Quantum gates

We need to figure out which part goes in evolutions and operations and which goes into computation (eg, the universal section) A '''quantum gate''' or '''quantum logic gate''' is a rudimentary [[quantum circuit]] operating on a small number of [[qubit]]s. They are the analogues for [[quantum computer]]s to classical [[logic gate]]s for conventional [[digital computer]]s. Quantum logic gates are [[reversible]], unlike many classical logic gates. Some universal classical logic gates, such as the [[Toffoli gate]], provide reversibility and can be directly mapped onto quantum logic gates. Quantum logic gates are represented by [[unitary]] matrices. The most common quantum gates operate on spaces of one or two qubits. This means that as matrices, quantum gates can be described by 2 x 2 or 4 x 4 matrices with [[orthonormal]] rows. '''Remark'''. The investigation of quantum logic gates is unrelated to [[quantum logic]], which is a foundational formalism for [[quantum mechanics]] based on a modification of some of the rules of [[propositional logic]]. == Examples == '''Hadamard gate'''. This gate operates on a single qubit. It is represented by the Hadamard matrix: [[Image:Hadamard-gate.png|thumb|Graphical representation of Hadamard gate]] :

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

Since the rows of the matrix are orthogonal, ''H'' is indeed a unitary matrix. '''Phase shifter gates'''. Gates in this class operate on a single qubit. They are represented by 2 x 2 matrices of the form :

R(\theta) =  \begin{bmatrix} 1 & 0\\ 0 & e^{i \theta} \end{bmatrix}

where θ is the ''phase shift''. '''Controlled gates'''. Suppose ''U'' is a gate that operates on single qubits with matrix representation :

U =  \begin{bmatrix} x_{00} & x_{10} \\ x_{01} & x_{11} \end{bmatrix}

The ''controlled-U gate'' is a gate that operates on two qubits in such a way that the first qubit serves as a control. [[Image:Controlled-gate.png|thumb|Graphical representation of controlled-''U'' gate]] :

| 0 0 \rangle \mapsto | 0 0 \rangle

| 0 1 \rangle \mapsto | 0 1 \rangle

| 1 0 \rangle \mapsto | 1 \rangle U |0 \rangle = | 1 \rangle \left(x_{00} |0 \rangle + x_{10} |1 \rangle\right)

| 1 1 \rangle \mapsto | 1 \rangle U |1 \rangle = | 1 \rangle \left(x_{01} |0 \rangle + x_{11} |1 \rangle\right)

Thus the matrix of the controlled ''U'' gate is as follows: :

\operatorname{C}(U) =  \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & x_{00} & x_{01} \\  0 & 0 & x_{10} & x_{11} \end{bmatrix}

'''Uncontrolled gate'''. We note the difference between the controlled-''U'' gate and an ''uncontrolled'' 2 qubit gate [[Image:Uncontrolled-gate.png|thumb|Graphical representation of uncontrolled ''U'' gate]]

I \otimes U

defined as follows: :

| 0 0 \rangle \mapsto | 0 \rangle U |0 \rangle

| 0 1 \rangle \mapsto | 0 \rangle U |1 \rangle

| 1 0 \rangle \mapsto | 1 \rangle U |0 \rangle

| 1 1 \rangle \mapsto | 1 \rangle U |1 \rangle

represented by the unitary matrix :

\begin{bmatrix}  x_{00} & x_{01} & 0 & 0 \\ x_{10} & x_{11} & 0 & 0 \\ 0 & 0 & x_{00} & x_{01} \\  0 & 0 & x_{10} & x_{11} \end{bmatrix}.

Since this gate is reducible to more elementary gates it is usually not included in the basic repertoire of quantum gates. It is mentioned here only to contrast it with the previous controlled gate. == Universal Quantum Gates == A set of universal quantum gates is any set of gates to which any operation possible on a quantum computer can be reduced. One simple set of two-qubit universal quantum gates is the Hadamard gate (

H

), a phase rotation gate

R(\cos^{-1}\begin{matrix} \frac{3}{5} \end{matrix}))

, and the [[CNOT|controlled-NOT gate]], a special case of controlled-U such that :

\operatorname{CNOT} =  \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\  0 & 0 & 1 & 0 \end{bmatrix}

. A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate,

D(\theta)

\operatorname{D(\theta)}: |i,j,k\rangle \rightarrow \begin{cases} i \cos(\theta) |i,j,k\rangle + \sin(\theta) |i,j,1-k\rangle & \mbox{for }i=j=1 \\ |i,j,k\rangle & \mbox{otherwise}\end{cases}

The universal classical logic gate, the [[Toffoli gate]], is reducible to the Deutsch gate,

D(\begin{matrix} \frac{\pi}{2} \end{matrix})

, thus showing that all classical logic operations can be performed on a universal quantum computer. == References == * M. Nielsen and I. Chuang, ''Quantum Computation and Quantum Information'', Cambridge University Press, 2000 {{FromWikipedia}} [[Category:Evolutions and Operations]]

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