# Gates¶

A Gate is an operation that can be applied to a collection of qubits (objects with a QubitId). 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.

from cirq.ops import CNOT
from cirq.devices import GridQubit
q0, q1 = (GridQubit(0, 0), GridQubit(0, 1))
print(CNOT.on(q0, q1))
print(CNOT(q0, q1))
# prints
# CNOT((0, 0), (0, 1))
# CNOT((0, 0), (0, 1))


## Gate Features¶

The raw Gate class itself simply describes that a Gate can be applied to qubits to produce an Operation. We then use marker classes for Gates indicated what additional features a Gate has.

For example, one feature is ReversibleEffect. A Gate that inherits this class is required to implement the method inverse which returns the inverse gate. Algorithms that operate on gates can use isinstance(gate, ReversibleEffect) to determine whether gates implements inverse method, and then use it. (Note that, even if the gate is not reversible, the algorithm may have been given an Extension with a cast from the gate to ReversibleEffect. See the extensions documentation for more information.)

We describe some gate features below.

### ReversibleEffect, SelfInverseGate¶

As described above, a ReversibleEffect implements the inverse method (returns a gatethat is the inverse of the receiving gate). SelfInverseGate is a Gate for which the inverse is simply the Gate itself (so the feature SelfInverseGate doesn’t need to implement inverse, it already just returns self).

### ExtrapolatableEffect¶

Represents an effect which can be scaled continuously up or down, or negated. Implementing gates and operations implement the extrapolate_effect method, which takes a single float parameter factor. This factor is the amount to scale the gate by. Roughly, one can think about this as applying the effect factor times. There is some subtlety in this definition since, for example, there are often two ways to define the square root of a gate. It is up to the implementation to define which root is chosen.

The primary use of ExtrapolatableEffect is to allow easy powering of gates. That is one can define for these gates a power

import numpy as np
from cirq.ops import X
print(np.around(X.matrix()))
# prints
# [[0.+0.j 1.+0.j]
#  [1.+0.j 0.+0.j]]

sqrt_x = X**0.5
print(sqrt_x.matrix())
# prints
# [[0.5+0.5j 0.5-0.5j]
#  [0.5-0.5j 0.5+0.5j]]


The Pauli gates included in Cirq use the convention Z**0.5 ≡ S ≡ np.diag(1, i), Z**-0.5 ≡ S**-1, X**0.5 ≡ H·S·H, and the square root of Y is inferred via the right hand rule. Note that it is often the case that (g**a)**b != g**(a * b), due to the intermediate values normalizing rotation angles into a canonical range.

### KnownMatrix¶

We’ve seen this above. These are Gate or Operation instances which implement the matrix method. This returns a numpy ndarray matrix which is the unitary gate for the gate/operation.

### CompositeGate and CompositeOperation¶

A CompositeGate is a gate which consists of multiple gates that can be applied to a given set of qubits. This is a manner in which one can decompose one gate into multiple gates. In particular CompositeGates implement the method default_decompose which acts on a sequence of qubits, and returns a list of the operations acting on these qubits for the constituents gates.

One thing about CompositeGates is that sometimes you want to modify the decomposition. Algorithms that allow this can take an Extension which allows for overriding the CompositeGate. An example of this is for in Simulators where an optional extension can be supplied that can be used to override the CompositeGate.

A CompositeOperation is just like a CompositeGate, except it already knows the qubits it should be applied to.

### TextDiagrammable¶

Text diagrams of Circuits are actually quite useful for visualizing the moment structure of a Circuit. Gates that implement this feature can specify compact representations to use in the diagram (e.g. ‘×’ instead of ‘SWAP’).

## XmonGates¶

Google’s Xmon devices support a specific gate set. Gates in this gate set operate on GridQubits, which are qubits arranged on a square grid and which have an x and y coordinate.

The XmonGates are

ExpWGate This gate is a rotation about a combination of a Pauli X and Pauli Y gates. The ExpWGate takes two parameters, half_turns and axis_half_turns. The later describes the angle of the operator that is being rotated about in the XY plane. In particular if we define W(theta) = cos(pi theta) X + sin (pi theta) Y then axis_half_turns is theta. And the full gate is exp(-i pi half_turns W(axis_half_turns) / 2).

ExpZGate This gate is a rotation about the Pauli Z axis. The gate is exp(-i pi Z half_turns / 2) where half_turns is the supplied parameter. Note that in quantum computing hardware, this gate is often compiled out of the circuit (TODO: explain this in more detail)

Exp11Gate This is a two qubit gate and is a rotation about the |11><11| projector. It takes a single parameter half_turns and is the gate exp(i pi |11><11| half_turns).

XmonMeasurementGate This is a single qubit measurement in the computational basis.

## CommonGates¶

XmonGates are hardware specific. In addition Cirq has a number of more commonly named gates that are then implemented as XmonGates via an extension or composite gates. Some of these are our old friends:

RotXGate, RotYGate, RotZGate, Rot11Gate. These are gates corresponding to the Pauli rotations or (in the case of Rot11Gate a two qubit rotation).

Our old friends the Paulis: X, Y, and Z. Some other two qubit fiends, CZ the controlled-Z gate, CNOT the controlled-X gate, and SWAP the swap gate. As well as some other Clifford friends, H and S, and our error correcting friend T.

TODO: describe these in more detail.

TODO