Editing Quantum logic gate (section)

== Representation ==
[[File:Bloch_sphere.svg|thumb|upright=1|Single [[qubit]] states that are not [[Quantum entanglement|entangled]] and lack [[List of quantum logic gates#Identity gate and global phase|global phase]] can be represented as points on the surface of the '''[[Bloch sphere]]''', written as <math>|\psi\rangle = \cos\left(\theta /2\right)|0 \rangle + e^{i\varphi}\sin\left(\theta /2\right) |1\rangle.</math><br>Rotations about the {{math|x, y, z}} axes of the Bloch sphere are represented by the [[List of quantum logic gates#Rotation operator gates|rotation operator gates]].]]
Quantum logic gates are represented by [[unitary matrix|unitary matrices]]. A gate that acts on <math>n</math> [[qubit]]s (a [[quantum register|register]]) is represented by a <math>2^n \times 2^n</math> unitary matrix, and the [[Set (mathematics)|set]] of all such gates with the group operation of [[matrix multiplication]]{{efn|Matrix multiplication of quantum gates is defined as [[#Serially wired gates|series circuits]].}} is the [[unitary group]] U(2<sup>''n''</sup>).<ref name="Barenco"/> The [[quantum state]]s that the gates act upon are [[unit vector]]s in <math>2^n</math> [[complex number|complex]] dimensions, with the [[Norm (mathematics)#Euclidean norm of complex numbers|complex Euclidean norm]] (the [[Norm (mathematics)#p-norm|2-norm]]).{{r|Nielsen-Chuang|page=66}}{{r|Yanofsky-Mannucci|pages=56,65}} The [[basis vectors]] (sometimes called ''[[eigenstate]]s'') are the possible outcomes if the state of the qubits is [[Quantum measurement|measured]], and a quantum state is a [[linear combination]] of these outcomes. The most common quantum gates operate on [[vector space]]s of one or two qubits, just like the common [[Logic gate|classical logic gate]]s operate on one or two [[bit]]s.

Even though the quantum logic gates belong to [[continuous symmetry|continuous symmetry group]]s, real [[computer hardware|hardware]] is inexact and thus limited in precision. The application of gates typically introduces errors, and the [[Fidelity of quantum states|quantum states' fidelities]] decrease over time. If [[Quantum error correction|error correction]] is used, the usable gates are further restricted to a finite set.{{r|Nielsen-Chuang|pages=ch. 10}}{{r|Williams|pages=ch. 14}} Later in this article, this is ignored as the focus is on the ideal quantum gates' properties.

Quantum states are typically represented by "kets", from a notation known as [[Bra–ket notation|bra–ket]].

The vector representation of a single [[qubit]] is
:<math>|a\rangle = v_0 | 0 \rangle + v_1 | 1 \rangle  \rightarrow \begin{bmatrix} v_0 \\ v_1 \end{bmatrix} .</math>

Here, <math>v_0</math> and <math>v_1</math> are the complex [[probability amplitude]]s of the qubit. These values determine the probability of measuring a 0 or a 1, when measuring the state of the qubit. See [[#Measurement|measurement]] below for details.

The value zero is represented by the ket {{nowrap|<math>|0\rangle = \begin{bmatrix} 1 \\ 0 \end{bmatrix}</math>,}} and the value one is represented by the ket {{nowrap|<math>|1\rangle = \begin{bmatrix} 0 \\ 1\end{bmatrix}</math>.}}

The [[tensor product]] (or [[Kronecker product]]) is used to combine quantum states. The combined state for a [[quantum register|qubit register]] is the tensor product of the constituent qubits. The tensor product is denoted by the symbol {{nowrap|<math>\otimes</math>.}}

The vector representation of two qubits is:<ref name="Preskill">{{cite arXiv|last=Preskill|first=John|title=Quantum computing 40 years later|eprint=2106.10522|class=quant-ph|date=2021-06-06|pages=10–15}}</ref>
:<math>| \psi \rangle = v_{00} | 00 \rangle + v_{01} | 0 1 \rangle  + v_{10} | 1 0 \rangle + v_{11} | 1 1 \rangle \rightarrow \begin{bmatrix} v_{00} \\ v_{01} \\ v_{10} \\ v_{11} \end{bmatrix}.</math>

The action of the gate on a specific quantum state is found by [[Matrix multiplication|multiplying]] the vector <math>|\psi_1\rangle</math>, which represents the state by the matrix <math>U</math> representing the gate. The result is a new quantum state {{nowrap|<math>|\psi_2\rangle</math>:}}

:<math>U|\psi_1\rangle = |\psi_2\rangle.</math>

=== Relation to the time evolution operator ===
The [[Schrödinger equation]] describes how quantum systems that are not [[Measurement in quantum mechanics|observed]] evolve over time, and is <math>i\hbar\frac{d}{dt}|\Psi\rangle = \hat{H}|\Psi\rangle.</math> When the system is in a stable environment, so it has a constant [[Hamiltonian (quantum mechanics)|Hamiltonian]], the solution to this equation is <math>U(t) = e^{-i\hat{H}t/\hbar}.</math>{{r|Williams|pages=24–25}} If the time <math>t</math> is always the same it may be omitted for simplicity, and the way quantum states evolve can be described as <math>U|\psi_1\rangle = |\psi_2\rangle,</math> just as in the above section.

That is, a quantum gate is how a quantum system that is not observed evolves over some specific time, or equivalently, a gate is the unitary [[time evolution]] operator <math>U</math> acting on a quantum state for a specific duration.