Quantum gate: Information from Answers.com

This article is about quantum computation. Quantum logic could also refer to a different formulation of propositional logic based on quantum mechanics.

In quantum computing and specifically the quantum circuit model of computation, a quantum gate (or quantum logic 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, classical computing can be performed using only reversible gates. For example, the reversible Toffoli gate can implement all Boolean functions. This gate has a direct quantum equivalent, showing that quantum circuits can perform all operations performed by classical circuits.

Quantum logic gates are represented by unitary matrices. The most common quantum gates operate on spaces of one or two qubits, just like the common classical logic gates operate on one or two bits. This means that as matrices, quantum gates can be described by 2 × 2 or 4 × 4 unitary matrices.

1 Commonly used gates
2 Universal quantum gates
3 History
4 See also
5 References

Commonly used gates

Quantum gates are usually represented as matrices. A gate which acts on k qubits is represented by a 2^k x 2^k unitary matrix. The number of qubits in the input and output of the gate have to be equal. The action of the quantum gate is found by multiplying the matrix representing the gate with the vector which represents the quantum state.

Hadamard gate

The Hadamard gate acts on a single qubit. It maps the basis state $|0\rangle$ to $\frac{|0\rangle + |1\rangle}{\sqrt{2}}$ and $|1\rangle$ to $\frac{|0\rangle - |1\rangle}{\sqrt{2}}$ . It is represented by the Hadamard matrix:

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.

Pauli-X gate

The Pauli-X gate acts on a single qubit. It is the quantum equivalent of a NOT gate. It maps $|0\rangle$ to $|1\rangle$ and $|1\rangle$ to $|0\rangle$ . It is represented by the Pauli X matrix:

$X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ .

Pauli-Y gate

The Pauli-Y gate acts on a single qubit. It is represented by the Pauli Y matrix:

$Y = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$ .

Pauli-Z gate

The Pauli-Z gate acts on a single qubit. It leaves the basis state $|0\rangle$ unchanged and maps $|1\rangle$ to $-|1\rangle$ . It is represented by the Pauli Z matrix:

$Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$ .

Phase shift gates

This is a family of single-qubit gates that leave the basis state $|0\rangle$ unchanged and map $|1\rangle$ to $e^{i\theta}|1\rangle$ .

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

where θ is the phase shift. Some common examples are the $\frac{\pi}{8}$ gate, where θ = $\frac{\pi}{4}$ , the phase gate where θ = $\frac{\pi}{2}$ and the Pauli-Z gate where θ = $π$ .

Swap gate

The swap gate swaps two qubits. It is represented by the matrix:

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

Controlled gates

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some operation. For example, the controlled-NOT (or CNOT) gate acts on 2 qubits, and performs the NOT operation on the second qubits only when the first qubit is $|1\rangle$ , and otherwise leaves it unchanged. It is represented by the matrix

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

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

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

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.

Graphical representation of controlled-U gate

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

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

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

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

The matrix representing the controlled U is

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

Toffoli gate

The Toffoli gate is a 3-bit gate, which is universal for classical computation. The quantum Toffoli gate is the same gate, defined for 3 qubits. If the first two bits are in the state $|1\rangle$ , it applies a Pauli-X 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.

INPUT			OUTPUT
0	0	0	0	0	0
0	0	1	0	0	1
0	1	0	0	1	0
0	1	1	0	1	1
1	0	0	1	0	0
1	0	1	1	0	1
1	1	0	1	1	1
1	1	1	1	1	0

It can be also described as the gate which maps $|a, b, c\rangle$ to $|a, b, c\oplus ab\rangle$ .

Universal quantum gates

Informally, 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 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, the Solovay–Kitaev theorem guarantees that this can be done efficiently.

One simple set of two-qubit universal quantum gates is the Hadamard gate ( $H$ ), a phase gate $R (π / 4)$ , and the controlled NOT gate.

A single-gate set of universal quantum gates can also be formulated using the three-qubit Deutsch gate, $D (θ)$ which performs the transformation (where $\iota=\sqrt{-1}$ )

$|i,j,k\rangle \mapsto \begin{cases} \iota \cos(\theta) |i,j,k\rangle + \sin(\theta) |i,j,1-k\rangle & \mbox{for }i=j=1 \\ |i,j,k\rangle & \mbox{otherwise.}\end{cases}$

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.

History

The current notation for quantum gates was developed by Barenco et al.,^[1] building on notation introduced by Feynman^[2].

References

M. Nielsen and I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000

^ Phys. Rev. A 52 3457–3467 (1995), DOI:10.1103/PhysRevA.52.3457; e-print arXiv:quant-ph/9503016
^ R. P. Feynman, “Quantum mechanical computers”, Optics News, February 1985, 11, p. 11; reprinted in Foundations of Physics 16(6) 507–531

v • d • e Quantum computing

General	Quantum computer • Qubit • Quantum information • Quantum programming • Quantum virtual machine • Timeline of quantum computing

Quantum communication	Quantum channel • Quantum cryptography • Quantum teleportation • Superdense coding • LOCC • Entanglement distillation

Quantum algorithms	Universal quantum simulator • Deutsch–Jozsa algorithm • Grover's search • Shor's factorization • Simon's Algorithm

Quantum complexity theory	Quantum Turing machine • BQP • QMA

Quantum computing models	Quantum circuit (quantum gate) • One-way quantum computer (cluster state) • Adiabatic quantum computation

Decoherence prevention	Quantum error correction • Topological quantum computer

Physical implementations

Quantum optics	Cavity QED

Ultracold atoms	Trapped ion quantum computer • Optical lattice

Spin-based	Nuclear magnetic resonance (NMR) quantum computing • Kane QC • Loss–DiVincenzo (quantum dot) QC

Other	Superconducting quantum computing (Charge qubit • Flux qubit • Phase qubit) • Nitrogen-vacancy center

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