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{{Short description|Basic circuit operation in quantum computing}}
{{Quantum methods backlink|Quantum information methods}}

In [[quantum computing]] and the [[quantum circuit]] model, a '''quantum gate''' is a basic circuit operation acting on a small number of [[Physics:Quantum methods/qubit|qubits]]. Quantum gates are the building blocks of [[Physics:Quantum methods/quantum circuit|quantum circuits]], in the same way that classical logic gates are building blocks of ordinary digital circuits.<ref>{{Cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |edition=10th anniversary |publisher=Cambridge University Press |year=2010 |isbn=978-1-10700-217-3}}</ref>

Unlike many classical logic gates, quantum gates are reversible. They are represented mathematically by [[unitary matrix|unitary matrices]], so their action preserves the norm of a quantum state and has an inverse operation.<ref>{{cite book |last=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=Springer |year=2011 |isbn=978-1-84628-887-6}}</ref> This reversibility is one of the main differences between classical digital logic and quantum logic.

A gate acting on <math>n</math> qubits is represented by a <math>2^n \times 2^n</math> unitary matrix. The state on which it acts is a vector in a complex Hilbert space. In the usual computational basis, qubit basis states are written as <math>|0\rangle</math> and <math>|1\rangle</math>, while multi-qubit states are written as tensor products such as <math>|00\rangle</math>, <math>|01\rangle</math>, <math>|10\rangle</math>, and <math>|11\rangle</math>.<ref>{{Cite book |last1=Yanofsky |first1=Noson S. |last2=Mannucci |first2=Mirco |title=Quantum Computing for Computer Scientists |publisher=Cambridge University Press |year=2013 |isbn=978-0-521-87996-5}}</ref>
[[File:Quantum Logic Gates-y.jpg|thumb|right|upright=2|Common quantum logic gates by name, circuit form, and corresponding unitary matrices.]]

== History ==

The modern notation for quantum gates was developed by many founders of quantum information science, including Adriano Barenco, Charles Bennett, Richard Cleve, David DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John Smolin, and Harald Weinfurter.<ref>{{cite journal |last1=Barenco |first1=Adriano |last2=Bennett |first2=Charles H. |last3=Cleve |first3=Richard |last4=DiVincenzo |first4=David P. |last5=Margolus |first5=Norman |last6=Shor |first6=Peter |last7=Sleator |first7=Tycho |last8=Smolin |first8=John A. |last9=Weinfurter |first9=Harald |title=Elementary gates for quantum computation |journal=Physical Review A |volume=52 |issue=5 |pages=3457–3467 |year=1995 |doi=10.1103/PhysRevA.52.3457 |arxiv=quant-ph/9503016}}</ref> Their work built on earlier ideas about quantum mechanical computers introduced by Richard Feynman.<ref>{{cite journal |last=Feynman |first=Richard P. |title=Quantum mechanical computers |journal=Foundations of Physics |volume=16 |issue=6 |pages=507–531 |year=1986 |doi=10.1007/BF01886518}}</ref>

== Representation ==

A single qubit may be written as

:<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 complex probability amplitudes. Their squared magnitudes determine the probabilities of measuring the qubit as <math>|0\rangle</math> or <math>|1\rangle</math>.

The computational basis states are

:<math>|0\rangle = \begin{bmatrix}1\\0\end{bmatrix}, \qquad |1\rangle = \begin{bmatrix}0\\1\end{bmatrix}.</math>

For two qubits, the state is written as

:<math>|\psi\rangle = v_{00}|00\rangle + v_{01}|01\rangle + v_{10}|10\rangle + v_{11}|11\rangle.</math>

A quantum gate transforms a state vector by matrix multiplication:

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

Because <math>U</math> is unitary, the total probability is conserved.<ref>{{Cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010 |isbn=978-1-10700-217-3}}</ref>

== Relation to time evolution ==

A quantum gate can also be interpreted as a controlled period of quantum time evolution. The Schrödinger equation describes how an unobserved quantum system evolves in time. If the Hamiltonian is constant, this evolution is represented by a unitary time-evolution operator. In circuit language, the same mathematical operation is treated as a quantum gate.<ref>{{cite book |last=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=Springer |year=2011 |isbn=978-1-84628-887-6}}</ref>

== Common gates ==

=== Identity gate ===

The identity gate leaves a qubit unchanged:

:<math>I=\begin{bmatrix}1&0\\0&1\end{bmatrix}.</math>

It is useful when describing larger circuits or tensor products in which one qubit is left untouched.

=== Pauli gates ===

The Pauli gates <math>X</math>, <math>Y</math>, and <math>Z</math> act on a single qubit and correspond to rotations about the axes of the Bloch sphere.

:<math>X=\begin{bmatrix}0&1\\1&0\end{bmatrix}, \qquad
Y=\begin{bmatrix}0&-i\\i&0\end{bmatrix}, \qquad
Z=\begin{bmatrix}1&0\\0&-1\end{bmatrix}.</math>

The Pauli-<math>X</math> gate is the quantum analogue of the classical NOT gate: it maps <math>|0\rangle</math> to <math>|1\rangle</math> and <math>|1\rangle</math> to <math>|0\rangle</math>. The Pauli-<math>Z</math> gate changes the phase of <math>|1\rangle</math> while leaving <math>|0\rangle</math> unchanged.

=== Hadamard gate ===

The Hadamard gate creates equal superpositions from computational basis states:

:<math>H=\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\1&-1\end{bmatrix}.</math>

It maps

:<math>|0\rangle \mapsto \frac{|0\rangle+|1\rangle}{\sqrt{2}}, \qquad
|1\rangle \mapsto \frac{|0\rangle-|1\rangle}{\sqrt{2}}.</math>

The Hadamard gate is central to many quantum algorithms because it prepares superposition states.<ref>{{Cite book |last1=Yanofsky |first1=Noson S. |last2=Mannucci |first2=Mirco |title=Quantum Computing for Computer Scientists |publisher=Cambridge University Press |year=2013 |isbn=978-0-521-87996-5}}</ref>

=== Phase shift gates ===

Phase shift gates change the relative phase of a qubit without changing the measurement probabilities in the computational basis:

:<math>P(\varphi)=\begin{bmatrix}1&0\\0&e^{i\varphi}\end{bmatrix}.</math>

Important examples include the <math>S</math> gate, the <math>T</math> gate, and the Pauli-<math>Z</math> gate.

=== Controlled gates ===

Controlled gates act on two or more qubits. One qubit acts as a control and determines whether an operation is applied to another qubit.<ref>{{cite journal |last1=Barenco |first1=Adriano |last2=Bennett |first2=Charles H. |last3=Cleve |first3=Richard |last4=DiVincenzo |first4=David P. |last5=Margolus |first5=Norman |last6=Shor |first6=Peter |last7=Sleator |first7=Tycho |last8=Smolin |first8=John A. |last9=Weinfurter |first9=Harald |title=Elementary gates for quantum computation |journal=Physical Review A |volume=52 |issue=5 |pages=3457–3467 |year=1995 |doi=10.1103/PhysRevA.52.3457}}</ref>

The controlled-NOT gate, or CNOT gate, applies <math>X</math> to the target qubit when the control qubit is <math>|1\rangle</math>. In the basis <math>|00\rangle, |01\rangle, |10\rangle, |11\rangle</math>, it is represented by

:<math>\mathrm{CNOT}=
\begin{bmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&0&1\\
0&0&1&0
\end{bmatrix}.</math>

CNOT is essential for creating entanglement and for building multi-qubit algorithms.

=== Swap gate ===

The SWAP gate exchanges two qubits:

:<math>\mathrm{SWAP}=
\begin{bmatrix}
1&0&0&0\\
0&0&1&0\\
0&1&0&0\\
0&0&0&1
\end{bmatrix}.</math>

It is useful in hardware architectures where only nearby qubits can interact.

=== Toffoli gate ===

The Toffoli gate, also called the controlled-controlled-NOT gate, acts on three qubits. It flips the third qubit if the first two qubits are both <math>|1\rangle</math>. It is universal for reversible classical computation and has a direct quantum version.<ref>{{cite arXiv |last=Aharonov |first=Dorit |title=A Simple Proof that Toffoli and Hadamard are Quantum Universal |eprint=quant-ph/0301040 |year=2003}}</ref>

== Universal quantum gates ==

A universal quantum gate set is a set of gates from which any quantum operation can be approximated to arbitrary precision. Common universal sets include single-qubit rotations together with CNOT, or the Clifford gates supplemented by the <math>T</math> gate.<ref>{{cite book |last=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=Springer |year=2011 |isbn=978-1-84628-887-6}}</ref>

The Solovay–Kitaev theorem shows that, under suitable conditions, arbitrary quantum gates can be efficiently approximated using a finite universal set.<ref>{{cite journal |last1=Dawson |first1=Christopher M. |last2=Nielsen |first2=Michael A. |title=The Solovay-Kitaev algorithm |journal=Quantum Information and Computation |volume=6 |issue=1 |year=2006 |doi=10.26421/QIC6.1-6 |arxiv=quant-ph/0505030}}</ref>

== Circuit composition ==

Quantum gates can be composed in series or in parallel.

If gate <math>A</math> is followed by gate <math>B</math>, the combined operation is

:<math>C = B A.</math>

The order is reversed compared with the visual left-to-right reading of some circuit diagrams because matrices act on state vectors from the right.

Parallel composition is represented by the tensor product. If one gate acts on one qubit and another gate acts on a different qubit, the combined operation is written as

:<math>A \otimes B.</math>

Tensor products are also used to extend a single-qubit gate so that it acts on one qubit within a larger multi-qubit state.

== Measurement ==

Measurement is not a quantum gate because it is generally irreversible. A measurement projects a quantum state onto one of the basis states, with probabilities determined by the squared magnitudes of the probability amplitudes.<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |publisher=Pearson |year=2018 |isbn=978-1-107-18963-8}}</ref>

For a qubit

:<math>a|0\rangle+b|1\rangle,</math>

the probability of measuring <math>|0\rangle</math> is <math>|a|^2</math>, and the probability of measuring <math>|1\rangle</math> is <math>|b|^2</math>.

Measurement plays a central role in quantum algorithms, but it is separated conceptually from gate evolution.

== Quantum gates and entanglement ==

Multi-qubit gates such as CNOT can create entanglement. For example, applying a Hadamard gate to the first qubit of <math>|00\rangle</math>, followed by CNOT, gives the Bell state

:<math>\frac{|00\rangle+|11\rangle}{\sqrt{2}}.</math>

This state cannot be written as a tensor product of two independent single-qubit states. Entanglement produced by quantum gates is essential for quantum teleportation, quantum error correction, and many quantum algorithms.<ref>{{cite journal |last1=Yin |first1=Juan |display-authors=etal |title=Satellite-based entanglement distribution over 1200 kilometers |journal=Science |volume=356 |issue=6343 |pages=1140–1144 |year=2017 |doi=10.1126/science.aan3211 |arxiv=1707.01339}}</ref>

== Logic synthesis ==

Quantum algorithms are built from sequences of gates. More complex unitary transformations can be synthesized from elementary gates. In practice, quantum programming languages and circuit libraries provide standard gate sets and decomposition tools.<ref>{{cite web |title=Circuit Library |publisher=IBM Qiskit |url=https://docs.quantum.ibm.com/api/qiskit/circuit_library}}</ref>

Because quantum gates are unitary, quantum functions must be reversible. Irreversible classical logic can be embedded into reversible quantum circuits by adding ancilla qubits and preserving enough information to reconstruct the input.

== Applications ==

Quantum gates are used in:
* [[Physics:Quantum methods/quantum circuit|quantum circuits]]
* [[Physics:Quantum methods/algorithm|quantum algorithms]]
* [[Physics:Quantum methods/error correction|quantum error correction]]
* [[Physics:Quantum methods/simulation|quantum simulation]]
* [[Physics:Quantum methods/communication|quantum communication]]
* [[Physics:Quantum methods/entanglement|entanglement generation]]
* quantum hardware control

They form the operational language of the quantum circuit model.

=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also/Methods}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Physics:Quantum methods/quantum gate|1}}

Physics:Quantum methods/quantum gate - Revision history

imported>WikiHarold: Replace raw Quantum Collection backlink with BT backlink template

imported>WikiHarold: Replace raw Quantum Collection backlink with BT backlink template