Quantum logic gate

In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.

Unlike many classical logic gates, quantum logic gates are reversible. However, it is possible to perform classical computing using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions, often at the cost of having to use ancilla bits. The Toffoli gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits.

Quantum gates are unitary operators, and are described as unitary matrices relative to some basis. Usually we use the computational basis, which unless we compare it with something, just means that for a d-level quantum system (such as a qubit, a quantum register, or qutrits and qudits^{[1]: 22–23}) we have labeled the orthogonal basis vectors ${\displaystyle |0\rangle ,|1\rangle ,\dots ,|d-1\rangle }$, or use binary notation.

Common quantum logic gates by name (including abbreviation), circuit form(s) and the corresponding unitary matrices.

History[]

A quantum full adder, given by Feynman in 1986.^[2] It consists of only Toffoli and CNOT gates. The gate that is surrounded by the dotted square in this picture can be omitted if uncomputation to restore the B output is not required.

The current notation for quantum gates was developed by many of the founders of quantum information science including Adriano Barenco, Charles Bennett, Richard Cleve, David P. DiVincenzo, Norman Margolus, Peter Shor, Tycho Sleator, John A. Smolin, and Harald Weinfurter,^[3] building on notation introduced by Richard Feynman.^[2]

Representation[]

Single qubit states that are not entangled and lack global phase can be represented as points on the surface of the Bloch sphere, written as ${\displaystyle |\psi \rangle =\cos \left(\theta /2\right)|0\rangle +e^{i\varphi }\sin \left(\theta /2\right)|1\rangle .}$
Rotations about the x, y, z-axes of the Bloch sphere are represented by the rotation operator gates.

Quantum logic gates are represented by unitary matrices. A gate which acts on ${\displaystyle n}$ qubits is represented by a ${\displaystyle 2^{n}\times 2^{n}}$ unitary matrix, and the set of all such gates with the group operation of matrix multiplication^[a] is the symmetry group U(2ⁿ). The quantum states that the gates act upon are unit vectors in ${\displaystyle 2^{n}}$ complex dimensions, with the complex Euclidean norm (the 2-norm).^{[4]: 66 [5]: 56, 65} The basis vectors (sometimes called eigenstates) are the possible outcomes if measured, and a quantum state is a linear combination of these outcomes. The most common quantum gates operate on vector spaces of one or two qubits, just like the common classical logic gates operate on one or two bits.

Even though the quantum logic gates belong to a continuous symmetry group, real hardware is inexact and thus limited in precision. The application of gates typically introduces errors, and the quantum states fidelities decreases over time. If error correction is used, the usable gates are further restricted to a finite set.^{[4]: ch. 10 [1]: ch. 14} Later in this article, this is sometimes ignored as focus lies on the quantum gates' mathematical properties.

Quantum states are typically represented by "kets", from a notation known as bra-ket.

The vector representation of a single qubit is:

${\displaystyle |a\rangle =v_{0}|0\rangle +v_{1}|1\rangle \rightarrow {\begin{bmatrix}v_{0}\\v_{1}\end{bmatrix}},}$

Here, ${\displaystyle v_{0}}$ and ${\displaystyle v_{1}}$ are the complex probability amplitudes of the qubit. These values determine the probability of measuring a 0 or a 1, when measuring the state of the qubit. See measurement below for details.

The value zero is represented by the ket ${\displaystyle |0\rangle ={\begin{bmatrix}1\\0\end{bmatrix}}}$, and the value one is represented by the ket ${\displaystyle |1\rangle ={\begin{bmatrix}0\\1\end{bmatrix}}}$.

The tensor product (or Kronecker product) is used to combine quantum states. The combined state of two qubits is the tensor product of the two qubits. The tensor product is denoted by the symbol ${\displaystyle \otimes }$.

The vector representation of two qubits is:^[6]

${\displaystyle |ab\rangle =|a\rangle \otimes |b\rangle =v_{00}|00\rangle +v_{01}|01\rangle +v_{10}|10\rangle +v_{11}|11\rangle \rightarrow {\begin{bmatrix}v_{00}\\v_{01}\\v_{10}\\v_{11}\end{bmatrix}},}$

The action of the gate on a specific quantum state is found by multiplying the vector ${\displaystyle |\psi _{1}\rangle }$ which represents the state by the matrix ${\displaystyle U}$ representing the gate. The result is a new quantum state ${\displaystyle |\psi _{2}\rangle }$:

${\displaystyle U|\psi _{1}\rangle =|\psi _{2}\rangle .}$

Notable examples[]

There exists an uncountably infinite number of gates. Some of them have been named by various authors,^{[1][4][5][7][8][9]} and below follow some of those most often used in the literature.

Identity gate[]

The identity gate is the identity matrix, usually written as I, and is defined for a single qubit as

${\displaystyle I={\begin{bmatrix}1&0\\0&1\end{bmatrix}},}$

where I is basis independent and does not modify the quantum state. The identity gate is most useful when describing mathematically the result of various gate operations or when discussing multi-qubit circuits.

Pauli gates (X,Y,Z)[]

The Pauli gates ${\displaystyle (X,Y,Z)}$ are the three Pauli matrices ${\displaystyle (\sigma _{x},\sigma _{y},\sigma _{z})}$ and act on a single qubit. The Pauli X,Y and Z equate, respectively, to a rotation around the x, y and z axes of the Bloch sphere by ${\displaystyle \pi }$ radians.^[b]

The Pauli-X gate is the quantum equivalent of the NOT gate for classical computers with respect to the standard basis ${\displaystyle |0\rangle }$, ${\displaystyle |1\rangle }$, which distinguishes the z-axis on the Bloch sphere. It is sometimes called a bit-flip as it maps ${\displaystyle |0\rangle }$ to ${\displaystyle |1\rangle }$ and ${\displaystyle |1\rangle }$ to ${\displaystyle |0\rangle }$. Similarly, the Pauli-Y maps ${\displaystyle |0\rangle }$ to ${\displaystyle i|1\rangle }$ and ${\displaystyle |1\rangle }$ to ${\displaystyle -i|0\rangle }$. Pauli Z leaves the basis state ${\displaystyle |0\rangle }$ unchanged and maps ${\displaystyle |1\rangle }$ to ${\displaystyle -|1\rangle }$. Due to this nature, it is sometimes called phase-flip.

These matrices are usually represented as

${\displaystyle X=\sigma _{x}=\operatorname {NOT} ={\begin{bmatrix}0&1\\1&0\end{bmatrix}},}$

${\displaystyle Y=\sigma _{y}={\begin{bmatrix}0&-i\\i&0\end{bmatrix}},}$

${\displaystyle Z=\sigma _{z}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}.}$

The Pauli matrices are involutory, meaning that the square of a Pauli matrix is the identity matrix.

${\displaystyle I^{2}=X^{2}=Y^{2}=Z^{2}=-iXYZ=I}$

The Pauli matrices also anti-commute, for example ${\displaystyle ZX=iY=-XZ}$ .

Square root of NOT gate (√NOT)[]

Quantum circuit diagram of a square-root-of-NOT gate

The square root of NOT gate (or square root of Pauli-X, ${\displaystyle {\sqrt {X}}}$) acts on a single qubit. It maps the basis state ${\displaystyle |0\rangle }$ to ${\displaystyle {\frac {(1+i)|0\rangle +(1-i)|1\rangle }{2}}}$ and ${\displaystyle |1\rangle }$ to ${\displaystyle {\frac {(1-i)|0\rangle +(1+i)|1\rangle }{2}}}$. In matrix form it is given by

${\displaystyle {\sqrt {X}}={\sqrt {\text{NOT}}}={\frac {1}{2}}{\begin{bmatrix}1+i&1-i\\1-i&1+i\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}e^{i\pi /4}&e^{-i\pi /4}\\e^{-i\pi /4}&e^{i\pi /4}\end{bmatrix}}}$,

such that

${\displaystyle ({\sqrt {X}})^{2}=\left({\sqrt {\text{NOT}}}\right)^{2}={\begin{bmatrix}0&1\\1&0\end{bmatrix}}=X.}$

This operation represents a rotation of π/2 about the x-axis at the Bloch sphere.

Controlled gates[]

Circuit diagram of controlled Pauli gates (from left to right): CNOT (or controlled-X), controlled-Y and controlled-Z.

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation.^[3] For example, the controlled NOT gate (or CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is ${\displaystyle |1\rangle }$, and otherwise leaves it unchanged. With respect to the basis ${\displaystyle |00\rangle }$, ${\displaystyle |01\rangle }$, ${\displaystyle |10\rangle }$, ${\displaystyle |11\rangle }$, it is represented by the matrix:

${\displaystyle {\mbox{CNOT}}={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&1&0\end{bmatrix}}.}$

The CNOT (or controlled Pauli-X) gate can be described as the gate that maps the basis states ${\displaystyle |a,b\rangle \mapsto |a,a\oplus b\rangle }$, where ${\displaystyle \oplus }$ is XOR.

More generally if U is a gate that operates on a single qubit with matrix representation

${\displaystyle U={\begin{bmatrix}u_{00}&u_{01}\\u_{10}&u_{11}\end{bmatrix}},}$

then the controlled-U gate is a gate that operates on two qubits in such a way that the first qubit serves as a control. It maps the basis states as follows.

Circuit representation of controlled-U gate

${\displaystyle |00\rangle \mapsto |00\rangle }$

${\displaystyle |01\rangle \mapsto |01\rangle }$

${\displaystyle |10\rangle \mapsto |1\rangle \otimes U|0\rangle =|1\rangle \otimes (u_{00}|0\rangle +u_{10}|1\rangle )}$

${\displaystyle |11\rangle \mapsto |1\rangle \otimes U|1\rangle =|1\rangle \otimes (u_{01}|0\rangle +u_{11}|1\rangle )}$

The matrix representing the controlled U is

${\displaystyle {\mbox{C}}U={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&u_{00}&u_{01}\\0&0&u_{10}&u_{11}\end{bmatrix}}.}$

When U is one of the Pauli operators, X,Y, Z, the respective terms "controlled-X", "controlled-Y", or "controlled-Z" are sometimes used.^{[4]: 177–185} Sometimes this is shortened to just CX, CY and CZ.

Control can be extended to gates with arbitrary number of qubits^[3] and functions in programming languages.^[10][11]

A controlled function or gate behaves conceptually differently depending on whether one approaches it from before measurement, or after measurement: From the perspective of before measurement, the controlled function simultaneously executes on all those basis states, of a superposition (i.e. linear combination) of possible outcomes,^[12]: 7 that match the controls. If one instead approaches it from the perspective of after measurement, the controlled function has either executed, or not, depending on which basis state that was measured.^[c]

Phase shift gates[]

The phase shift is a family of single-qubit gates that map the basis states ${\displaystyle |0\rangle \mapsto |0\rangle }$ and ${\displaystyle |1\rangle \mapsto e^{i\varphi }|1\rangle }$. The probability of measuring a ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$ is unchanged after applying this gate, however it modifies the phase of the quantum state. This is equivalent to tracing a horizontal circle (a line of latitude) on the Bloch sphere by ${\displaystyle \varphi }$ radians. The phase shift gate is represented by the matrix:

${\displaystyle P(\varphi )={\begin{bmatrix}1&0\\0&e^{i\varphi }\end{bmatrix}}}$

where ${\displaystyle \varphi }$ is the phase shift with the period 2π. Some common examples are the T gate where ${\textstyle \varphi ={\frac {\pi }{4}}}$, the phase gate (written S, though S is sometimes used for SWAP gates) where ${\textstyle \varphi ={\frac {\pi }{2}}}$ and the Pauli-Z gate where ${\displaystyle \varphi =\pi }$.

The phase shift gates are related to each other as follows:

${\displaystyle Z={\begin{bmatrix}1&0\\0&e^{i\pi }\end{bmatrix}}={\begin{bmatrix}1&0\\0&-1\end{bmatrix}}=P\left(\pi \right)}$

${\displaystyle S={\begin{bmatrix}1&0\\0&e^{i{\frac {\pi }{2}}}\end{bmatrix}}={\begin{bmatrix}1&0\\0&i\end{bmatrix}}=P\left({\frac {\pi }{2}}\right)={\sqrt {Z}}}$

${\displaystyle T={\begin{bmatrix}1&0\\0&e^{i{\frac {\pi }{4}}}\end{bmatrix}}=P\left({\frac {\pi }{4}}\right)={\sqrt {S}}={\sqrt[{4}]{Z}}}$

${\displaystyle P\left({\frac {\pi }{\xi }}\right)={\sqrt[{\xi }]{Z}}}$ for all real ${\displaystyle \xi }$ except 0^[d]

The argument to the phase shift gate is in U(1), and the gate performs a phase rotation in U(1) along the specified basis state (e.g. ${\displaystyle P(\varphi )}$ rotates the phase about ${\displaystyle |1\rangle }$). Extending ${\displaystyle P(\varphi )}$ to a rotation about a generic phase of both basis states of a 2-level quantum system (a qubit) can be done with a series circuit: ${\displaystyle P(\beta )\cdot X\cdot P(\alpha )\cdot X={\begin{bmatrix}e^{i\alpha }&0\\0&e^{i\beta }\end{bmatrix}}}$. When ${\displaystyle \alpha =-\beta }$ this gate is the rotation operator ${\displaystyle R_{z}(2\beta )}$ gate.^[e]

Introducing the global phase gate ${\displaystyle \operatorname {Ph} (\delta )={\begin{bmatrix}e^{i\delta }&0\\0&e^{i\delta }\end{bmatrix}}=\left(P(\delta )\cdot X\right)^{2}=\exp \left(i\delta I\right)=e^{i\delta }I}$^[f] we also have the identity ${\displaystyle R_{z}(-\gamma )\operatorname {Ph} \left({\frac {\gamma }{2}}\right)=P(\gamma )}$.^{[3]: 11 [1]: 77–83}

Arbitrary single-qubit phase shift gates ${\displaystyle P(\varphi )}$ are natively available for transmon quantum processors through timing of microwave control pulses.^[13] It can be explained in terms of change of frame.^[14][15]

Controlled phase shift[]

The 2-qubit controlled phase shift gate is:

${\displaystyle \mathrm {CPHASE} (\varphi )={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&e^{i\varphi }\end{bmatrix}}}$

With respect to the computational basis, it shifts the phase with ${\displaystyle \varphi }$ only if it acts on the state ${\displaystyle |11\rangle }$:

${\displaystyle |a,b\rangle \mapsto {\begin{cases}|a,e^{i\varphi }b\rangle &{\mbox{for }}a=b=1\\|a,b\rangle &{\mbox{otherwise.}}\end{cases}}}$

The CZ gate is the special case where ${\displaystyle \varphi =\pi }$.

Rotation operator gates[]

The rotation operator gates ${\displaystyle R_{x}(\theta ),R_{y}(\theta )}$ and ${\displaystyle R_{z}(\theta )}$ are the analog rotation matrices in three Cartesian axes of SO(3), the axes on the Bloch sphere projection:

${\displaystyle R_{x}(\theta )=\exp(-iX\theta /2)={\begin{bmatrix}\cos(\theta /2)&-i\sin(\theta /2)\\-i\sin(\theta /2)&\cos(\theta /2)\end{bmatrix}},}$

${\displaystyle R_{y}(\theta )=\exp(-iY\theta /2)={\begin{bmatrix}\cos(\theta /2)&-\sin(\theta /2)\\\sin(\theta /2)&\cos(\theta /2)\end{bmatrix}},}$

${\displaystyle R_{z}(\theta )=\exp(-iZ\theta /2)={\begin{bmatrix}\exp(-i\theta /2)&0\\0&\exp(i\theta /2)\end{bmatrix}}.}$

As Pauli matrices are related to the generator of rotations, these rotation operators can be written as matrix exponentials with Pauli matrices in the argument. Any ${\displaystyle 2\times 2}$ unitary matrix in SU(2) can be written as a product (i.e. series circuit) of three rotation gates or less. Note that for two-level systems such as qubits and spinors, these rotations have a period of 4π. A rotation of 2π (360 degrees) returns the same statevector with a different phase.^[16]

We also have ${\displaystyle R_{b}(-\theta )=R_{b}(\theta )^{\dagger }}$ and ${\displaystyle R_{b}(0)=I}$ for all ${\displaystyle b\in \{x,y,z\}.}$

The rotation matrices are related to the Pauli matrices in the following way : ${\displaystyle R_{x}(\pi )=-iX,R_{y}(\pi )=-iY,R_{z}(\pi )=-iZ.}$

Similar rotation operator gates exist for SU(3). They are the rotation operators used with qutrits.

Hadamard gate[]

The Hadamard gate (French: [adamaʁ]) acts on a single qubit. It maps the basis states ${\textstyle |0\rangle \mapsto {\frac {|0\rangle +|1\rangle }{\sqrt {2}}}}$ and ${\textstyle |1\rangle \mapsto {\frac {|0\rangle -|1\rangle }{\sqrt {2}}}}$ (i.e. creates a superposition if given a basis state). It represents a rotation of ${\displaystyle \pi }$ about the axis ${\displaystyle ({\hat {x}}+{\hat {z}})/{\sqrt {2}}}$ at the Bloch sphere. It is represented by the Hadamard matrix:

Circuit representation of Hadamard gate

${\displaystyle H={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}.}$

H is an involutory matrix. Using rotation operators, we have the identities: ${\displaystyle R_{y}(\pi /2)Z=H}$ and ${\displaystyle XR_{y}(\pi /2)=H.}$ Controlled-H gate can also be defined as explained in the section of controlled gates.

The Hadamard gate can be thought as a unitary transformation that maps qubit operations in z-axis to the x-axis and viceversa. For example, ${\displaystyle HZH=X}$, ${\displaystyle H{\sqrt {X}}\;H={\sqrt {Z}}=S}$, and ${\displaystyle HR_{z}(\theta )H=R_{x}(\theta )}$.

Swap gate[]

Circuit representation of SWAP gate

The swap gate swaps two qubits. With respect to the basis ${\displaystyle |00\rangle }$, ${\displaystyle |01\rangle }$, ${\displaystyle |10\rangle }$, ${\displaystyle |11\rangle }$, it is represented by the matrix:

${\displaystyle {\mbox{SWAP}}={\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{bmatrix}}.}$

Square root of swap gate[]

Circuit representation of √SWAP gate

The √SWAP gate performs half-way of a two-qubit swap. It is universal such that any many-qubit gate can be constructed from only √SWAP and single qubit gates. The √SWAP gate is not, however maximally entangling; more than one application of it is required to produce a Bell state from product states. With respect to the basis ${\displaystyle |00\rangle }$, ${\displaystyle |01\rangle }$, ${\displaystyle |10\rangle }$, ${\displaystyle |11\rangle }$, it is represented by the matrix:

${\displaystyle {\sqrt {\text{SWAP}}}={\begin{bmatrix}1&0&0&0\\0&{\frac {1}{2}}(1+i)&{\frac {1}{2}}(1-i)&0\\0&{\frac {1}{2}}(1-i)&{\frac {1}{2}}(1+i)&0\\0&0&0&1\\\end{bmatrix}}.}$

This gate arises naturally in systems that exploit exchange interaction.^[17][18]

Toffoli (CCNOT) gate[]

Circuit representation of Toffoli gate

The Toffoli gate, named after Tommaso Toffoli; also called CCNOT gate or Deutsch gate ${\displaystyle D(\pi /2)}$; is a 3-bit gate, which is universal for classical computation but not for quantum computation. The quantum Toffoli gate is the same gate, defined for 3 qubits. If we limit ourselves to only accepting input qubits that are ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$, then if the first two bits are in the state ${\displaystyle |1\rangle }$ it applies a Pauli-X (or NOT) on the third bit, else it does nothing. It is an example of a controlled gate. Since it is the quantum analog of a classical gate, it is completely specified by its truth table. The Toffoli gate is universal when combined with the single qubit Hadamard gate.^[19]

The Toffoli gate is related to the classical AND (${\displaystyle \land }$) and XOR (${\displaystyle \oplus }$) operations as it performs the mapping ${\displaystyle |a,b,c\rangle \mapsto |a,b,c\oplus (a\land b)\rangle }$ on states in the computational basis.

Fredkin (CSWAP) gate[]

Circuit representation of Fredkin gate

The Fredkin gate (also CSWAP or CS gate), named after Edward Fredkin, is a 3-bit gate that performs a controlled swap. It is universal for classical computation. It has the useful property that the numbers of 0s and 1s are conserved throughout, which in the billiard ball model means the same number of balls are output as input.

Ising coupling gates[]

The Ising coupling gates R_xx, R_yy and R_zz are 2-qubit gates that are implemented natively in some trapped-ion quantum computers.^[20][21] These gates are defined as

${\displaystyle R_{xx}(\phi )=\exp \left(-i{\frac {\phi }{2}}X\otimes X\right)=\cos \left({\frac {\phi }{2}}\right)I\otimes I-i\sin \left({\frac {\phi }{2}}\right)X\otimes X={\begin{bmatrix}\cos \left({\frac {\phi }{2}}\right)&0&0&-i\sin \left({\frac {\phi }{2}}\right)\\0&\cos \left({\frac {\phi }{2}}\right)&-i\sin \left({\frac {\phi }{2}}\right)&0\\0&-i\sin \left({\frac {\phi }{2}}\right)&\cos \left({\frac {\phi }{2}}\right)&0\\-i\sin \left({\frac {\phi }{2}}\right)&0&0&\cos \left({\frac {\phi }{2}}\right)\\\end{bmatrix}},}$

${\displaystyle R_{yy}(\phi )=\exp \left(-i{\frac {\phi }{2}}Y\otimes Y\right)={\begin{bmatrix}\cos \left({\frac {\phi }{2}}\right)&0&0&i\sin \left({\frac {\phi }{2}}\right)\\0&\cos \left({\frac {\phi }{2}}\right)&-i\sin \left({\frac {\phi }{2}}\right)&0\\0&-i\sin \left({\frac {\phi }{2}}\right)&\cos \left({\frac {\phi }{2}}\right)&0\\i\sin \left({\frac {\phi }{2}}\right)&0&0&\cos \left({\frac {\phi }{2}}\right)\\\end{bmatrix}},}$

${\displaystyle R_{zz}(\phi )=\exp \left(-i{\frac {\phi }{2}}Z\otimes Z\right)={\begin{bmatrix}e^{-i\phi /2}&0&0&0\\0&e^{i\phi /2}&0&0\\0&0&e^{i\phi /2}&0\\0&0&0&e^{-i\phi /2}\\\end{bmatrix}}.}$^[22]

Imaginary swap (iSWAP)[]

For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap^[23] or iSWAP gate defined as^[24][25]

${\displaystyle i{\mbox{SWAP}}=\exp \left[i{\frac {\pi }{4}}(X\otimes X+Y\otimes Y)\right]={\begin{bmatrix}1&0&0&0\\0&0&i&0\\0&i&0&0\\0&0&0&1\end{bmatrix}},}$

where its squared root version is given by

${\displaystyle {\sqrt {i{\mbox{SWAP}}}}=\exp \left[i{\frac {\pi }{8}}(X\otimes X+Y\otimes Y)\right]={\begin{bmatrix}1&0&0&0\\0&{\frac {1}{\sqrt {2}}}&{\frac {i}{\sqrt {2}}}&0\\0&{\frac {i}{\sqrt {2}}}&{\frac {1}{\sqrt {2}}}&0\\0&0&0&1\end{bmatrix}}.}$

Note that ${\displaystyle X\otimes X}$ and ${\displaystyle Y\otimes Y}$ commute, so ${\displaystyle i{\mbox{SWAP}}=R_{xx}(-\pi /2)R_{yy}(-\pi /2)}$ and ${\displaystyle {\sqrt {i{\mbox{SWAP}}}}=R_{xx}(-\pi /4)R_{yy}(-\pi /4)}$, or more generally ${\displaystyle {\sqrt[{n}]{i{\mbox{SWAP}}}}=R_{xx}(-\pi /2n)R_{yy}(-\pi /2n)}$ for all real n except 0.

Deutsch gate[]

The Deutsch (or ${\displaystyle D_{\theta }}$) gate, named after physicist David Deutsch is a three-qubit gate. It is defined as

${\displaystyle |a,b,c\rangle \mapsto {\begin{cases}i\cos(\theta )|a,b,c\rangle +\sin(\theta )|a,b,1-c\rangle &{\mbox{for }}a=b=1\\|a,b,c\rangle &{\mbox{otherwise.}}\end{cases}}}$

Unfortunately, a working Deutsch gate has remained out of reach, due to lack of a protocol. There are some proposals to realize a Deutsch gate with dipole-dipole interaction in neutral atoms.^[26]

Universal quantum gates[]

Both CNOT and ${\displaystyle {\sqrt {\mbox{SWAP}}}}$ are universal two-qubit gates and can be transformed into each other.

A set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced, that is, any other unitary operation can be expressed as a finite sequence of gates from the set. Technically, this is impossible with anything less than an uncountable set of gates since the number of possible quantum gates is uncountable, whereas the number of finite sequences from a finite set is countable. To solve this problem, we only require that any quantum operation can be approximated by a sequence of gates from this finite set. Moreover, for unitaries on a constant number of qubits, the Solovay–Kitaev theorem guarantees that this can be done efficiently.

The rotation operators R_x(θ), R_y(θ), R_z(θ), the phase shift gate P(φ) and CNOT form a widely used universal set of quantum gates.^[18]

A common universal gate set is the Clifford + T gate set, which is composed of the CNOT, H, S and T gates. The Clifford set alone is not universal and can be efficiently simulated classically by the Gottesman–Knill theorem.

The Toffoli gate forms a set of universal gates for reversible boolean algebraic logic circuits. A two-gate set of universal quantum gates containing a Toffoli gate can be constructed by adding the Hadamard gate to the set.^[27]

A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate ${\displaystyle D(\theta )}$.^[28]

A universal logic gate for reversible classical computing, the Toffoli gate, is reducible to the Deutsch gate, ${\displaystyle D({\begin{matrix}{\frac {\pi }{2}}\end{matrix}})}$, thus showing that all reversible classical logic operations can be performed on a universal quantum computer.

There also exists a single two-qubit gate sufficient for universality, given it can be applied to any pairs of qubits ${\displaystyle (k,k+1)\mod n}$ on a circuit of width ${\displaystyle n}$.^[29]

Circuit composition[]

Serially wired gates[]

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 ${\displaystyle n}$ 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.

${\displaystyle C=B\cdot A}$

Where ${\displaystyle \cdot }$ is 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.^{[4]: 17–18,22–23,62–64 [5]: 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:

${\displaystyle C=X\cdot Y={\begin{bmatrix}0&1\\1&0\end{bmatrix}}\cdot {\begin{bmatrix}0&-i\\i&0\end{bmatrix}}={\begin{bmatrix}i&0\\0&-i\end{bmatrix}}=iZ}$

The product symbol (${\displaystyle \cdot }$) is often omitted.

Exponents of quantum gates[]

All real exponents of 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. ${\displaystyle X^{3}=X\cdot X\cdot X}$), and the real exponents is a generalization of the series circuit. For example, ${\displaystyle X^{\pi }}$ and ${\displaystyle {\sqrt {X}}=X^{1/2}}$ are both valid quantum gates.

${\displaystyle U^{0}=I}$ for any unitary matrix ${\displaystyle U}$. The identity matrix (${\displaystyle I}$) behaves like a NOP and can be represented as bare wire in quantum circuits, or not shown at all.

All gates are unitary matrices, so that ${\displaystyle U^{\dagger }U=UU^{\dagger }=I}$ and ${\displaystyle U^{\dagger }=U^{-1}}$, where ${\displaystyle \dagger }$ is the conjugate transpose. This means that negative exponents of gates are unitary inverses of their positively exponentiated counterparts: ${\displaystyle U^{-n}=(U^{n})^{\dagger }}$. For example, some negative exponents of the phase shift gates are ${\displaystyle T^{-1}=T^{\dagger }}$ and ${\displaystyle T^{-2}=(T^{2})^{\dagger }=S^{\dagger }}$.^[g]

Parallel gates[]

Two gates ${\displaystyle Y}$ and ${\displaystyle X}$ in parallel is equivalent to the gate ${\displaystyle Y\otimes X}$

The tensor product (or Kronecker product) of two quantum gates is the gate that is equal to the two gates in parallel.^{[4]: 71–75 [5]: 148}

If we, as in the picture, combine the Pauli-Y gate with the Pauli-X gate in parallel, then this can be written as:

${\displaystyle C=Y\otimes X={\begin{bmatrix}0&-i\\i&0\end{bmatrix}}\otimes {\begin{bmatrix}0&1\\1&0\end{bmatrix}}={\begin{bmatrix}0{\begin{bmatrix}0&1\\1&0\end{bmatrix}}&-i{\begin{bmatrix}0&1\\1&0\end{bmatrix}}\\i{\begin{bmatrix}0&1\\1&0\end{bmatrix}}&0{\begin{bmatrix}0&1\\1&0\end{bmatrix}}\end{bmatrix}}={\begin{bmatrix}0&0&0&-i\\0&0&-i&0\\0&i&0&0\\i&0&0&0\end{bmatrix}}}$

Both the Pauli-X and the Pauli-Y gate act on a single qubit. The resulting gate ${\displaystyle C}$ act on two qubits.

Hadamard transform[]

The gate ${\displaystyle H_{2}=H\otimes H}$ is the Hadamard gate (${\displaystyle H}$) applied in parallel on 2 qubits. It can be written as:

${\displaystyle H_{2}=H\otimes H={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\otimes {\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}}}$

This "two-qubit parallel Hadamard gate" will when applied to, for example, the two-qubit zero-vector (${\displaystyle |00\rangle }$), create a quantum state that have equal probability of being observed in any of its four possible outcomes; ${\displaystyle |00\rangle }$, ${\displaystyle |01\rangle }$, ${\displaystyle |10\rangle }$, and ${\displaystyle |11\rangle }$. We can write this operation as:

${\displaystyle H_{2}|00\rangle ={\frac {1}{2}}{\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\end{bmatrix}}{\begin{bmatrix}1\\0\\0\\0\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1\\1\\1\\1\end{bmatrix}}={\frac {1}{2}}|00\rangle +{\frac {1}{2}}|01\rangle +{\frac {1}{2}}|10\rangle +{\frac {1}{2}}|11\rangle ={\frac {|00\rangle +|01\rangle +|10\rangle +|11\rangle }{2}}}$

Example: The Hadamard transform on a 3-qubit register ${\displaystyle |\psi \rangle }$.

Here the amplitude for each measurable state is 1⁄2. The probability to observe any state is the square of the absolute value of the measurable states amplitude, which in the above example means that there is one in four that we observe any one of the individual four cases. See measurement for details.

${\displaystyle H_{2}}$ performs the Hadamard transform on two qubits. Similarly the gate ${\displaystyle \underbrace {H\otimes H\otimes \dots \otimes H} _{n{\text{ times}}}=\bigotimes _{1}^{n}H=H^{\otimes n}=H_{n}}$ performs a Hadamard transform on a register of ${\displaystyle n}$ qubits.

When applied to a register of ${\displaystyle n}$ qubits all initialized to ${\displaystyle |0\rangle }$, the Hadamard transform puts the quantum register into a superposition with equal probability of being measured in any of its ${\displaystyle 2^{n}}$ possible states:

${\displaystyle \bigotimes _{0}^{n-1}H|0\rangle ={\Big (}\bigotimes _{0}^{n-1}H{\Big )}{\Big (}\bigotimes _{0}^{n-1}|0\rangle {\Big )}={\frac {1}{\sqrt {2^{n}}}}{\begin{bmatrix}1\\1\\\vdots \\1\end{bmatrix}}={\frac {1}{\sqrt {2^{n}}}}{\Big (}|0\rangle +|1\rangle +\dots +|2^{n}-1\rangle {\Big )}={\frac {1}{\sqrt {2^{n}}}}\sum _{i=0}^{2^{n}-1}|i\rangle }$

This state is a uniform superposition and it is generated as the first step in some search algorithms, for example in amplitude amplification and phase estimation.

Measuring this state results in a random number between ${\displaystyle |0\rangle }$ and ${\displaystyle |2^{n}-1\rangle }$.^[h] How random the number is depends on the fidelity of the logic gates. If not measured, it is a quantum state with equal probability amplitude ${\displaystyle {\frac {1}{\sqrt {2^{n}}}}}$ for each of its possible states.

The Hadamard transform acts on a register ${\displaystyle |\psi \rangle }$ with ${\displaystyle n}$ qubits such that ${\textstyle |\psi \rangle =\bigotimes _{i=0}^{n-1}|\psi _{i}\rangle }$ as follows:

${\displaystyle \bigotimes _{0}^{n-1}H|\psi \rangle =\bigotimes _{i=0}^{n-1}{\frac {|0\rangle +(-1)^{\psi _{i}}|1\rangle }{\sqrt {2}}}={\frac {1}{\sqrt {2^{n}}}}\bigotimes _{i=0}^{n-1}{\Big (}|0\rangle +(-1)^{\psi _{i}}|1\rangle {\Big )}=H|\psi _{0}\rangle \otimes H|\psi _{1}\rangle \otimes \cdots \otimes H|\psi _{n-1}\rangle }$

Application on entangled states[]

If two or more qubits are viewed as a single quantum state, this combined state is equal to the tensor product of the constituent qubits. Any state that can be written as a tensor product from the constituent subsystems are called separable states. On the other hand, an entangled state is any state that cannot be tensor-factorized, or in other words: An entangled state can not be written as a tensor product of its constituent qubits states. Special care must be taken when applying gates to constituent qubits that make up entangled states.

If we have a set of N qubits that are entangled and wish to apply a quantum gate on M < N qubits in the set, we will have to extend the gate to take N qubits. This application can be done by combining the gate with an identity matrix such that their tensor product becomes a gate that act on N qubits. The identity matrix (${\displaystyle I}$) is a representation of the gate that maps every state to itself (i.e., does nothing at all). In a circuit diagram the identity gate or matrix will often appear as just a bare wire.

The example given in the text. The Hadamard gate ${\displaystyle H}$ only act on 1 qubit, but ${\displaystyle |\psi \rangle }$ is an entangled quantum state that spans 2 qubits. In our example, ${\displaystyle |\psi \rangle ={\frac {|00\rangle +|11\rangle }{\sqrt {2}}}}$

For example, the Hadamard gate (${\displaystyle H}$) acts on a single qubit, but if we feed it the first of the two qubits that constitute the entangled Bell state ${\displaystyle {\frac {|00\rangle +|11\rangle }{\sqrt {2}}}}$, we cannot write that operation easily. We need to extend the Hadamard gate ${\displaystyle H}$ with the identity gate ${\displaystyle I}$ so that we can act on quantum states that span two qubits:

${\displaystyle K=H\otimes I={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&1\\1&-1\end{bmatrix}}\otimes {\begin{bmatrix}1&0\\0&1\end{bmatrix}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&0&1&0\\0&1&0&1\\1&0&-1&0\\0&1&0&-1\end{bmatrix}}}$

The gate ${\displaystyle K}$ can now be applied to any two-qubit state, entangled or otherwise. The gate ${\displaystyle K}$ will leave the second qubit untouched and apply the Hadamard transform to the first qubit. If applied to the Bell state in our example, we may write that as:

${\displaystyle K{\frac {|00\rangle +|11\rangle }{\sqrt {2}}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&0&1&0\\0&1&0&1\\1&0&-1&0\\0&1&0&-1\end{bmatrix}}{\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\0\\0\\1\end{bmatrix}}={\frac {1}{2}}{\begin{bmatrix}1\\1\\1\\-1\end{bmatrix}}={\frac {|00\rangle +|01\rangle +|10\rangle -|11\rangle }{2}}}$

Computational complexity and the tensor product[]

The time complexity for multiplying two ${\displaystyle n\times n}$-matrices is at least ${\displaystyle \Omega (n^{2}\log n)}$,^[30] if using a classical machine. Because the size of a gate that operates on ${\displaystyle q}$ qubits is ${\displaystyle 2^{q}\times 2^{q}}$ it means that the time for simulating a step in a quantum circuit (by means of multiplying the gates) that operates on generic entangled states is ${\displaystyle \Omega ({2^{q}}^{2}\log({2^{q}}))}$. For this reason it is believed to be intractable to simulate large entangled quantum systems using classical computers. Subsets of the gates, such as the Clifford gates, or the trivial case of circuits that only implement classical boolean functions (e.g. combinations of X, CNOT, Toffoli), can however be efficiently simulated on classical computers.

The state vector of a quantum register with ${\displaystyle n}$ qubits is ${\displaystyle 2^{n}}$ complex entries. Storing the probability amplitudes as a list of floating point values is not tractable for large ${\displaystyle n}$.

Unitary inversion of gates[]

Example: The unitary inverse of the Hadamard-CNOT product. The three gates ${\displaystyle H}$, ${\displaystyle I}$ and ${\displaystyle \mathrm {CNOT} }$ are their own unitary inverses.

Because all quantum logical gates are reversible, any composition of multiple gates is also reversible. All products and tensor products (i.e. series and parallel combinations) of 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, I/O and spontaneous decoherence are side effects in quantum computers. Gates however are purely functional and bijective.

If ${\displaystyle U}$ is a unitary matrix, then ${\displaystyle U^{\dagger }U=UU^{\dagger }=I}$ and ${\displaystyle U^{\dagger }=U^{-1}}$. The dagger (${\displaystyle \dagger }$) denotes the conjugate transpose. It is also called the Hermitian adjoint.

If a function ${\displaystyle F}$ is a product of ${\displaystyle m}$ gates, ${\displaystyle F=A_{1}\cdot A_{2}\cdot \dots \cdot A_{m}}$, the unitary inverse of the function ${\displaystyle F^{\dagger }}$ can be constructed:

Because ${\displaystyle (UV)^{\dagger }=V^{\dagger }U^{\dagger }}$ we have, after repeated application on itself

${\displaystyle F^{\dagger }=\left(\prod _{0<i\leq m}A_{i}\right)^{\dagger }=\prod _{0\leq i<m}A_{m-i}^{\dagger }=A_{m}^{\dagger }\cdot \dots \cdot A_{2}^{\dagger }\cdot A_{1}^{\dagger }}$

Similarly if the function ${\displaystyle G}$ consists of two gates ${\displaystyle A}$ and ${\displaystyle B}$ in parallel, then ${\displaystyle G=A\otimes B}$ and ${\displaystyle G^{\dagger }=(A\otimes B)^{\dagger }=A^{\dagger }\otimes B^{\dagger }}$.

Gates that are their own unitary inverses are called Hermitian or self-adjoint operators. Some elementary gates such as the Hadamard (H) and the Pauli gates (I, X, Y, Z) are Hermitian operators, while others like the phase shift (S, T, P, CPHASE) gates generally are not. Gates that are not Hermitian are called skew-Hermitian, or adjoint operators.

For example, an algorithm for addition can be used for subtraction, if it is being "run in reverse", as its unitary inverse. The 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#^[10] and Bernhard Ömer's QCL^[31]: 61, contain function inversion as programming concepts.

Measurement[]

Circuit representation of measurement. The two lines on the right hand side represent a classical bit, and the single line on the left hand side represents a qubit.

Measurement (sometimes called observation) is irreversible and therefore not a quantum gate, because it assigns the observed quantum state to a single value. Measurement takes a quantum state and projects it to one of the basis vectors, with a likelihood equal to the square of the vector's length (in the 2-norm^{[4]: 66 [5]: 56, 65}) along that basis vector.^{[1]: 15–17 [32][33][34]} This is known as the Born rule and appears^[h] as a stochastic non-reversible operation as it probabilistically sets the quantum state equal to the basis vector that represents the measured state. At the instant of measurement, the state is said to "collapse" to the definite single value that was measured. Why and how, or even if^[35][36] the quantum state collapses at measurement, is called the measurement problem.

The probability of measuring a value with probability amplitude ${\displaystyle \phi }$ is ${\displaystyle 1\geq |\phi |^{2}\geq 0}$, where ${\displaystyle |\cdot |}$ is the modulus.

Measuring a single qubit, whose quantum state is represented by the vector ${\displaystyle a|0\rangle +b|1\rangle ={\begin{bmatrix}a\\b\end{bmatrix}}}$, will result in ${\displaystyle |0\rangle }$ with probability ${\displaystyle |a|^{2}}$, and in ${\displaystyle |1\rangle }$ with probability ${\displaystyle |b|^{2}}$.

For example, measuring a qubit with the quantum state ${\displaystyle {\frac {|0\rangle -i|1\rangle }{\sqrt {2}}}={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1\\-i\end{bmatrix}}}$ will yield with equal probability either ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$.

For a single qubit, we have a unit sphere in ${\displaystyle \mathbb {C} ^{2}}$ with the quantum state ${\displaystyle a|0\rangle +b|1\rangle }$ such that ${\displaystyle |a|^{2}+|b|^{2}=1}$. The state can be re-written as ${\displaystyle |\cos \theta |^{2}+|\sin \theta |^{2}=1}$, or ${\displaystyle |a|^{2}=\cos ^{2}\theta }$ and ${\displaystyle |b|^{2}=\sin ^{2}\theta }$.
Note: ${\displaystyle |a|^{2}}$ is the probability of measuring ${\displaystyle |0\rangle }$ and ${\displaystyle |b|^{2}}$ is the probability of measuring ${\displaystyle |1\rangle }$.