Applies arithmetic to a target and some inputs.

Inherits From: ArithmeticOperation, Operation

Implements Quirk-specific implicit effects like assuming that the presence of an 'r' input implies modular arithmetic.

In Quirk, modular operations have no effect on values larger than the modulus. This convention is used because unitarity forces some convention on out-of-range values (they cannot simply disappear or raise exceptions), and the simplest is to do nothing. This call handles ensuring that happens, and ensuring the new target register value is normalized modulo the modulus.




tags Returns a tuple of the operation's tags.
untagged Returns the underlying operation without any tags.



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Returns the result of the operation 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 operation, 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 operation 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 ArithmeticOperation class:

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


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Returns a controlled version of this operation. If no control_qubits are specified, returns self.

control_qubits Qubits to control the operation by. Required.
control_values For which control qubit values to apply the operation. A sequence of the same length as control_qubits where each entry is an integer (or set of integers) corresponding to the qubit value (or set of possible values) where that control is enabled. When all controls are enabled, the operation is applied. If unspecified, control values default to 1.


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The data acted upon by the arithmetic operation.

Each register in the list can either be a classical constant (an int), or else a list of qubits/qudits (a List[cirq.Qid]). Registers that are set to a classical constant must not be mutated by the arithmetic operation (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 operation will act upon.


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Returns the same operation, but with different qubits.

func The function to use to turn each current qubit into a desired new qubit.

The receiving operation but with qubits transformed by the given function.


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Raises an exception if the qubits don't match this operation's qid shape.

Call this method from a subclass's with_qubits method.

qubits The new qids for the operation.

ValueError The operation had qids that don't match it's qid shape.


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Returns the same operation, but applied to different qubits.

new_qubits The new qubits to apply the operation to. The order must exactly match the order of qubits returned from the operation's qubits property.


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Returns the same operation targeting different registers.

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

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


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Creates a new TaggedOperation, with this op and the specified tags.

This method can be used to attach meta-data to specific operations without affecting their functionality. The intended usage is to attach classes intended for this purpose or strings to mark operations for specific usage that will be recognized by consumers. Specific examples include ignoring this operation in optimization passes, hardware-specific functionality, or circuit diagram customizability.

Tags can be a list of any type of object that is useful to identify this operation as long as the type is hashable. If you wish the resulting operation to be eventually serialized into JSON, you should also restrict the operation to be JSON serializable.

new_tags The tags to wrap this operation in.


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