Editing Quantum logic gate (section)

=== Serially wired gates ===
[[File:Serially_wired_quantum_logic_gates.png|thumb|right|upright=2|Two gates ''Y'' and ''X'' in series. The order in which they appear on the wire is reversed when multiplying them together.]]Assume that we have two gates ''A'' and ''B'' that both act on <math>n</math> qubits. When ''B'' is put after ''A'' in a series circuit, then the effect of the two gates can be described as a single gate ''C''.

: <math>C = B \cdot A</math>

where <math>\cdot</math> is [[Matrix multiplication#Definition|matrix multiplication]]. The resulting gate ''C'' will have the same dimensions as ''A'' and ''B''. The order in which the gates would appear in a circuit diagram is reversed when multiplying them together.{{r|Nielsen-Chuang|p=17–18,22–23,62–64}}{{r|Yanofsky-Mannucci|p=147–169}}

For example, putting the Pauli ''X'' gate after the Pauli ''Y'' gate, both of which act on a single qubit, can be described as a single combined gate ''C'':

: <math>C = X \cdot Y = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \cdot \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} = iZ</math>

The product symbol (<math>\cdot</math>) is often omitted.

==== Exponents of quantum gates ====
All [[real number|real]] exponents of [[unitary matrix|unitary matrices]] are also unitary matrices, and all quantum gates are unitary matrices.

Positive integer exponents are equivalent to sequences of serially wired gates (e.g. {{nowrap|<math>X^3 = X \cdot X \cdot X</math>),}} and the real exponents is a generalization of the series circuit. For example, <math>X^\pi</math> and <math>\sqrt{X}=X^{1/2}</math> are both valid quantum gates.

<math>U^0=I</math> for any unitary matrix <math>U</math>. The [[identity matrix]] (<math>I</math>) behaves like a [[NOP (code)|NOP]]<ref>{{cite web|url=https://docs.microsoft.com/en-us/qsharp/api/qsharp/microsoft.quantum.intrinsic.i|title=I operation|website=docs.microsoft.com|date=28 July 2023 }}</ref><ref>{{cite web|url=https://qiskit.org/documentation/stubs/qiskit.circuit.library.IGate.html#qiskit.circuit.library.IGate|title=IGate|website=qiskit.org}} [[Qiskit]] online documentation.</ref> and can be represented as bare wire in quantum circuits, or not shown at all.

All gates are unitary matrices, so that <math>U^\dagger U = UU^\dagger = I</math> and {{nowrap|<math>U^\dagger = U^{-1}</math>,}} where <math>\dagger</math> is the [[conjugate transpose]]. This means that negative exponents of gates are [[#Unitary inversion of gates|unitary inverses]] of their positively exponentiated counterparts: {{nowrap|<math>U^{-n} = (U^n)^{\dagger}</math>.}} For example, some negative exponents of the [[#Phase gate|phase shift gates]] are <math>T^{-1}=T^{\dagger}</math> and {{nowrap|<math>T^{-2}=(T^2)^{\dagger}=S^{\dagger}</math>.}}

Note that for a [[Hermitian matrix]] <math>H^\dagger=H,</math> and because of unitarity, <math>HH^\dagger=I,</math> so <math>H^2 = I</math> for all Hermitian gates. They are [[Involutory matrix|involutory]]. Examples of Hermitian gates are the [[#X|Pauli gates]], [[#Hadamard gate|Hadamard]], [[#Controlled gates|CNOT]], [[#Swap gate|SWAP]] and [[#Toffoli|Toffoli]]. Each Hermitian unitary matrix <math>H</math> [[Sylvester's formula#Special case|has the property]] that <math>e^{i\theta H}=(\cos \theta)I+(i\sin \theta) H</math> where <math>H=e^{i\frac{\pi}{2}(I-H)}=e^{-i\frac{\pi}{2}(I-H)}.</math>

The exponent of a gate is a multiple of the duration of time that the [[#Relation to the time evolution operator|time evolution operator]] is applied to a quantum state. E.g. in a [[spin qubit quantum computer]] the <math>\sqrt{\mathrm{SWAP}}</math> gate could be realized via [[exchange interaction]] on the [[Spin (physics)|spin]] of two [[electron]]s for half the duration of a full exchange interaction.<ref name="Loss-DiVincenzo"/>