# 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))


## Magic Methods¶

A class that implements Gate can be applied to qubits to produce an Operation. In order to support functionality beyond that basic task, it is necessary to implement several magic methods.

Standard magic methods in python are __add__, __eq__, and __len__. Cirq defines several additional magic methods, for functionality such as parameterization, diagramming, and simulation. For example, if a gate specifies a _unitary_ method that returns a matrix for the gate, then simulators will be able to simulate applying the gate. Or, if a gate specifies a __pow__ method that works for an exponent of -1, then cirq.inverse will start to work on lists including the gate.

We describe some magic methods below.

### cirq.inverse and __pow__¶

Gates and operations are considered to be invertable when they implement a __pow__ method that returns a result besides NotImplemented for an exponent of -1. This inverse can be accessed either directly as value**-1, or via the utility method cirq.inverse(value). If you are sure that value has an inverse, saying value**-1 is more convenient than saying cirq.inverse(value). cirq.inverse is for cases where you aren’t sure if value is invertable, or where value might be a sequence of invertible operations.

cirq.inverse has a default parameter used as a fallback when value isn’t invertable. For example, cirq.inverse(value, default=None) returns the inverse of value, or else returns None if value isn’t invertable. (If no default is specified and value isn’t invertible, a TypeError is raised.)

When you give cirq.inverse a list, or any other kind of iterable thing, it will return a sequence of operations that (if run in order) undoes the operations of the original sequence (if run in order). Basically, the items of the list are individually inverted and returned in reverse order. For example, the expression cirq.inverse([cirq.S(b), cirq.CNOT(a, b)]) will return the tuple (cirq.CNOT(a, b), cirq.S(b)**-1).

Gates and operations can also return values beside NotImplemented from their __pow__ method for exponents besides -1. This pattern is used often by Cirq. For example, the square root of X gate can be created by raising cirq.X to 0.5:

import cirq
print(cirq.unitary(cirq.X))
# prints
# [[0.+0.j 1.+0.j]
#  [1.+0.j 0.+0.j]]

sqrt_x = cirq.X**0.5
print(cirq.unitary(sqrt_x))
# 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.

### cirq.unitary and def _unitary_¶

When objects can be described by a unitary matrix, they let Cirq know by implementing the _unitary_ method. This method should return a numpy ndarray matrix and this array should be the unitary matrix corresponding to the object. The method may also return NotImplemented, in which case cirq.unitary behaves as if the method is not implemented.

### cirq.decompose and def _decompose_¶

A cirq.Operation indicates that it can be broken down into smaller simpler operations by implementing a def _decompose_(self): method. Code that doesn’t understand a particular operation can call cirq.decompose_once or cirq.decompose on that operation in order to get a set of simpler operations that it does understand.

One useful thing about cirq.decompose is that it will decompose recursively, until only operations meeting a keep predicate remain. You can also give an intercepting_decomposer to cirq.decompose, which will take priority over operations’ own decompositions.

For cirq.Gates, the decompose method is slightly different; it takes qubits: def _decompose_(self, qubits). Callers who know the qubits that the gate is being applied to will use cirq.decompose_once_with_qubits to trigger this method.

### _circuit_diagram_info_(self, args) and cirq.circuit_diagram_info(val, [args], [default])¶

Circuit diagrams are useful for visualizing the structure of a Circuit. Gates can specify compact representations to use in diagrams by implementing a _circuit_diagram_info_ method. For example, this is why SWAP gates are shown as linked ‘×’ characters in diagrams.

The _circuit_diagram_info_ method takes an args parameter of type cirq.CircuitDiagramInfoArgs and returns either a string (typically the gate’s name), a sequence of strings (a label to use on each qubit targeted by the gate), or an instance of cirq.CircuitDiagramInfo (which can specify more advanced properties such as exponents and will expand in the future).

You can query the circuit diagram info of a value by passing it into cirq.circuit_diagram_info.

## Xmon gates¶

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 native Xmon gates are

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.Z / cirq.Rz Rotations about the Pauli Z axis. The matrix of cirq.Z**t is exp(i pi |1><1| t) whereas the matrix of cirq.Rz(θ) is exp(-i Z θ/2). Note that in quantum computing hardware, this gate is often implemented in the classical control hardware as a phase change on later operations, instead of as a physical modification applied to the qubits. (TODO: explain this in more detail)

cirq.CZ The controlled-Z gate. A two qubit gate that phases the |11> state. The matrix of cirq.CZ**t is exp(i pi |11><11| t).

cirq.MeasurementGate This is a single qubit measurement in the computational basis.

## Other Common Gates¶

Cirq comes with a number of common named gates:

CNOT the controlled-X gate

SWAP the swap gate