Quantum gate

Quantum gate

A quantum gate or quantum logic gate is a basic quantum circuit operating on a small number of qubits. They are the analogues for quantum computers to classical logic gates for conventional digital computers. 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 × 2 or 4 × 4 matrices with orthonormal rows.

Remark. "Quantum logic" can refer either to the performance of quantum logic gates or to a foundational formalism for quantum mechanics called quantum logic 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:

: H = frac{1}{sqrt{2 egin{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 × 2 matrices of the form

: R( heta) = egin{bmatrix} 1 & 0 \ 0 & e^{2 pi i heta} end{bmatrix}

where θ is the "phase shift".

Controlled gates. Suppose "U" is a gate that operates on single qubits with matrix representation

: U = egin{bmatrix} x_{00} & x_{01} \ x_{10} & 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.

: | 0 0 angle mapsto | 0 0 angle

: | 0 1 angle mapsto | 0 1 angle

: | 1 0 angle mapsto | 1 angle U |0 angle = | 1 angle left(x_{00} |0 angle + x_{10} |1 angle ight)

: | 1 1 angle mapsto | 1 angle U |1 angle = | 1 angle left(x_{01} |0 angle + x_{11} |1 angle ight)

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

: operatorname{C}(U) = egin{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 I otimes U defined as follows:

: | 0 0 angle mapsto | 0 angle U |0 angle

: | 0 1 angle mapsto | 0 angle U |1 angle

: | 1 0 angle mapsto | 1 angle U |0 angle

: | 1 1 angle mapsto | 1 angle U |1 angle

represented by the unitary matrix

: egin{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, that is, any other unitary operation can be expressed as a finite sequence of gates from the set. Equivalently, a set of universal quantum gates is a set of generators for the group of unitary matrices. One simple set of two-qubit universal quantum gates is the Hadamard gate (H), a phase rotation gate R(cos^{-1}egin{matrix} frac{3}{5} end{matrix})), and the controlled NOT gate, a special case of controlled-U such that

: operatorname{CNOT} = egin{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( heta)

: operatorname{D( heta)}: |i,j,k angle ightarrow egin{cases} i cos( heta) |i,j,k angle + sin( heta) |i,j,1-k angle & mbox{for }i=j=1 \ |i,j,k angle & mbox{otherwise}end{cases}.

The universal classical logic gate, the Toffoli gate, is reducible to the Deutsch gate, D(egin{matrix} frac{pi}{2} end{matrix}), thus showing that all classical logic operations can be performed on a universal quantum computer.

See also

*Pauli matrices
*Quantum finite automata

References

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


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Quantum Gate (video game) — Infobox VG| title = Quantum Gate developer = Hyperbole Studios publisher = Media Vision Technology designer = Greg Roach engine = Virtual Cinema version = released = 1993 genre = FMV game, adventure modes = ratings = platforms = Windows 3.x,… …   Wikipedia

  • Quantum teleportation — Quantum teleportation, or entanglement assisted teleportation, is a technique used to transfer information on a quantum level, usually from one particle (or series of particles) to another particle (or series of particles) in another location via …   Wikipedia

  • Quantum error correction — is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault tolerant quantum computation that can deal not only with noise on …   Wikipedia

  • Quantum circuit — In quantum information theory, a quantum circuit is a model for quantum computation in which a computation is a sequence of reversible transformations on a quantum mechanical analog of an n bit register. This analogous structure is referred to as …   Wikipedia

  • Quantum computer — A quantum computer is a device for computation that makes direct use of distinctively quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. In a classical (or conventional) computer, information is… …   Wikipedia

  • Quantum information — For the journal with this title, see Historical Social Research. In quantum mechanics, quantum information is physical information that is held in the state of a quantum system. The most popular unit of quantum information is the qubit, a two… …   Wikipedia

  • Quantum channel — In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information… …   Wikipedia

  • Quantum finite automata — In quantum computing, quantum finite automata or QFA are a quantum analog of probabilistic automata. They are related to quantum computers in a similar fashion as finite automata are related to Turing machines. Several types of automata may be… …   Wikipedia

  • Quantum dot cellular automaton — Quantum Dot Cellular Automata (sometimes referred to simply as quantum cellular automata, or QCA) Any device designed to represent data and perform computation, regardless of the physics principles it exploits and materials used to build it, must …   Wikipedia

  • Quantum optics — is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. History of quantum optics Light is made up of particles called photons and hence inherently is… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”