A helper gate for implementing reversible classical arithmetic.

Inherits From: Gate

Used in the notebooks

Used in the tutorials

Child classes must override the registers, with_registers, and apply methods.

This class handles the details of ensuring that the scaling of implementing the gate is O(2^n) instead of O(4^n) where n is the number of qubits being acted on, by implementing an _apply_unitary_ function in terms of the registers and the apply function of the child class.


class Add(cirq.ArithmeticGate):
    def __init__(
        target_register: '[int, Sequence[int]]',
        input_register: 'Union[int, Sequence[int]]',
        self.target_register = target_register
        self.input_register = input_register

    def registers(self) -> 'Sequence[Union[int, Sequence[int]]]':
        return self.target_register, self.input_register

    def with_registers(
        self, *new_registers: 'Union[int, Sequence[int]]'
    ) -> 'Add':
        return Add(*new_registers)

    def apply(self, *register_values: int) -> 'Union[int, Iterable[int]]':
        return sum(register_values)
    Add(target_register=[2, 2],
array([[0, 0, 0, 1],
       [1, 0, 0, 0],
       [0, 1, 0, 0],
       [0, 0, 1, 0]], dtype=int32)
c = cirq.Circuit(
   cirq.measure(*cirq.LineQubit.range(4, 8), key='before_in'),
   cirq.measure(*cirq.LineQubit.range(4), key='before_out'),

   Add(target_register=[2] * 4,
       input_register=[2] * 4).on(*cirq.LineQubit.range(8)),

   cirq.measure(*cirq.LineQubit.range(4, 8), key='after_in'),
   cirq.measure(*cirq.LineQubit.range(4), key='after_out'),
   before_in  before_out  after_in  after_out
0          2           3         2          5



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Returns the result of the gate operating on classical values.

For example, an addition takes two values (the target and the source), adds the source into the target, then returns the target and source as the new register values.

The apply method is permitted to be sloppy in three ways:

  1. The apply method is permitted to return values that have more bits than the registers they will be stored into. The extra bits are simply dropped. For example, if the value 5 is returned for a 2 qubit register then 5 % 22 = 1 will be used instead. Negative values are also permitted. For example, for a 3 qubit register the value -2 becomes -2 % 23 = 6.
  2. When the value of the last k registers is not changed by the gate, the apply method is permitted to omit these values from the result. That is to say, when the length of the output is less than the length of the input, it is padded up to the intended length by copying from the same position in the input.
  3. When only the first register's value changes, the apply method is permitted to return an int instead of a sequence of ints.

The apply method must be reversible. Otherwise the gate will not be unitary, and incorrect behavior will result.

A fully detailed adder:

def apply(self, target, offset):
    return (target + offset) % 2**len(self.target_register), offset

The same adder, with less boilerplate due to the details being handled by the ArithmeticGate class:

def apply(self, target, offset):
    return target + offset


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Returns a controlled version of this gate. If no arguments are specified, defaults to a single qubit control.

num_controls Total number of control qubits.
control_values Which control computational basis state to apply the sub gate. A sequence of length num_controls where each entry is an integer (or set of integers) corresponding to the computational basis state (or set of possible values) where that control is enabled. When all controls are enabled, the sub gate is applied. If unspecified, control values default to 1.
control_qid_shape The qid shape of the controls. A tuple of the expected dimension of each control qid. Defaults to (2,) * num_controls. Specify this argument when using qudits.

A cirq.Gate representing self controlled by the given control values and qubits. This is a cirq.ControlledGate in the base implementation, but subclasses may return a different gate type.


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The number of qubits this gate acts on.


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Returns an application of this gate to the given qubits.

*qubits The collection of qubits to potentially apply the gate to.

Returns: a cirq.Operation which is this gate applied to the given qubits.


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Returns a list of operations applying the gate to all targets.

*targets The qubits to apply this gate to. For single-qubit gates this can be provided as varargs or a combination of nested iterables. For multi-qubit gates this must be provided as an Iterable[Sequence[Qid]], where each sequence has num_qubits qubits.

Operations applying this gate to the target qubits.

ValueError If targets are not instances of Qid or Iterable[Qid]. If the gate qubit number is incompatible.
TypeError If a single target is supplied and it is not iterable.


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The data acted upon by the arithmetic gate.

Each register in the list can either be a classical constant (an int), or else a list of qubit/qudit dimensions. Registers that are set to a classical constant must not be mutated by the arithmetic gate (their value must remain fixed when passed to apply).

Registers are big endian. The first qubit is the most significant, the last qubit is the 1s qubit, the before last qubit is the 2s qubit, etc.

A list of constants and qubit groups that the gate will act upon.


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Checks if this gate can be applied to the given qubits.

By default checks that:

  • inputs are of type Qid
  • len(qubits) == num_qubits()
  • qubit_i.dimension == qid_shape[i] for all qubits

Child classes can override. The child implementation should call super().validate_args(qubits) then do custom checks.

qubits The sequence of qubits to potentially apply the gate to.

ValueError The gate can't be applied to the qubits.


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Creates a probabilistic channel with this gate.

probability floating point value between 0 and 1, giving the probability this gate is applied.

cirq.RandomGateChannel that applies self with probability probability and the identity with probability 1-p.


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Returns the same fate targeting different registers.

*new_registers The new values that should be returned by the registers method.

An instance of the same kind of gate, but acting on different registers.


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Returns a LinearCombinationOfGates with this gate.

coefficient number coefficient to use in the resulting cirq.LinearCombinationOfGates object.

cirq.LinearCombinationOfGates containing self with a coefficient of coefficient.


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Call self as a function.


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