# Functor Application

Functors are factories that allow to access particular specialization implementations of a callable. Q# currently supports two functors; the Adjoint and the Controlled, both of which can be applied to operations that provide the necessary specialization(s).

The Controlled and Adjoint functors commute; if ApplyUnitary is an operation that supports both functors, then there is no difference between Controlled Adjoint ApplyUnitary and Adjoint Controlled ApplyUnitary. Both have the same type and upon invocation execute the implementation defined for the controlled adjoint specialization.

If the operation ApplyUnitary defines a unitary transformation U of the quantum state, Adjoint ApplyUnitary accesses the implementation of U†. The Adjoint functor is its own inverse, since (U†)† = U by definition; i.e. Adjoint Adjoint ApplyUnitary is the same as ApplyUnitary.

The expression Adjoint ApplyUnitary is an operation of the same type as ApplyUnitary; it has the same argument and return type and supports the same functors. Like any operation, it can be invoked with an argument of suitable type. The following expression will apply the adjoint specialization of ApplyUnitary to an argument arg:

Adjoint ApplyUnitary(arg)


## Controlled Functor

For an operation ApplyUnitary that defines a unitary transformation U of the quantum state, Controlled ApplyUnitary accesses the implementation that applies U conditional on all qubits in an array of control qubits being in the |1⟩ state.

The expression Controlled ApplyUnitary is an operation with the same return type and operation characteristics as ApplyUnitary, meaning it supports the same functors. It takes an argument of type (Qubit[], <TIn>), where <TIn> should be replaced with the argument type of ApplyUnitary, taking singleton tuple equivalence into account.

Operation Argument Type Controlled Argument Type
X Qubit (Qubit[], Qubit)
SWAP (Qubit, Qubit) (Qubit[], (Qubit, Qubit))
ApplyQFT LittleEndian (Qubit[], LittleEndian)

Concretely, if cs contains an array of qubits, q1 and q2 are two qubits, and the operation SWAP is as defined here, then the following expression exchanges the state of q1 and q2 if all qubits in cs are in the |1⟩ state:

Controlled SWAP(cs, (q1, q2))


### Discussion

Conditionally applying an operation based on the control qubits being in another state than a zero-state may be achieved by applying the appropriate adjointable transformation to the control qubits before invocation, and applying is inverses after. Conditioning the transformation on all control qubits being in the |0⟩ state, for example, can be achieved by applying the X operation before and after. This can be conveniently expressed using a conjugation. Nonetheless, the verbosity of such a construct may merit additional support for a more compact syntax in the future.