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Concept
Quantum logic gate
Summary
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. 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 the computational basis is used, which unless comparing it with something, just means that for a d-level quantum system (such as a qubit, a quantum register, or qutrits and qudits) the orthogonal basis vectors are labeled , or use binary notation.
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, building on notation introduced by Richard Feynman in 1986.
Quantum logic gates are represented by unitary matrices. A gate which acts on qubits is represented by a unitary matrix, and the set of all such gates with the group operation of matrix multiplication is the symmetry group U(2n). The quantum states that the gates act upon are unit vectors in complex dimensions, with the complex Euclidean norm (the 2-norm). 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.
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