Quantum gate

Quantum gates are the basic operations that can perform a quantum computer on his qubits. They are comparable with electronic gates, which perform the basic operations of a classical computer. However, a quantum gate operates with quantum mechanical systems such as the spin. Although their name might suggest, provide quantum gate is usually no physical components such as transistors represents a quantum gate is rather a time- controlled interaction of the qubits with each other or with the environment.

From a mathematical perspective, a quantum gate is a unitary transformation that is applied to the state of the qubit, and generates the state. The unitarity of this transformation follows from the requirement that a quantum gate must obtain the normalization of the wave function.

• 5.1 1- qubit gates
• 5.2 2- qubit gates

Representation

To write as a matrix, is chosen as the basis states usually just the qubit states corresponding to classical figures, for two qubits ie

A quantum gate which simply swaps the two qubits, would then have the matrix representation

For specific calculations, such a matrix representation is useful. But in order not to lose the overview at several successively applied to the system quantum gates, one introduces analogously to a classical logic gates circuit symbols that are connected to a quantum circuit. Each circuit symbol corresponds to the fact of a unitary operation.

The graphs shown are intended to represent the Bloch sphere for different initial and final states, which are represented in a different color. Thus, one can imagine the rotations better. The probabilities of each superposed states, however, can not be considered in this presentation.

Universal Gates

For ease of implementation, it is desirable to limit similar to the classical computer on a handful of basic, easy to be realized gate in a quantum computer. There is, for example, the NAND gate alone is sufficient to build any conceivable circuit.

A set of quantum gates is called universal if any unitary transformation can be represented as a product of gates from the observed set.

Examples

Special

Quantum gates have, in addition to the properties mentioned at the beginning to yet further features that you want to distinguish them from the classical gates and therefore underlines again.

Reversibility

The realized from the quantum gate operation is a unitary transformation and thus in particular also an irreversible or reversible transformation. This means that the effect of each quantum gate can be reversed with another quantum gate. A consequence of this is that a quantum gate can not have more inputs than outputs, because it would mean yes one of the Eingangsqubits lost.

Non - copyability

Since a quantum gate is performed on the qubit operation, a quantum gate can not generate more qubits than are present from the beginning. In particular, the state of a qubit can not be copied without destroying the initial state. This implies the important no- cloning theorem. Thus, while in a conventional circuit diagram of a bit line into two lines may branch, this is not possible for the quantum computer.

Therefore, there are exactly in a quantum circuit one line per qubit. This is drawn continuously from left to right through the schematic and includes the one - qubit gates and the connections of the multi - qubit gate.

Realization

The physical realization of a quantum gate, of course, depends on how the qubit itself is physically realized. In an ion trap captured particles can be manipulated, for example by means of photons with a given Quantisierungszustand.

1- qubit gates

A single qubit with the states can be always written as a purely formal spin state of a spin- ½ particle. The states can therefore always be represented as elements of the so-called Bloch sphere. A gate, which operates on a single qubit, it can be formally described as a rotation on the Bloch sphere at a certain angle.

Two - qubit gates

For quantum gates that operate on two qubits, an interaction between the qubit in question is required. For spin qubits this can be done, inter alia, the exchange interaction. Atoms in an ion trap could exchange photons.

Since gates with more than two inputs are theoretically conceivable, but are much more complex to implement due to the necessary for Mehrteilcheneffekte, we restrict ourselves the case of proposals for quantum computers usually on the 1 - and 2- qubit gates. It suffices indeed to have a universal set of gates with these gates.

See also: List of quantum gates, Optimal Quantum Circuits for General Two - qubit gates

Effect

Quantum gate having a single input is to change a single qubit able. This qubit can only either logic 1 or logic 0 represent. That alone is thus no advantage compared to the previous electronic gates. However, the phase angle is an indication of how likely the respective states. It is said here that overlap the two states and and the qubit is in superposition. For example, in a phase shift of 90 °, the values ​​measured at 50% of a logic 1 and the other 50 % of the measurement values ​​logic 0 an arithmetic operation on such a qubit is therefore applied to the status and the condition of the same time.

The disadvantage is that, for a measurement due to the collapse of the wave function, only a single possible result is returned. A useful result is therefore usually possible only by multiple repetition of the arithmetic operation and a subsequent statistical evaluation of the measurement results. However, if multiple qubits is expected at the same time, it is sometimes useful results can occur at a trick, for example, the quantum fourier transform, already with only one calculation.