Quantum gate

Quantum gates are the elementary operations that a quantum computer can perform on its qubits . They are comparable to electronic gates that carry out the elementary operations of a classic computer. However, a quantum gate works with quantum mechanical systems such as spin . Even if their name suggests it, quantum gates are usually not physical components like transistors . A quantum gate is rather a time-controllable interaction of the qubits with one another or with the environment.

From a mathematical point of view, a quantum gate is a unitary transformation that is applied to the state of the qubits and creates the state . The unitarity this transformation follows from the requirement that a quantum gates must be given the normalization of the wave function: . ${\ displaystyle U}$ ${\ displaystyle \ Psi}$ ${\ displaystyle U \ Psi}$ ${\ displaystyle | U \ Psi | ^ {2} = | \ Psi | ^ {2}}$

presentation

In order to be able to write as a matrix , one usually selects the arithmetic basis as the basic states, i.e. precisely the qubit states that correspond to classical numbers. For example, for two qubits, the calculation base consists of ${\ displaystyle U}$

${\ displaystyle | 00 \ rangle, | 01 \ rangle, | 10 \ rangle, | 11 \ rangle.}$

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

${\ displaystyle {\ mathsf {SWAP}} = {\ begin {pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \ end {pmatrix}}}$

Such a matrix representation is useful for concrete calculations. However, in order not to lose the overview when several quantum gates are applied one after the other to the system, switching symbols are introduced analogous to the classic logic gates , which are connected to form a quantum circuit. Each circuit symbol corresponds to a unitary operation.

The graphs shown are intended to represent the Bloch sphere for various initial and final states, each of which is shown in a different color. This makes it easier to imagine the rotations. However, the probabilities of the individual superimposed states cannot be taken into account in this form of representation.

Universal gates

For reasons of simpler implementation, it is desirable in a quantum computer, similar to the classical computer, to limit oneself to a handful of elementary, easy-to-implement gates. There, for example, the NAND gate alone is sufficient to build every conceivable circuit.

A set of quantum gates is called universal if every unitary transformation can be represented as the product of gates from the set under consideration. ${\ displaystyle U}$

It could be shown that the CNOT gate together with all 1-qubit gates is such a universal set.

Examples

Two commonly used quantum gates
Surname	matrix	symbol	description
Hadamard Gate	${\ displaystyle {\ frac {1} {\ sqrt {2}}} {\ begin {pmatrix} 1 & 1 \\ 1 & -1 \ end {pmatrix}}}$		Transferred and in superimposed states. ${\ displaystyle \| 0 \ rangle}$ ${\ displaystyle \| 1 \ rangle}$
Controlled NOT (CNOT)	${\ displaystyle {\ begin {pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \ end {pmatrix}}}$		The first qubit is called the control qubit and the second is called the target qubit. The target qubit is inverted exactly when the control bit is on . ${\ displaystyle \| 1 \ rangle}$

particularities

In addition to the properties mentioned at the beginning, quantum gates have other peculiarities that distinguish them from classic gates and should therefore be emphasized again.

Reversibility

The operation realized by the quantum gate is a unitary transformation and thus in particular also a reversible or reversible transformation. That means: The effect of each quantum gate can be reversed with another quantum gate. One consequence of this is that a quantum gate cannot have more inputs than outputs, because then one of the input qubits would be lost.

Non-copyability

Since a quantum gate is an operation performed on the qubits, a quantum gate cannot generate more qubits than are initially present. In particular, the state of a qubit cannot be copied without destroying the initial state. This states the important no-cloning theorem . While the line can branch from one bit into two lines in a classic circuit diagram, this is not possible with the quantum computer.

Therefore there is exactly one line per qubit in a quantum circuit. This is drawn continuously from left to right through the circuit diagram and includes the 1-qubit gates and the connections of the multi-qubit gates.

realization

The physical realization of a quantum gate naturally depends on how the qubit itself is physically realized. Particles held in an ion trap are manipulated, for example, with the aid of photons with a predetermined quantization state.

How a quantum gate works as an ion trap
Step 1	step 2	step 3

An ion (yellow) is held in an ion trap by an electromagnetic field (blue) and cooled by a laser . A polarized photon (green) is "shot" at this ion .	As soon as the photon hits the ion, an interaction occurs between the two particles. This is the actual arithmetic operation on the quantum system.	When the photon leaves the ion trap, the ion has assumed a state that results from the superposition of the quantum state of the ion and the photon.

1-qubit gate

A single qubit with the states can always be written purely formally as the spin state of a spin ½ particle. The states can therefore always be represented as elements on the so-called Bloch sphere . A gate that works on a single qubit can then be formally described as a rotation on the Bloch sphere through a certain angle. ${\ displaystyle | 0 \ rangle, | 1 \ rangle}$

2-qubit gate

For quantum gates that work on two qubits, an interaction between the qubits in question is required. In the case of spin qubits, this can be done through the exchange interaction, among other things . 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 multiple-particle effects required for them, proposals for quantum computers are usually limited to 1 and 2-qubit gates. It is enough 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 gates with a single input are able to change a single qubit. This qubit can only represent either logic 1 or logic 0. So that alone is not an advantage compared to the previous electronic gates . However, the phase position is an indicator of how likely the respective states are. One speaks here of the fact that the two states and are superimposed and that the qubit is in superposition . For example, with a phase shift of 90 °, 50% of the measured values are logical 1 and the other 50% of the measured values are logical 0. An arithmetic operation on such a qubit is therefore applied to the state and the state at the same time. ${\ displaystyle \ left | 0 \ right \ rangle}$ ${\ displaystyle \ left | 1 \ right \ rangle}$ ${\ displaystyle \ left | 0 \ right \ rangle}$ ${\ displaystyle \ left | 1 \ right \ rangle}$

The disadvantage is that, due to the collapse of the wave function, only a single possible result is returned during a measurement . A usable result is therefore usually only possible by repeating the arithmetic operation several times and a subsequent statistical evaluation of the measurement results. However, if you are calculating with several qubits at the same time, you can sometimes use a trick, for example the quantum Fourier transform , to get useful results with just one calculation.

Web links

List of simulators

Individual evidence

↑ Farrokh Vatan, Colin Williams: Optimal Quantum Circuits for General Two-Qubit Gates ( English , PDF) arxiv.org. January. Retrieved September 24, 2019.

[1] Farrokh Vatan, Colin Williams: Optimal Quantum Circuits for General Two-Qubit Gates ( English , PDF) arxiv.org. January. Retrieved September 24, 2019.