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```
try:
import cirq
except ImportError:
print("installing cirq...")
!pip install --quiet cirq
print("installed cirq.")
import cirq
```

A `Gate`

is an effect that can be applied to a collection of
qubits (objects with a `Qid`

). `Gates`

can be applied
to qubits by calling their `on`

method, or, alternatively
calling the gate on the qubits. The object created by such calls
is an `Operation`

. Alternatively, a `Gate`

can be thought of
as a factory
that, given input qubits, generates an associated
`GateOperation`

object.

## Gates versus operations

The above example shows the first half of a quantum teleportation circuit,
found in many quantum computation textbooks. This example uses three different
gates: a Hadamard (H), controlled-Not (CNOT), and measurement. These are
represented in cirq by `cirq.H`

, `cirq.CNOT`

, and `cirq.measure`

,
respectively.

In this example, a Hadamard is an example of a `Gate`

object that can be applied
in many different circumstances and to many different qubits. Note that the
above example has two instances of an H gate but applied to different qubits.
This is an example of one `Gate`

type with two `Operation`

instances, one
applied to the qubit '|b⟩' and the other applied to qubit '|a⟩'.

Gates can generally be applied to any type of qubit (`NamedQubit`

,
`LineQubit`

, `GridQubit`

, etc. - see Qubits for more details) to create an Operation.

The following example shows how to construct each of these gates and operations.

```
# This examples uses named qubits to remain abstract.
# However, we can also use LineQubits or GridQubits to specify a geometry
a = cirq.NamedQubit('a')
b = cirq.NamedQubit('b')
c = cirq.NamedQubit('c')
# Example Operations, that correspond to the moments above
print(cirq.H(b))
print(cirq.CNOT(b, c))
print(cirq.CNOT(a, b))
print(cirq.H(a))
print(cirq.measure(a,b))
```

H(b) CNOT(b, c) CNOT(a, b) H(a) cirq.MeasurementGate(2, cirq.MeasurementKey(name='a,b'), ())(a, b)

This would create the operations needed to comprise the circuit from the above diagram. The next step would be composing these operations into moments and circuits. For more on those types, see the documentation on Circuits.

## Immutability of Gates and Operations

Gates and Operations in Cirq are considered to be immutable objects. This means that a `cirq.Gate`

or `cirq.Operation`

should not be modified after its creation. If attributes of these objects need to be modified, a new object should be created.

Modifying these objects in-place could cause unexpected behavior. For instance, changing the qubits of an existing `cirq.Operation`

object could cause an existing `cirq.Moment`

that contains this object to have operations with overlapping qubits.

## Other gate features

Most `Gate`

s operate on a specific number of qubits, which can be accessed
by the `num_qubits()`

function. One notable exception is the
`MeasurementGate`

which can be applied to a variable number of qubits.

Most gates also have a unitary matrix representation, which can be accessed
by `cirq.unitary(gate)`

.

Not all `Gate`

s correspond to unitary evolution. They may represent a
probabilistic mixture of unitaries, or a general quantum channel. The component
unitaries and associated probabilities of a mixture can be accessed by
`cirq.mixture(gate)`

. The Kraus operator representation of a channel can be
accessed by `cirq.kraus(gate)`

. Non-unitary gates are often used in the
simulation of noise. See noise documentation for more details.

Many arithmetic operators will work in the expected way when applied to
gates. For instance, `cirq.X**0.5`

represents a square root of X gate.
These can also be applied to Operators for a more compact representation, such
as `cirq.X(q1)**0.5`

will be a square root of X gate applied to the q1
qubit. This functionality depends on the "magic methods" of the gate
being defined (see below for details).

Gates can be converted to a controlled version by using `Gate.controlled()`

.
In general, this returns an instance of a `ControlledGate`

. However, for
certain special cases where the controlled version of the gate is also a
known gate, this returns the instance of that gate.
For instance, `cirq.X.controlled()`

returns a `cirq.CNOT`

gate.
Operations have similar functionality `Operation.controlled_by()`

, such as
`cirq.X(q0).controlled_by(q1)`

.

## Common gates

Cirq supports a number of gates natively, with the opportunity to extend these gates for more advanced use cases.

### Measurement gate

**cirq.MeasurementGate** This is a measurement in the computational basis.
This gate can be applied to a variable number of qubits. The function
`cirq.measure(q0, q1, ...)`

can also be used as a short-hand to create a
`MeasurementGate`

.

### Single qubit gates

Most single-qubit gates can be thought of as rotation around an axis in the Bloch Sphere representation and are usually referred to by their axis of rotation. Some operators use the notation of a 'half-turn' which is defined as a 180 degree (pi radians) rotation around the axis.

**cirq.X / cirq.Y / cirq.Z** The Pauli gates X, Y, and Z which rotate the state around the associated axis by one half-turn.

**cirq.rx(rads)** A rotation about the Pauli 'X' axis in terms of radians. This is equivalent to `exp(-i X rads / 2) = cos(rads/2) I - i sin(rads/2) X`

**cirq.ry(rads)** A rotation about the Pauli 'Y' axis in terms of radians. This is equivalent to `exp(-i Y rads / 2) = cos(rads/2) I - i sin(rads/2) Y`

**cirq.rz(rads)** A rotation about the Pauli 'Z' axis in terms of radians. This is equivalent to `exp(-i Z rads / 2) = cos(rads/2) I - i sin(rads/2) Z`

**cirq.XPowGate(exponent=t)** Rotations about the Pauli `X`

axis, equivalent to `cirq.X**t`

. See `cirq.XPowGate`

for its unitary matrix. Note that this has a global phase of e^{i·π·t/2} versus the traditionally defined rotation matrix, which can be modified by the optional parameter `global_shift`

.

**cirq.YPowGate(exponent=t)** Rotations about the Pauli `Y`

axis, equivalent to `cirq.Y**t`

. See `cirq.YPowGate`

for its unitary matrix. Note that this has a global phase of e^{i·π·t/2} versus the traditionally defined rotation matrix, which can be modified by the optional parameter `global_shift`

.

**cirq.ZPowGate(exponent=t)** Rotations about the Pauli `Z`

axis, equivalent to `cirq.Z**t`

. See `cirq.ZPowGate`

for its unitary matrix. Note that this has a global phase of e^{i·π·t/2} versus the traditionally defined rotation matrix, which can be modified by the optional parameter `global_shift`

.

**cirq.PhasedXPowGate**
This gate is a rotation about an axis in the XY plane of the Bloch sphere.
The `PhasedXPowGate`

takes two parameters, `exponent`

and `phase_exponent`

.
The gate is equivalent to the circuit `───Z^-p───X^t───Z^p───`

where `p`

is the
`phase_exponent`

and `t`

is the `exponent`

.

**cirq.PhasedXZGate**
This gate is like a `cirq.PhasedXPowGate`

above, but it also includes an
extra phase about the z axis. `PhasedXZGate`

takes three parameters, `x_exponent`

, `z_exponent`

, and `axis_phase_exponent`

. The gate is equivalent to the circuit `───Z^(-a)──X^x──Z^a────Z^z───`

where `x`

is the `x_exponent`

, `z`

is the `z_exponent`

, and `a`

is the `axis_phase_exponent`

.

**cirq.H / cirq.HPowGate** The Hadamard gate is a rotation around the X+Z axis.
`cirq.HPowGate(exponent=t)`

is a variable rotation of t turns around this
axis. `cirq.H`

is a π rotation and is equivalent to
`cirq.HPowGate(exponent=1)`

**S** The square root of Z gate, equivalent to `cirq.Z**0.5`

**T** The fourth root of Z gate, equivalent to `cirq.Z**0.25`

.

### Two qubit gates

**cirq.CZ / cirq.CZPowGate** The controlled-Z gate. A two qubit gate that
phases the |11⟩ state. `cirq.CZPowGate(exponent=t)`

is equivalent to
`cirq.CZ**t`

and has a matrix representation of `exp(i pi |11⟩⟨11| t)`

.

**cirq.CNOT / cirq.CNotPowGate** The controlled-X gate. This gate swaps the
|11⟩ and |10⟩ states. `cirq.CNotPowGate(exponent=t)`

is equivalent
to `cirq.CNOT**t`

.

**cirq.SWAP / cirq.SwapPowGate** The swap gate swaps the |01⟩ and |10⟩ states.
`cirq.SWAP**t`

is the same as `cirq.SwapPowGate(exponent = t)`

**cirq.ISWAP / cirq.ISwapPowGate** The iSwap gate swaps the |01⟩ and |10⟩
states and adds a relative phase of i. `cirq.ISWAP**t`

is the same as
`cirq.ISwapPowGate(exponent = t)`

**Parity gates**: The gates `cirq.XX`

, `cirq.YY`

, and `cirq.ZZ`

are equivalent to
performing the equivalent one-qubit Pauli gates on both qubits. The gates
`cirq.XXPowGate`

, `cirq.YYPowGate`

, and `cirq.ZZPowGate`

are the powers of
these gates.

### Other gates

**cirq.MatrixGate**: A gate defined by its unitary matrix in the form of a
numpy ndarray.

**cirq.WaitGate**: This gate does nothing for a specified `cirq.Duration`

amount of time. This is useful for conducting T1 and T2 decay experiments.

**cirq.CCNOT, cirq.CCX, cirq.TOFFOLI, cirq.CCXPowGate**: Three qubit gates
representing a controlled-controlled-X gate and powers of this gate.

**cirq.CCZ, cirq.CCZPowGate**: Three qubit gates representing a
controlled-controlled-Z gate and power of this gate.

**CSWAP, CSwapGate, FREDKIN**: Three qubit gates representing a controlled-SWAP
gate.

**TwoQubitDiagonalGate, ThreeQubitDiagonalGate**: Two and three qubit gates which are diagonal in the computational basis.

**QubitPermutationGate** A gate that permutes a given set of qubits.

## Advanced: create your own gates

If the above gates are not sufficient for your use case, it is fairly simple
to create your own gate. In order to do so, you can define your class and
inherit the `cirq.Gate`

class and define the functionality in your class.
For more information, see this guide.