# Gates¶

A Gate is an operation 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.

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))


Gates operate on a specific number of qubit and classes that implement Gate must supply the num_qubits method. For convenience one can use the SingleQubitGate, TwoQubitGate, and ThreeQubitGate classes for these common gate sizes.

The most common type of Gate is one that corresponds to applying a unitary evolution on the qubits that the gate acts on. Gates can also correspond to noisy evolution on the qubits. This version of a gate is not used when sending the circuit to a quantum computer for execution, but it can be used with various simulators. See noise documentation .

## 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.unitary and def _unitary_¶

When an object can be described by a unitary matrix, it can expose that unitary matrix by implementing a _unitary_(self) -> np.ndarray method. Callers can query whether or not an object has a unitary matrix by calling cirq.unitary on it. The _unitary_ method may also return NotImplemented, in which case cirq.unitary behaves as if the method is not implemented.

### cirq.decompose and def _decompose_¶

Operations and gates can be defined in terms of other operations by implementing a _decompose_ method that returns those other operations. Operations implement _decompose_(self) whereas gates implement _decompose_(self, qubits) (since gates don’t know their qubits ahead of time).

The main requirements on the output of _decompose_ methods are:

1. DO NOT CREATE CYCLES. The cirq.decompose method will iterative decompose until it finds values satisfying a keep predicate. Cycles cause it to enter an infinite loop.
2. Head towards operations defined by Cirq, because these operations have good decomposition methods that terminate in single-qubit and two qubit gates. These gates can be understood by the simulator, optimizers, and other code.
3. All that matters is functional equivalence. Don’t worry about staying within or reaching a particular gate set; it’s too hard to predict what the caller will want. Gate-set-aware decomposition is useful, but this is not the protocol that does that. Gate-set-aware decomposition may be added in the future, but doesn’t exist within Cirq at the moment.

For example, cirq.CCZ decomposes into a series of cirq.CNOT and cirq.T operations. This allows code that doesn’t understand three-qubit operation to work with cirq.CCZ; by decomposing it into operations they do understand. As another example, cirq.TOFFOLI decomposes into a cirq.H followed by a cirq.CCZ followed by a cirq.H. Although the output contains a three qubit operation (the CCZ), that operation can be decomposed into two qubit and one qubit operations. So code that doesn’t understand three qubit operations can deal with Toffolis by decomposing them, and then decomposing the CCZs that result from the initial decomposition.

In general, decomposition-aware code consuming operations is expected to recursively decompose unknown operations until the code either hits operations it understands or hits a dead end where no more decomposition is possible. The cirq.decompose method implements logic for performing exactly this kind of recursive decomposition. Callers specify a keep predicate, and optionally specify intercepting and fallback decomposers, and then cirq.decompose will repeatedly decompose whatever operations it was given until the operations satisfy the given keep. If cirq.decompose hits a dead end, it raises an error.

Cirq doesn’t make any guarantees about the “target gate set” decomposition is heading towards. cirq.decompose is not a method Decompositions within Cirq happen to converge towards X, Y, Z, CZ, PhasedX, specified-matrix gates, and others. But this set will vary from release to release, and so it is important for consumers of decompositions to look for generic properties of gates, such as “two qubit gate with a unitary matrix”, instead of specific gate types such as CZ gates.

### 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.

### _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