Editing Quantum logic gate (section)

=== Unitary inversion of gates ===
[[File:The_unitary_inverse_of_hadamard-cnot.png|thumb|right|upright=0.6|'''Example:''' The unitary inverse of the Hadamard-CNOT product. The three gates {{nowrap|<math>H</math>,}} <math>I</math> and <math>\mathrm{CNOT}</math> are their own unitary inverses.]]Because all quantum logical gates are [[reversible computing|reversible]], any composition of multiple gates is also reversible. All products and tensor products (i.e. [[#Serially wired gates|series]] and [[#Parallel gates|parallel]] combinations) of [[unitary matrix|unitary matrices]] are also unitary matrices. This means that it is possible to construct an inverse of all algorithms and functions, as long as they contain only gates.

Initialization, measurement, [[Input/Output|I/O]] and spontaneous [[quantum decoherence|decoherence]] are [[Side effect (computer science)|side effects]] in quantum computers. Gates however are [[pure function|purely functional]] and [[bijective]].

If <math>U</math> is a [[unitary matrix]], then <math>U^\dagger U = UU^\dagger = I</math> and {{nowrap|<math>U^\dagger = U^{-1}</math>.}} The dagger (<math>\dagger</math>) denotes the [[conjugate transpose]]. It is also called the [[Hermitian adjoint]].

If a function <math>F</math> is a product of <math>m</math> gates, {{nowrap|<math>F = A_1 \cdot A_2 \cdot \dots \cdot A_m</math>,}} the unitary inverse of the function <math>F^\dagger</math> can be constructed:

Because <math>(UV)^\dagger = V^\dagger U^\dagger</math> we have, after repeated application on itself

:<math>F^\dagger = \left(\prod_{i=1}^{m} A_i\right)^\dagger = \prod_{i=m}^{1} A^\dagger_{i} = A_m^\dagger \cdot \dots \cdot A_2^\dagger \cdot A_1^\dagger</math>

Similarly if the function <math>G</math> consists of two gates <math>A</math> and <math>B</math> in parallel, then <math>G=A\otimes B</math> and {{nowrap|<math>G^\dagger = (A \otimes B)^\dagger = A^\dagger \otimes B^\dagger</math>.}}

Gates that are their own unitary inverses are called [[Hermitian matrix|Hermitian]] or [[self-adjoint operator]]s. Some elementary gates such as the [[#Hadamard|Hadamard]] (''H'') and the [[Pauli matrices|Pauli gates]] (''I'', ''X'', ''Y'', ''Z'') are Hermitian operators, while others like the [[#Phase shift|phase shift]] (''S'', ''T'', ''P'', [[List of quantum logic gates#Relative phase gates|CPhase]]) gates generally are not.

For example, an algorithm for addition can be used for subtraction, if it is being "run in reverse", as its unitary inverse. The [[Quantum Fourier transform#Unitarity|inverse quantum Fourier transform]] is the unitary inverse. Unitary inverses can also be used for [[uncomputation]]. Programming languages for quantum computers, such as [[Microsoft]]'s [[Q Sharp|Q#]],<ref name="adjoint-controlled-qsharp">[https://docs.microsoft.com/en-us/quantum/user-guide/using-qsharp/operations-functions?view=qsharp-preview#controlled-and-adjoint-operations Operations and Functions (Q# documentation)]</ref> Bernhard Ömer's [[Quantum Computation Language|QCL]],{{r|Oemer|page=61}} and [[IBM]]'s [[Qiskit]],<ref>{{cite web|url=https://docs.quantum.ibm.com/api/qiskit/qiskit.circuit.library.UnitaryGate#adjoint|title=UnitaryGate § UnitaryGate adjoint()|website=docs.quantum.ibm.com}}</ref> contain function inversion as programming concepts.