# Gates¶

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) to create an Operation. However, depending on the application, you may prefer a specific type of qubit. For instance, Google devices generally use GridQubits. Other devices may have connectivity constraints that further restrict the set of qubits that can be used, especially in multi- qubit operations.

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

[2]:

import cirq

# 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))
# prints
# H(b)
# CNOT(b, c)
# CNOT(a, b)
# H(a)
# cirq.MeasurementGate(2, 'a,b', ())(a, b)

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

## Other gate features¶

Most Gates 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 Gates 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.channel(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 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 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 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.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 the controlled-controlled-X gates.

cirq.CCZ, cirq.CCZPowGate: Three qubit gates representing a controlled-controlled-Z gate.

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

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.

Much of cirq relies on “magic methods”, which are methods prefixed with one or two underscores and used by cirq’s protocols or built-in python methods. For instance, python translates cirq.Z**0.25 into cirq.Z.__pow__(0.25). Other uses are specific to cirq and are found in the protocols subdirectory. They are defined below.

At minimum, you will need to define either the _num_qubits_ or _qid_shape_ magic method to define the number of qubits (or qudits) used in the gate. For convenience one can use the SingleQubitGate, TwoQubitGate, and ThreeQubitGate classes for these common gate sizes.

### Magic Methods¶

#### Standard python magic methods¶

There are many standard magic methods in python. Here are a few of the most important ones used in cirq: * __str__ for user-friendly string output and __repr__ is the python-friendly string output, meaning that eval(repr(y))==y should always be true. * __eq__ and __hash__ which define whether objects are equal or not. You can also use cirq.value.value_equality for objects that have a small list of sub-values that can be compared for equality. * Arithmetic functions such as __pow__, __mul__, __add__ define the action of **, *, and + respectively.

#### cirq.num_qubits and def _num_qubits_¶

A Gate must implement the _num_qubits_ (or _qid_shape_) method. This method returns an integer and is used by cirq.num_qubits to determine how many qubits this gate operates on.

#### cirq.qid_shape and def _qid_shape_¶

A qudit gate or operation must implement the _qid_shape_ method that returns a tuple of integers. This method is used to determine how many qudits the gate or operation operates on and what dimension each qudit must be. If only the _num_qubits_ method is implemented, the object is assumed to operate only on qubits. Callers can query the qid shape of the object by calling cirq.qid_shape on it. See qudit documentation for more information.

#### 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:

[3]:

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]]

[[0.+0.j 1.+0.j]
[1.+0.j 0.+0.j]]
[[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.