Quantum logic 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.

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.[lower-alpha 2]

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, Pauli Z 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 ${\displaystyle ZX=iY=-XZ}$ .

In general, any Hermitian unitary gate ${\displaystyle U}$ only has eigenvalues of ${\displaystyle \pm 1}$, and has the property ${\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U}$ and ${\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}}$.

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 ${\displaystyle {\sqrt {X}}=e^{i\pi /4}R_{x}(\pi /2)}$, where ${\displaystyle R_{x}(\pi /2)={\frac {1}{\sqrt {2}}}{\begin{bmatrix}1&-i\\-i&1\end{bmatrix}}}$ is a rotation operator gate.

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 Hermitian unitary 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.

The CNOT can be expressed in the Pauli basis as:

${\displaystyle {\mbox{CNOT}}=e^{i{\frac {\pi }{4}}(I-Z_{1})(I-X_{2})}=e^{-i{\frac {\pi }{4}}(I-Z_{1})(I-X_{2})}.}$

Being a Hermitian unitary operator, CNOT has the property ${\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U}$ and ${\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}}$, and is involutory.

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.

In general, any single qubit unitary gate can be expressed as ${\displaystyle U=e^{iH}}$, where H is a Hermitian matrix, and then the controlled U is ${\displaystyle CU=e^{i{\frac {1}{2}}(I-Z_{1})H_{2}}}$.

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. Such an operation is a unitary transformation on both the control qubit and the target qubit, and is described by a controlled-U gate.
• 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.[lower-alpha 3] Such an operation is a measurement on the control qubit, with its result controlling the operations on the target qubit (i.e. a classically controlled function). For example, this is used in the typical quantum teleportation protocol where Alice measures her system and then transmits the result through classical channel to Bob, so that Bob can perform the corrsponding operation to retrieve the unknown quantum state.

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 ${\textstyle \varphi ={\frac {\pi }{4}}}$ (historically known as the ${\displaystyle \pi /8}$ gate), the phase gate (also known as the S gate, written as 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 Hermitian adjoint of the T gate is the T-dagger (${\displaystyle T^{\dagger }}$) gate, or TdgGate in IBM Qiskit;[13] Similarly, the adjoint of the S gate is the S-dagger (${\displaystyle S^{\dagger }}$) gate, or SdgGate in Qiskit.[14]

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[lower-alpha 4]

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.[lower-alpha 5]

Introducing the global phase gate ${\displaystyle \operatorname {Ph} (\delta )={\begin{bmatrix}e^{i\delta }&0\\0&e^{i\delta }\end{bmatrix}}=\left(P(\delta )\cdot X\right)^{2}=\exp \left(i\delta I\right)=e^{i\delta }I}$[lower-alpha 6] we also have the identity ${\displaystyle R_{z}(\gamma )\operatorname {Ph} \left({\frac {\gamma }{2}}\right)=P(\gamma )}$.[3]^: 11[1]^: 77–83 The T gate's historic name of ${\displaystyle \pi /8}$ gate comes from the identity ${\displaystyle R_{z}(\pi /4)\operatorname {Ph} \left({\frac {\pi }{8}}\right)=P(\pi /4)}$, where ${\displaystyle R_{z}(\pi /4)={\begin{bmatrix}e^{-i\pi /8}&0\\0&e^{i\pi /8}\end{bmatrix}}}$.

Arbitrary single-qubit phase shift gates ${\displaystyle P(\varphi )}$ are natively available for transmon quantum processors through timing of microwave control pulses.[15] It can be explained in terms of change of frame.[16][17]

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 ${\displaystyle |11\rangle }$:

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

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

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.[18]

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 : ${\displaystyle R_{x}(\pi )=-iX,R_{y}(\pi )=-iY,R_{z}(\pi )=-iZ.}$

It's straightforward to work out the adjoint action of rotations on the Pauli vector, namely rotation effectively by double the angle a to apply Rodrigues' rotation formula:

${\displaystyle R_{n}(-a){\vec {\sigma }}R_{n}(a)=e^{i{\frac {a}{2}}\left({\hat {n}}\cdot {\vec {\sigma }}\right)}~{\vec {\sigma }}~e^{-i{\frac {a}{2}}\left({\hat {n}}\cdot {\vec {\sigma }}\right)}={\vec {\sigma }}\cos(a)+{\hat {n}}\times {\vec {\sigma }}~\sin(a)+{\hat {n}}~{\hat {n}}\cdot {\vec {\sigma }}~(1-\cos(a))~.}$

Taking the dot product of any unit vector with the above formula generates the expression of any single qubit gate when sandwiched within adjoint rotation gates. For example, it can be shown that ${\displaystyle R_{y}(-\pi /2)XR_{y}(\pi /2)={\hat {x}}\cdot ({\hat {y}}\times {\vec {\sigma }})=Z}$.

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

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 a Hermitian unitary matrix, and has the property ${\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U}$ and ${\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}}$. It is also involutory.

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 ${\displaystyle HR_{z}(\theta )H=R_{x}(\theta )}$.[lower-alpha 7]

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:

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

The swap gate can be decomposed into summation form:

${\displaystyle {\mbox{SWAP}}={\frac {I\otimes I+X\otimes X+Y\otimes Y+Z\otimes Z}{2}}}$

The swap gate is Hermitian and unitary; Therefore we have ${\displaystyle {\text{SWAP}}=e^{i{\frac {\pi }{2}}(I-{\text{SWAP}})}=e^{i{\frac {\pi }{4}}}R_{xx}(\pi /2)R_{yy}(\pi /2)R_{zz}(\pi /2)}$.

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.[19][20]

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 CC-U (controlled-controlled Unitary) 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.[21]

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.

The Toffoli gate can be expressed in the Pauli basis as:

${\displaystyle {\mbox{Toff}}=e^{i{\frac {\pi }{8}}(I-Z_{1})(I-Z_{2})(I-X_{3})}=e^{-i{\frac {\pi }{8}}(I-Z_{1})(I-Z_{2})(I-X_{3})}.}$

Being a Hermitian unitary operator, the Toffoli gate has the property ${\displaystyle e^{i\theta U}=(\cos \theta )I+(i\sin \theta )U}$ and ${\displaystyle U=e^{i{\frac {\pi }{2}}(I-U)}=e^{-i{\frac {\pi }{2}}(I-U)}}$, and is involutory.

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.

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 (where it's known as Mølmer–Sørensen gate).[22][23] These gates are defined as

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

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

${\displaystyle R_{zz}(\phi )=\exp \left(-i{\frac {\phi }{2}}Z\otimes Z\right)={\begin{bmatrix}e^{-i\phi /2}&0&0&0\\0&e^{i\phi /2}&0&0\\0&0&e^{i\phi /2}&0\\0&0&0&e^{-i\phi /2}\\\end{bmatrix}}.}$[24]

The CNOT gate can be further decomposed as products of rotation operator gates and exactly one Ising coupling gate (or Mølmer–Sørensen gate), for example

${\displaystyle {\mbox{CNOT}}=e^{-i{\frac {\pi }{4}}}R_{y_{1}}(-\pi /2)R_{x_{1}}(-\pi /2)R_{x_{2}}(-\pi /2)R_{xx}(\pi /2)R_{y_{1}}(\pi /2).}$

For systems with Ising like interactions, it is sometimes more natural to introduce the imaginary swap[25] or iSWAP gate defined as[26][27]

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

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.

The Deutsch (or ${\displaystyle D_{\theta }}$) gate, named after physicist David Deutsch is a three-qubit gate. It is a general case of CC-U or controlled-controlled-Unitary gate, and 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.[28]

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.

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 ${\displaystyle T^{-2}=(T^{2})^{\dagger }=S^{\dagger }}$.[lower-alpha 8]

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.

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 ${\displaystyle |2^{n}-1\rangle }$.[lower-alpha 9] 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:

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

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: