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Repair Quantum Collection B backlink template

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{{Quantum book backlink|Quantum information and computing}}
'''Quantum gates''' are the fundamental operations that act on qubits. They are the building blocks of '''quantum circuits''', which define computations in a quantum computer.<ref name="NielsenChuang2010">{{cite book |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>
<ref>{{cite book
| last = Williams
| first = Colin P.
| title = Explorations in Quantum Computing
| publisher = Springer
| year = 2011
| isbn = 978-1-84628-887-6
}}</ref>

[[File:Quantum_circuits.jpg|thumb|400px|Example of a quantum circuit composed of quantum gates acting on qubits.]]

== Quantum operations ==
Quantum states evolve according to unitary transformations:<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
| bibcode = 1986FoPh...16..507F
| issn = 0015-9018
| s2cid = 122076550
}}</ref>

<math>
|\psi\rangle \rightarrow U |\psi\rangle,
</math>

where <math>U</math> is a unitary operator satisfying

<math>
U^\dagger U = I.
</math>

These transformations preserve normalization and correspond to reversible evolution in quantum mechanics.

== Single-qubit gates ==

Single-qubit gates act on individual qubits and correspond to rotations on the Bloch sphere.

Common examples include:

* '''Pauli-X gate'''
<math>
X = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}
</math>

* '''Pauli-Y gate'''
<math>
Y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}
</math>

* '''Pauli-Z gate'''
<math>
Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}
</math>

* '''Hadamard gate'''
<math>
H = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\ 1 & -1\end{pmatrix}
</math>

These gates create superposition and control phase relationships.

== Multi-qubit gates ==

Multi-qubit gates act on multiple qubits and can generate entanglement.<ref>{{cite web
| title = UnitaryGate adjoint()
| website = IBM Quantum Documentation
| publisher = IBM
| url = https://quantum.cloud.ibm.com/docs/api/qiskit/qiskit.circuit.library.UnitaryGate
| access-date = 2025
}}</ref>

A key example is the '''controlled-NOT (CNOT) gate''', which flips a target qubit if the control qubit is in the state <math>|1\rangle</math>.

The CNOT gate acts on basis states as:

<math>
|00\rangle \rightarrow |00\rangle
</math>

<math>
|01\rangle \rightarrow |01\rangle
</math>

<math>
|10\rangle \rightarrow |11\rangle
</math>

<math>
|11\rangle \rightarrow |10\rangle
</math>

Together with single-qubit gates, the CNOT gate forms a universal set for quantum computation.<ref name="NielsenChuang2010" />

== Quantum circuits ==

A '''quantum circuit''' is a sequence of quantum gates applied to a set of qubits.

A typical circuit consists of:

* initialization of qubits
* application of unitary gates
* measurement of the final state

Circuits are usually represented diagrammatically, with time progressing from left to right.
== Universal gate sets ==

A set of quantum gates is called '''universal''' if any unitary operation can be approximated to arbitrary accuracy using only gates from that set.<ref name="NielsenChuang2010" />

A commonly used universal set consists of:

* all single-qubit gates
* the controlled-NOT (CNOT) gate

In practice, finite gate sets such as {H, T, CNOT} are used, where the T gate introduces a nontrivial phase. These sets allow efficient approximation of arbitrary quantum operations.
== Circuit depth and complexity ==

The complexity of a quantum circuit is characterized by measures such as:

* '''circuit depth''' — the number of sequential layers of gates
* '''gate count''' — the total number of gates used

Circuit depth is particularly important, as it determines how long a computation takes and how susceptible it is to noise.

Efficient quantum algorithms aim to minimize both gate count and depth while achieving the desired transformation.
== Example: Bell state circuit ==

Entanglement can be generated using a simple circuit:

1. Start with <math>|0\rangle \otimes |0\rangle</math>
2. Apply a Hadamard gate to the first qubit
3. Apply a CNOT gate

This produces the entangled state

<math>
\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle).
</math>
== Noise and decoherence ==

Real quantum systems are affected by noise and interactions with their environment, leading to decoherence and errors in quantum operations.

Noise limits the size and depth of quantum circuits that can be reliably executed. This is a central challenge in current quantum computing devices.

These limitations define the so-called noisy intermediate-scale quantum (NISQ) regime (see [[Physics:Quantum Noisy Qubits]]).
== Measurement ==

Measurement converts a quantum state into classical information.

For a qubit

<math>
|\psi\rangle = \alpha |0\rangle + \beta |1\rangle,
</math>

measurement yields:

* <math>|0\rangle</math> with probability <math>|\alpha|^2</math>
* <math>|1\rangle</math> with probability <math>|\beta|^2</math>

Measurement is irreversible and collapses the quantum state.

== Physical significance ==

Quantum gates and circuits:

* define how quantum computations are performed
* enable superposition and entanglement to be controlled
* provide the framework for quantum algorithms

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

= References =
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum gates and circuits|1}}

Physics:Quantum Gates and circuits - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template