrigues' rotation formula
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform al ...
:
:
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. For example, it can be shown that
. Also, using the anticommuting relation we have
.
Rotation operators have interesting identities. For example,
and
Also, using the anticommuting relations we have
and
Global phase and phase shift can be transformed into each others with the Z-rotation operator:
.
The
gate represents a rotation of about the ''x'' axis at the Bloch sphere
.
Similar rotation operator gates exist for
SU(3)
In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...
using
Gell-Mann matrices
The Gell-Mann matrices, developed by Murray Gell-Mann, are a set of eight linearly independent 3×3 traceless Hermitian matrices used in the study of the strong interaction in particle physics.
They span the Lie algebra of the SU(3) group in th ...
. They are the rotation operators used with
qutrits.
Two qubit interaction gates
The qubit-qubit Ising coupling or Heisenberg interaction gates ''R
xx'', ''R
yy'' and ''R
zz'' are 2-qubit gates that are implemented natively in some
trapped-ion quantum computers, using for example the
Mølmer–Sørensen gate procedure.
Note that these gates can be expressed in sinusoidal form also, for example
.
The CNOT gate can be further decomposed as products of rotation operator gates and exactly a single two qubit interaction gate, for example
:
The SWAP gate can be constructed from other gates, for example using the
two qubit interaction gates:
.
Non-Clifford swap gates
The gate performs half-way of a two-qubit swap (see Clifford gates). It is universal such that any many-qubit gate can be constructed from only and single qubit gates. The gate is not, however maximally entangling; more than one application of it is required to produce a
Bell state
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form o ...
from product states. The gate arises naturally in systems that exploit
exchange interaction
In chemistry and physics, the exchange interaction (with an exchange energy and exchange term) is a quantum mechanical effect that only occurs between identical particles. Despite sometimes being called an exchange force in an analogy to classical ...
.
For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap or iSWAP.
Note that
and
, or more generally
for all real ''n'' except 0.
SWAP
''α'' arises naturally in spintronic quantum computers.
The
Fredkin gate
The Fredkin gate (also CSWAP gate and conservative logic gate) is a computational circuit suitable for reversible computing, invented by Edward Fredkin. It is ''universal'', which means that any logical or arithmetic operation can be constructed en ...
(also CSWAP or CS gate), named after
Edward Fredkin
Edward Fredkin (born October 2, 1934) is a distinguished career professor at Carnegie Mellon University (CMU), and an early pioneer of digital physics.
Fredkin's primary contributions include work on reversible computing and cellular automata. ...
, is a 3-bit gate that performs a
controlled swap
Swap or SWAP may refer to:
Finance
* Swap (finance), a derivative in which two parties agree to exchange one stream of cash flows against another
* Barter
Science and technology
* Swap (computer programming), exchanging two variables in t ...
. It is
universal
Universal is the adjective for universe.
Universal may also refer to:
Companies
* NBCUniversal, a media and entertainment company
** Universal Animation Studios, an American Animation studio, and a subsidiary of NBCUniversal
** Universal TV, a t ...
for classical computation. It has the useful property that the numbers of 0s and 1s are conserved throughout, which in the
billiard ball model means the same number of balls are output as input.
Other named gates
Notes
{{notelist
References
Quantum computing
Quantum gates