Custom gates

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

import numpy as np

Standard gates such as Pauli gates and CNOTs are defined in cirq.ops as described here. To use a unitary which is not a standard gate in a circuit, one can create a custom gate as described in this guide.

General pattern

Gates are classes in Cirq. To define custom gates, we inherit from a base gate class and define a few methods.

The general pattern is to:

  • Inherit from cirq.Gate.
  • Define one of the _num_qubits_ or _qid_shape_ methods.
  • Define one of the _unitary_ or _decompose_ methods.

We demonstrate these patterns via the following examples.

From a unitary

One can create a custom Cirq gate from a unitary matrix in the following manner. Here, we define a gate which corresponds to the unitary

\[ U = \frac{1}{\sqrt{2} } \left[ \begin{matrix} 1 & 1 \\ -1 & 1 \end{matrix} \right] . \]

"""Define a custom single-qubit gate."""
class MyGate(cirq.Gate):
    def __init__(self):
        super(MyGate, self)

    def _num_qubits_(self):
        return 1

    def _unitary_(self):
        return np.array([
            [1.0,  1.0],
            [-1.0, 1.0]
        ]) / np.sqrt(2)

    def _circuit_diagram_info_(self, args):
        return "G"

my_gate = MyGate()

In this example, the _num_qubits_ method tells Cirq that this gate acts on a single-qubit, and the _unitary_ method defines the unitary of the gate. The _circuit_diagram_info_ method tells Cirq how to display the gate in a circuit, as we will see below.

Once this gate is defined, it can be used like any standard gate in Cirq.

"""Use the custom gate in a circuit."""
circ = cirq.Circuit(

print("Circuit with custom gates:")
Circuit with custom gates:
0: ───G───

When we print the circuit, we see the symbol we specified in the _circuit_diagram_info_ method.

Circuits with custom gates can be simulated in the same manner as circuits with standard gates.

"""Simulate a circuit with a custom gate."""
sim = cirq.Simulator()

res = sim.simulate(circ)
measurements: (no measurements)

qubits: (cirq.LineQubit(0),)
output vector: 0.707|0⟩ - 0.707|1⟩

output vector: |⟩
"""Define a custom two-qubit gate."""
class AnotherGate(cirq.Gate):
    def __init__(self):
        super(AnotherGate, self)

    def _num_qubits_(self):
        return 2

    def _unitary_(self):
        return np.array([
            [1.0, -1.0, 0.0,  0.0],
            [0.0,  0.0, 1.0,  1.0],
            [1.0,  1.0, 0.0,  0.0],
            [0.0,  0.0, 1.0, -1.0]
        ]) / np.sqrt(2)

    def _circuit_diagram_info_(self, args):
        return "Top wire symbol", "Bottom wire symbol"

this_gate = AnotherGate()

Here, the _circuit_diagram_info_ method returns two symbols (one for each wire) since it is a two-qubit gate.

"""Use the custom two-qubit gate in a circuit."""
circ = cirq.Circuit(

print("Circuit with custom two-qubit gate:")
Circuit with custom two-qubit gate:
0: ───Top wire symbol──────
1: ───Bottom wire symbol───

As above, this circuit can also be simulated in the expected way.

With parameters

Custom gates can be defined and used with parameters. For example, to define the gate

\[ R(\theta) = \left[ \begin{matrix} \cos \theta & \sin \theta \\ \sin \theta & - \cos \theta \end{matrix} \right], \]

we can do the following.

"""Define a custom gate with a parameter."""
class RotationGate(cirq.Gate):
    def __init__(self, theta):
        super(RotationGate, self)
        self.theta = theta

    def _num_qubits_(self):
        return 1

    def _unitary_(self):
        return np.array([
            [np.cos(self.theta), np.sin(self.theta)],
            [np.sin(self.theta), -np.cos(self.theta)]
        ]) / np.sqrt(2)

    def _circuit_diagram_info_(self, args):
        return f"R({self.theta})"

This gate can be used in a circuit as shown below.

"""Use the custom gate in a circuit."""
circ = cirq.Circuit(

print("Circuit with a custom rotation gate:")
Circuit with a custom rotation gate:
0: ───R(0.1)───

From a known decomposition

Custom gates can also be defined from a known decomposition (of gates). This is useful, for example, when groups of gates appear repeatedly in a circuit, or when a standard decomposition of a gate into primitive gates is known.

We show an example below of a custom swap gate defined from a known decomposition of three CNOT gates.

class MySwap(cirq.Gate):
    def __init__(self):
        super(MySwap, self)

    def _num_qubits_(self):
        return 2

    def _decompose_(self, qubits):
        a, b = qubits
        yield cirq.CNOT(a, b)
        yield cirq.CNOT(b, a)
        yield cirq.CNOT(a, b)

    def _circuit_diagram_info_(self, args):
        return ["CustomSWAP"] * self.num_qubits()

my_swap = MySwap()

The _decompose_ method yields the operations which implement the custom gate. (One can also return a list of operations instead of a generator.)

When we use this gate in a circuit, the individual gates in the decomposition do not appear in the circuit. Instead, the _circuit_diagram_info_ appears in the circuit. As mentioned, this can be useful for interpreting circuits at a higher level than individual (primitive) gates.

"""Use the custom gate in a circuit."""
qreg = cirq.LineQubit.range(2)
circ = cirq.Circuit(

0: ───X───CustomSWAP───
1: ───────CustomSWAP───

We can simulate this circuit and verify it indeed swaps the qubits.

"""Simulate the circuit."""
measurements: (no measurements)

qubits: (cirq.LineQubit(0), cirq.LineQubit(1))
output vector: |01⟩

output vector: |⟩

More on magic methods and protocols

As mentioned, methods such as _unitary_ which we have seen are known as "magic methods." 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.

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. Instead, use features available in the transformer API.

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 invertible 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 invertible, or where value might be a sequence of invertible operations.

cirq.inverse has a default parameter used as a fallback when value isn't invertible. For example, cirq.inverse(value, default=None) returns the inverse of value, or else returns None if value isn't invertible. (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:

# prints
# [[0.+0.j 1.+0.j]
#  [1.+0.j 0.+0.j]]

sqrt_x = cirq.X**0.5
# 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.