Qubit

A qubit ( for " quantum bit " rarely qubit ) ([' kju.bɪt ] or [ k'bɪt ] ) is a manipulable any two-state quantum system. That is, it is a system that is correctly described only by quantum mechanics and the safe distinguishable only by measuring two states has.

Form qubits in quantum computer science the basis for quantum computing and quantum cryptography. The qubit plays the role analogous to the classical bit in conventional computers: it serves as the smallest possible storage unit, and also defines a measure of the quantum information.

• 4.1 Description of individual qubits
• 4.2 Bloch sphere
• 4.3 Description of systems of several qubits

Properties

As a two-state quantum system qubit is the simplest non-trivial quantum system at all. The term " two-state system " does not refer to this as the number of states that the system may assume. In fact you can take infinitely many different states of each non-trivial quantum mechanical system in principle. However, in general, the state of a quantum system through measurement can not be determined with certainty, but by measuring random one of the possible values ​​of the observables measured is selected, the probability of each measured value is determined by the present prior to the measurement condition. Moreover, since the measurement changes the state, this problem can not be circumvented by repeated reading on the same system.

However, there are certain states for each measurement, if they are present prior to the measurement, the measured value can be predicted with absolute certainty, the so-called eigenstates of the measurement and the measured observables. In this case, for every possible outcome at least one such state. The maximum number of measured values ​​obtained here for measurements in which there is only exactly one state providing this value safely. In addition, after each measurement is an associated to the obtained measurement eigenstate before ( collapse of the wave function); However, the measurement is already available before a proper state of the measurement, it will not be changed.

Two states that one can reliably differentiate by measurement, also called orthogonal. The maximum number of measured values ​​in a measurement, and thus the maximum number of orthogonal states, is a property of the quantum system. In the qubit as a two-state system can accurately distinguish two different states so safely by measurement. Will you therefore use a qubit simply as classical memory, so you can save a classical bit is accurate. However, the benefits of the qubit are just in the existence of the other states.

An example is the polarization of a photon ( " light particle "). The polarization of light indicates the direction in which light vibrates. Although the polarization is actually a wave property, it can also be defined for the single photon, and all polarizations (linear in any direction, circular, elliptical) are also possible for single photons. Linear polarization can be measured, for example, a birefringent crystal. Where in the crystal emerges entering photon at a specific location, depending on whether it is polarized parallel or perpendicular to the optical axis of the crystal. There are so to speak, two " outputs", one for parallel and perpendicular polarized for photons. If, at both sites a photon detector, then one can thus determine whether the photon was polarized parallel or perpendicular to the optical axis.

Photons having a different polarization (linear at a different angle, circular or elliptical), but also occur in these " output " out. To which " output", such a photon coming out, but it is unpredictable in this case; only the likelihood may be predicted. However, afterwards, it has the polarization associated with the respective output, as can be demonstrated, for example, in that instead of the detector more crystals (with collimated optical axis) with two detectors mounted on the " outputs ": Only those detectors of the second crystals, respectively belonging to the correct polarization for the output of the first crystal register photons.

The crystal is thus distinguished from a polarization direction. Which is, however, one can thus determine, by rotating the crystal. Two linear polarization states are therefore orthogonal to each other, when the polarization directions are orthogonal to each other. However, can not be directly applied to other states of polarization of that correspondence; Thus, for example, the polarized state rechtszirkulär linkszirkulär and also orthogonal to each other.

As with classical bits also several qubits can be combined to save larger values. One - qubit system has it exactly mutually orthogonal states. In the qubit can thus be exactly classical bits store so that the complete information can be read reliably again; For example, a " quantum byte " from 8 qubits store 256 different again reliably readable values ​​.

Much more important for use in quantum computers is the existence of entangled states of multiple qubits. In such states of a single qubit has absolutely no defined state, the whole of the qubits, however, already. This leads to non-local correlations, as occur in the Einstein - Podolsky-Rosen paradox.

The entanglement of the qubits has surprising consequences. For example, you can store in a pair of entangled qubits two classical bits such that both bits can be manipulated separately by manipulating only one of the qubits. However, it requires both the qubit in order to read the information.

Also on the non-locality of entanglement based quantum teleportation, leave with which quantum states transmitted by transmitting classical bits.

Important for quantum computer is the fact that an arbitrary set of sequences of classical bits can be displayed simultaneously by the interleaving of a set of qubits. For example, can be produced with a 4 qubit state exactly the bit sequences 0000, 0101, 1011 and 1110 containing, and no other. In the extreme case, all possible bit sequences contained therein, eg contains an appropriately crafted " quantum byte " all the numbers from 0 to 255 at the same time. Is introduced with the aid now of quantum mechanical calculations of operations in this state, these calculations are carried out effectively to all these bit strings simultaneously. This so-called quantum parallelism is the reason that quantum computers can solve certain problems faster than classical computers. However, one can not read individually stored bit pattern; each measurement provides only a randomly selected one of the stored values ​​. To use the quantum parallelism, therefore additionally specifically quantum mechanical transformations need to be made that have no classical equivalents, ie states which correspond to exactly one bit pattern that can be converted into superpositions of several bit patterns and vice versa.

Implementation of qubits

In theory, each quantum mechanical two-state system can be used as a qubit. In practice, however, many systems are unsuitable because they can not be manipulated sufficiently or too much disturbed by the environment. In addition, there is the problem of scalability: Some implementations, such as the use of nuclear magnetic resonance in molecules, are inherently only for a very limited number of qubits.

For the usability of a system as a qubit David P. DiVincenzo has identified seven criteria. The first five criteria also apply to the use in quantum computers, the last two are specific for quantum communication.

The five general criteria are:

The two additional criteria for quantum communication are:

In practice, among others, the following systems are studied:

Ions in ion traps

A promising approach for quantum computers is the use of ions in ion traps. This individual ions are lined up by electromagnetic fields in the vacuum like pearls on a chain.

The qubits are formed by two long-lived states of the individual ions; the number of qubits is identical to the number of ions in the trap. The manipulation of the qubit is done by laser, which interact with the individual ions. About the movement of ions in the trap can be the qubits coupled together and fold so.

With this technology, up to 14 qubits have already been entangled

Electrons in quantum dots

Another approach is the use of quantum dots. Quantum dots are quasi- zero-dimensional semiconductor structures in which electrons can occupy only discrete states; one speaks therefore often of designer atoms. An advantage of quantum dot technology is that in the manufacture of semiconductor proven methods may be used.

SQUIDs

Even with SQUID qubits can be implemented. SQUIDs are systems consisting of superconductors, which are connected by two parallel Josephson junctions. The manipulation of the qubit is done via the applied voltage and the magnetic field.

Nuclear magnetic resonance

The spins of the nuclei of molecules can represent qubits that can be manipulated via nuclear magnetic resonance / read. This is a particularly technically simple method that does not meet the DiVincenzo criteria. In particular, the method is not scalable, also in this case ( a single molecule that is ) can not be a single system to be measured, but one has to deal with many of the same molecules at once.

Photon states

The polarization of photons described above is an example of moving qubits. It is not suitable for quantum computations, but it can be easily transferred via optical fibers over long distances. Experiments on quantum communication and quantum cryptography therefore use almost exclusively photon polarization states.

Quantum coding

Similar to classical information can be compressed and quantum information. It is assumed that the signal state of a purely random "alphabet", wherein these states need not necessarily be orthogonal to each other, that it may not be possible to distinguish the states secure measurement. These states are coded in a system of qubit (which can be the original state necessarily destroyed ), and this is sent to the receiver, which then reconstructs an approximation of the original state of the transmitted qubits.

The accuracy ( fidelity) of such coding is defined by the expected match of the reconstructed state of the original. That is, assuming the receiver knows which characters are transmitted, and shall keep at its reconstructed state one measurement for which the initial state is an eigenstate, then the accuracy of the encoding is given by the proportion of measurements that the transmitted state result.

An ideal encoding is now analogous to the classical theory of transfer of information, in which the minimum number to be transmitted to the qubit in order to achieve an arbitrarily high probability of transmission at a sufficiently large number of transmitted symbols.

It now appears that the minimum number of qubits to transmit such a state is just the von Neumann entropy defined by the "alphabet " and the associated probabilities density matrix. Thus, the qubit, analogous to classical bit be regarded as an information unit of the quantum information; the von Neumann entropy of a quantum system then provides just the information content in qubits.

In fact, the term qubit BW Schumacher was coined in this context.

Mathematical Description

Description of individual qubits

For the description of a qubit, take any measure used (parallel and perpendicular to the optical axis of a birefringent crystal, the polarization, for example in the example with the photons ) and calls the corresponding eigenstates and ( The notation to designate that it is a quantum state concerns, see also Bra- Ket notation). The quantum mechanical superposition principle is now demanding that there are infinitely many states of this system, which is formally known as

Can be written, where a and b are complex numbers with

Are. The state can thus, more precisely as normalized vector in a complex vector space, describe a Hilbert space. ( In the case of the photon is just around the Jones vector, which describes the polarization ). However, the description is not unique; two vectors that differ only by a factor of the form are different ( " phase factor " ), describe the same state. Note, however, that such a phase factor certainly makes for one of the components is a difference: The vectors and describe generally different states.

The system is in this state, then the probability to be found the state after the measurement, straight, and according to the probability of the state to be found are the same.

Alternatively, the qubit also describe about its density matrix. For the qubit in the state is the projection operator

In contrast to the state vector of the density matrix is uniquely defined. With the help of the density matrix can be described also qubits whose state is not completely known ( so-called " mixed states "). Generally leaves the density matrix of a qubit by state

Wherein the 2 × 2 unit matrix and the Pauli matrices. The probability to find the state with a corresponding measurement is given by.

Bloch sphere

The states of an individual ( unverschränkten ) qubits can be represented as points on the surface of a sphere in three-dimensional space. This surface is called after Felix Bloch Bloch sphere or sphere. Particularly clearly seen on the Spin-1/2-Teilchen where the point on the sphere indicates the direction in which it will be measured with certainty spin up. The equivalence however, applies to all two-state systems. The picture on the right shows how the above described polarization states on the Bloch sphere can be arranged. For example, corresponds to the " North Pole " here the vertical and the "south pole " of the horizontal polarization. General correspond to mutually orthogonal states of opposite points on the Bloch sphere.

Substituting the state to the " north pole " of the sphere, and are and the angle of the point in spherical coordinates (see picture left), the corresponding state is represented by the vector

Described.

The points inside the sphere can be interpreted: you can assign them qubit, whose state you do not have full information. The Cartesian coordinates of the point in the ball are then just the coefficients of the Pauli matrices in the equation (*). The center of the sphere thus corresponds to a qubit, about which you know nothing; The farther one moves away from the center, the greater the knowledge about the state of the qubit. This ball is in some ways the analogue of the probability interval [ 0,1] for the classical bit: Put the points on the edge of the possible states of the exact bits (0 or 1) or the qubits (in quantum mechanics is also called of "pure states " ), while the dots represent incomplete knowledge about the bit / qubit inside ( in quantum mechanics this is called " mixed states "). The point in the middle represents in both cases complete ignorance about the system ( in bits: a probability of 1 /2).

Also, the process of measuring can be illustrated nicely by means of the Bloch sphere: In the picture of the little red dot indicates a possible state of the qubit. In this case, the point is located on the outside of the sphere, so it is a pure state; but the method also works for mixed states. Since the natural conditions of the measurement are mutually orthogonal, in other words on the Bloch sphere opposite to each other, defining the measurement, a straight line through the center of the sphere ( in the image by the blue line ). Consider, now along these lines the diameter (in the picture green / white) through the ball and projects the point that represents the current knowledge of the qubit, perpendicular to this line ( the projection is here by the red plane and the yellow line marks, the intersection of the line with the yellow diameter is the projected point). This route can then be viewed directly as a probability interval for the measurement result. If it does not read the measuring result, then this point is within the ball, in fact, the new description of the system; by reading out the measurement result of the point is, of course (such as during a normal bit ) at one end of the track. If, for example, in the picture at the " north pole " of the sphere of the state and to the "south pole " of the state, then the ratio of the length of the white part of the diameter is ( from the South Pole to the intersection with the plane) to the overall diameter just the probability of finding the qubit after the measurement in the state when the state was previously passed through the red dot ( sitting behind the state in this case, of course, on the North Pole ).

Some physicists suspect in this context between qubits and points in three-dimensional space the reason that our space is three-dimensional. Prominent representatives of this idea is the original hypothesis of Carl Friedrich von Weizsäcker. Weizsäcker Ur is essentially what is now called qubit.

Description of systems of several qubits

The states of a system of multiple qubits form a Hilbert space due to the superposition principle. This is the tensor product of Hilbert spaces of the individual qubits. That is, a system is described by a qubit -dimensional Hilbert space, the base of states can be written as a direct product of the single qubit states, eg

Where the indices indicate to which of the qubit state each part. Each direct product of 1- qubit states results in a qubit state, e.g.

Conversely, this is not true: Some qubit states can not be written as a product of single- qubit states. An example of such a state is the 2- qubit state. Such conditions, which can not be written as a product of individual states are called entangled. The description of a single qubit in an entangled state is only possible via a density matrix, which indicates again the ignorance (or ignoring ) of information about the qubit: In this case, it is in the lack of information just to the entanglement with other qubits. However, the complete state can not be described by the density matrices are specified for each qubit. The entanglement is rather a non-local property, which is reflected in the correlations between the entangled qubits.

Complementary observables of qubit

Two observables are complementary if the full knowledge of the value of an observable the complete ignorance of the other implied. Since complete ignorance about the value is equivalent to projection onto the center of the Bloch sphere in the above description of the measurement, it is immediately evident that mutually complementary observables are described by mutually orthogonal directions in the Bloch sphere. Accordingly, it is always for a single qubit exactly three pairs of mutually complementary observables, corresponding to the three spatial directions.

If you have many identical copies of a qubit prepared, so the condition can be determined by measuring the probabilities of a set of three pairs of complementary observables (each measurement on a new copy must be made ​​, since the measurement has destroyed the original state ). From the probabilities are then obtained directly the coordinates of the state descriptive point on the Bloch sphere, and thus the state.

Error correction

As with classical bits external influences can not be completely eliminated even with qubits. Therefore, you need error correction codes here. In contrast to classical error correction codes are available for error correction of qubits, however, important limitations:

• The collapse of the wave function ensures that each measurement, the information about the state of a qubit provides destroys this condition.
• The no -cloning theorem forbids to copy the state of a qubit.
• Since qubits, unlike classical bits, allowing a continuum of states, and errors can be continuous.

Despite these limitations, however, an error correction possible. This is possible because you do not really need the result for the correction of an error, but just need to know which error occurred. For example, if a so-called bit - flip occurred, the reversed and each other, it is clear that the problem is resolved by a further bit flip is made; it is not necessary to know the actual state of the qubit.

The restriction of the no -cloning theorem is not as serious as it seems, because you can represent a qubit with two states of a system of multiple qubits anyway. Only one has then just generally no copies, but a set of entangled states.

The problem of continuous error is achieved by the superposition principle: The result of a small perturbation by a particular type of error can be quantum mechanically interpreted as a superposition of two states: one where this error has not occurred, and one in which this error is maximum occurred. The collapse of the wave function to measure now whether the error has occurred, ensures that exactly one of these two cases is found; one has therefore to be done only with a limited number of discrete errors.

Types of errors that can occur in a single qubit are

• No error: The qubit is not changed. Presented by the unit operator.
• Bit flip: interchanging and. Is from the state. Presented by the operator
• Phase: the sign is reversed for the state. Is from the state. Shown by the operator.
• Bit phase: combination of the two upper error. Is from the state. Shown by the operator.

General one - qubit errors can be described by linear combinations of these errors.

The elementary - qubit errors are combinations of these types of errors for each qubit ( ie, for example, qubit 1 is without error, but qubit 2 has made a bit flip ). Again, is described by a linear combination of a common mistake; thus can also be so complicated error types describe as " qubit 1 had a phase error, if qubit 2 was ".

A simple example is the repetition code. Here, the information is simply symmetrically distributed on several qubits. For example, the value is encoded in three qubits. In this coding, it is already possible with three qubits to correct bit - flip error safely. If one uses instead two Bell states as a basis to phase error can be corrected. The combination of both mechanisms leads to the so-called developed by Peter Shor Shor code, which means 9 qubits all three elementary types of errors can be corrected. Error correction is, however, also possible with fewer qubits Andrew Steane has developed an error correction code, which runs on only 7 qubits per stored qubit.