Qubit
A qubit (for "quantum bit"), rarely qubit ( [ 'kju.bɪt] or [k'bɪt] ), an arbitrarily manipulable has two state - quantum system , ie a system that is properly described only by quantum mechanics and the only has two states that can be reliably distinguished by measurement.
Qubits are formed in the quantum computer science the basis for quantum computing and quantum cryptography . The qubit plays the same role as the classic bit in conventional computers: It serves as the smallest possible storage unit and at the same time defines a measure for quantum information .
properties
As a two-state quantum system, the qubit is the simplest nontrivial quantum system. The term “two-state system” does not refer to the number of states that the system can assume. In fact, every nontrivial quantum mechanical system can in principle assume an infinite number of different states. However, in general, the state of a quantum system can not always be reliably determined by measurement ; the measurement randomly selects one of the possible measured values of the observed observables , the probability of each measured value being determined by the state that existed before the measurement . Since the measurement usually changes its status, this problem cannot be avoided by measuring the same system several times.
However, there are certain states for every measurement, if they are present, the measured value can be predicted with absolute certainty before the measurement, the so-called eigenstates of the measurement or the measured observables. There is at least one such state for every possible result. The maximum number of possible measured values is obtained for measurements in which there is only one state that reliably delivers this measured value. In addition, after each measurement there is an intrinsic state associated with the measured value obtained ( collapse of the wave function ); however, if there is an intrinsic state of the measurement before the measurement, this is not changed.
Two states that can be reliably differentiated by measurement are also called orthogonal to each other. The maximum number of possible measured values in a measurement - and thus also the maximum number of orthogonal states - is a property of the quantum system. With the qubit as a two-state system, you can reliably differentiate between two different states by measuring. If you want to use a qubit simply as a classic memory, you can store exactly one classic bit in it . However, the advantage of the qubit lies precisely in the existence of the other states.
An example of this is the polarization of a photon (“light particle”). The polarization of light indicates the direction in which light oscillates. Although the polarization is actually a wave property, it can also be defined for the individual photon, and all polarizations (linear in any direction, circular, elliptical) are also possible for individual photons. For example, linear polarization can be measured using a birefringent crystal. Where a photon that enters the crystal at a certain point comes out depends on whether it is polarized parallel or perpendicular to the optical axis of the crystal. So there are, so to speak, two “outputs”, one for parallel and one for perpendicularly polarized photons. If you place a photon detector at both points, you can determine whether the photon was polarized parallel or perpendicular to the optical axis.
Photons that have a different polarization (linear at a different angle, circular or elliptical), however, also come out of these “exits”. In this case, however, it is not possible to predict at which “exit” such a photon will come out; only the probability can be predicted. Afterwards, however, it has the polarization that belongs to the corresponding output, as e.g. B. can be proven that instead of the detector further crystals (with a parallel aligned optical axis) with two detectors each are attached to the "outputs": Only those detectors on the second crystals that have the correct polarization for the output of the first Belong to crystals, register photons.
The crystal thus distinguishes a direction of polarization. However, which one it is can be determined by rotating the crystal. Two linearly polarized states are therefore orthogonal to one another if the directions of polarization are orthogonal to one another. However, this correspondence cannot be transferred directly to other polarization states; so are z. B. the left circular and right circular polarized state are also orthogonal to each other.
As with traditional bits, several qubits can be combined in order to store larger values. A qubit system has exactly mutually orthogonal states. Exactly classic bits can be stored in qubits in such a way that the complete information can be reliably read out again; For example, a “quantum byte” from eight qubits can store 256 different values that can be reliably read out again.
Much more important for the use in quantum computers is the existence of entangled states of several qubits. In such states, an individual qubit has no defined state at all , but the totality of the qubits does. This leads to nonlocal correlations such as occur in the Einstein-Podolsky-Rosen paradox .
The entanglement of the qubits has surprising consequences. For example, two classic bits can be stored in a pair of interleaved qubits in such a way that both bits can be manipulated separately by manipulating only one of the qubits. However, both qubits are required to read out the information.
Quantum teleportation is also based on the non-locality of entanglement , with which quantum mechanical states can be transmitted by transmitting classic bits.
What is important for quantum computers is the fact that by entangling a set of qubits, any set of sequences of classical bits can be represented simultaneously . For example, with four qubits a state can be established which contains exactly the bit sequences 0000, 0101, 1011 and 1110 and no others. In the extreme case, it contains all possible bit sequences, e.g. B. a correspondingly prepared "quantum byte" contains all numbers from 0 to 255 at the same time. If one now carries out calculations on this state with the help of quantum mechanical operations, then these calculations are effectively carried out on all these bit sequences at the same time. This so-called quantum parallelism is the reason why quantum computers can solve certain problems faster than classic computers. However, the stored bit patterns cannot be read out individually; each measurement delivers only one of the stored values selected at random. In order to use quantum parallelism, specific quantum mechanical transformations have to be carried out that have no classical equivalents, i.e. states that correspond exactly to one bit pattern can be converted into overlays of several bit patterns and vice versa.
Implementation of qubits
Theoretically, any quantum mechanical two-state system can be used as a qubit. In practice, however, many systems are unsuitable because they cannot be manipulated to a sufficient extent or are too disturbed by the environment. There is also the problem of scalability: Some implementations, such as B. the use of nuclear magnetic resonance in molecules, are in principle only suitable for a very limited number of qubits.
David DiVincenzo established seven criteria for the usability of a system as a qubit . The first five criteria also apply to use in quantum computers, the last two apply specifically to quantum communication.
The five general criteria are:
- The system must have well-defined qubits and be scalable, i. H. in principle it must be expandable to any number of qubits.
- It must be possible to prepare the qubits in a pure state (at least in the state ).
- The system must have a sufficiently long decoherence time .
- The system must allow the implementation of a universal set of quantum gates . An example would be B. all 1-qubit gates plus the CNOT gate.
- It must be possible to measure each and every one of the qubits in a targeted manner.
The two additional criteria for quantum communication are:
- It must be possible to transform stationary qubits into movable qubits and vice versa.
- It must be possible to exchange the movable qubits between remote locations.
In practice, the following systems, among others, are examined:
Ions in ion traps
A promising approach for quantum computers is the use of ions in ion traps . Here, individual ions are strung together by electromagnetic fields in a vacuum, like a string of pearls.
The qubits are each formed by two long-lived internal states of the individual ions, for example two hyperfine levels of the ground state . The number of qubits is identical to the number of ions in the trap. The qubits are manipulated using lasers that interact with the individual ions. The qubits can be coupled and entangled via the movement of the ions in the trap.
With this technology, up to twenty qubits could already be interlaced.
Electrons in quantum dots
Another approach is the use of quantum dots . Quantum dots are quasi-zero-dimensional semiconductor structures in which electrons can only assume discrete states; therefore one often speaks of designer atoms. One advantage of quantum dot technology is that tried and tested semiconductor methods can be used in production. Two orientations of the spin with a fixed number of electrons (“spin qubit”) or two different charge configurations (“charge qubit”; e.g. with an electron in either the first or second of two quantum dots) or a combination of these two possibilities can be used as the base states of the qubit . In practice, spin qubits dominate due to the much longer coherence times. So far, universal control of up to two spin qubits has been demonstrated.
SQUIDs
Qubits can also be implemented with SQUIDs . SQUIDs are systems of superconductors that are connected by two parallel Josephson junctions. The qubits are manipulated via the applied voltage and the magnetic field. The basic states can be determined here via the value of the relative phase, charge or magnetic flux through the SQUID. Quantum processors (with full control) of up to ten qubits have been implemented so far. Significantly larger Josephson quantum registers (with up to 72 qubits) were demonstrated, but without full control over their state.
Nuclear spins in molecules and solids
The spins of the atomic nuclei in molecules can also represent qubits. They can be manipulated and read out via nuclear magnetic resonance. This is a technically particularly simple method, but it does not meet the DiVincenzo criteria mentioned above. In particular, the method is not scalable, since the number of spins per molecule is limited and its addressing and controlled coupling are all the more difficult the more spins have to be addressed. In addition, it is not possible to measure a single system (i.e. a single molecule), but rather many similar molecules at once.
In contrast, with nuclear spins in solids, in principle, scalable architectures can be implemented. Particularly promising here are e.g. B. the nuclear spins of foreign atoms in silicon or nitrogen vacancy centers (or other color centers) in diamond.
Photons
Movable qubits can be defined particularly well with photons . As a rule, different particle number eigenstates of the electromagnetic field are used as basic states . A common realization is the polarization qubit, which is defined by two orthogonal polarizations of a photon. Another important implementation is the time-bin qubit , which is defined by a photon in either the first or second of two consecutive time windows. There are also qubits that are defined by many-photon states , such as, for example, by coherent states of opposite phase. Photonic qubits can easily be transmitted over long distances via fiber optics or through the air. Experiments on quantum communication and quantum cryptography therefore almost exclusively use photon states. However, since it is very difficult to bring photons into interaction with one another, they are less suitable for the implementation of a quantum computer, even if this is possible in principle.
Quantum coding
Similar to classical information, quantum information can also be compressed. It is assumed here that the signal consists of randomly selected pure states from an “alphabet”, although these states do not necessarily have to be orthogonal to one another. H. it does not have to be possible to reliably differentiate the states by measurement. These states are encoded in a system of qubits (the original state is necessarily destroyed in the process) and these are sent to the receiver, who then reconstructs an approximation of the original state from the sent qubits.
The accuracy (fidelity) of such a coding is defined by the expected correspondence between the reconstructed state and the original. That is, assuming the receiver knows which characters were sent and carries out a measurement on its reconstructed state, for which the original state is an intrinsic state, then the accuracy of the coding is given by the proportion of measurements that correspond to the transmitted state surrender.
Analogous to classical information theory, an ideal coding is a transmission in which the minimum number of qubits must be transmitted in order to achieve an arbitrarily high transmission probability given a sufficiently large number of transmitted characters.
It now turns out that the minimum number of qubits to transfer such a state is precisely the Von Neumann entropy of the density matrix defined by the “alphabet” and the associated probabilities. Thus the qubit, analogous to the classical bit, can be viewed as an information unit of quantum information; the Von Neumann entropy of a quantum system then indicates its information content in qubits.
Indeed, the term qubit was coined by BW Schumacher in this context.
Mathematical description
Description of individual qubits
To describe a qubit one takes any measurable variable (e.g. in the example with the photons the polarization parallel and perpendicular to the optical axis of a birefringent crystal) and names the associated eigenstates and (the notation is used to identify that it is a quantum state acts, see also Dirac notation ). The quantum mechanical principle of superposition now demands that there are infinitely many states of this system, which formally appear as
let write, where and complex numbers with
are. The state can thus be described as a normalized vector in a complex vector space , more precisely, a Hilbert space . (In the case of photons, it is precisely the Jones vector that describes the polarization). However, the description is not clear; two vectors that differ only by one factor of the shape (“phase factor”) describe the same state. However, it should be noted that such a phase factor makes a difference for only one of the components: The vectors and generally describe different states.
If the system is in this state, then the probability of finding the state after the measurement is just and, accordingly, the probability of finding the state is the same .
Alternatively, the qubit can also be described using its density matrix . For the qubit in state , this is the projection operator
In contrast to the state vector, the density matrix is clearly defined. The density matrix can also be used to describe qubits whose state is not fully known (so-called "mixed states"). In general, the density matrix for a qubit can be given by
- (*)
where are the 2 × 2 identity matrix and the Pauli matrices . The probability of finding the condition with a corresponding measurement is given by .
Bloch sphere
The states of a single (non-entangled) qubit can be represented as points on the surface of a sphere in three- dimensional space . This surface is called a Bloch sphere or sphere after Felix Bloch . This can be seen particularly clearly in the spin 1/2 particle, where the point on the sphere indicates in which direction you will definitely measure spin up. But the equivalence applies to all two-state systems. The picture on the right shows how the polarization states described above can be arranged on the Bloch sphere. For example, the “North Pole” here corresponds to the vertical polarization and the “South Pole” to the horizontal polarization. In general, mutually orthogonal states correspond to mutually opposite points on the Bloch sphere.
If you set the state to the “north pole” of the sphere, and if and the angles of the point are in spherical coordinates (see picture on the left), then the associated state is given by the vector
described.
The points inside the sphere can also be interpreted: You can assign them to qubits whose state you do not have complete information about . The Cartesian coordinates of the point in the sphere are then precisely the factors in front of the Pauli matrices in the equation (*). The center of the sphere thus corresponds to a qubit about which one knows nothing at all; the further one moves from the center, the greater the knowledge about the state of the qubit. This ball is in some ways analogous to the probability - interval [0,1] for the classical bit : The points on the edge is referred to give the exact possible states of the bit (0 or 1) or of the qubits (in quantum mechanics also of "pure states"), while the points inside represent incomplete knowledge about the bit / qubit (in quantum mechanics one speaks of "mixed states"). The point in the middle represents complete ignorance of the system in both cases (for the bit: probability 1/2).
The process of measuring can also be nicely represented using the Bloch sphere: In the "spherical coordinates" picture, the small red point indicates a possible state of the qubit. In this case the point is on the outside of the sphere, so it is a pure state; however, the procedure also works for mixed conditions. Since the eigenstates of the measurement are orthogonal to one another, i.e. they are opposite to one another on the Bloch sphere, the measurement defines a straight line through the center of the sphere (marked in the picture by the blue line). You now look at the diameter (green / white in the picture) through the sphere along this straight line and project the point that represents the current knowledge about the qubit perpendicularly onto this line (the projection is marked here by the red plane and the yellow line ; the intersection of the yellow line with the diameter is the projected point). This distance can then be viewed directly as a probability interval for the measurement result. If one does not read the measurement result, then this point within the sphere also gives the new description of the system; After reading out the measurement result, the point is of course (as with the normal bit) at one end of the line. If you put z. E.g. in the picture the state of the "north pole" of the sphere and the state of the "south pole" , then the ratio of the length of the white part of the diameter (from the south pole to the point of intersection with the plane) to the total diameter is the probability that Finding the qubit after the measurement in the state if the state was given by the red dot before (afterwards the state is of course on the North Pole in this case).
Some physicists suspect in this connection between qubits and points in three-dimensional space the reason why our space is three-dimensional. A prominent representative of this idea is the original theory by Carl Friedrich von Weizsäcker . Weizsäcker's Ur is essentially what is now called the qubit.
Description of systems made up of several qubits
The states of a system made up of several qubits also form a Hilbert space due to the superposition principle. This is the tensor product of the Hilbert spaces of the individual qubits. This means that a system of qubits is described by a -dimensional Hilbert space, the base states of which can be written as direct products of the individual qubit states, e.g. B.
where the indices indicate which qubit the state belongs to. Any direct product of 1 qubit states gives a qubit state, e.g. B.
However, the reverse is not true : some qubit states cannot be written as the product of one-qubit states . An example of such a state is the 2-qubit state (one of the bell states ). Such states that cannot be written as the product of individual states are called entangled . The description of an individual qubit in an entangled state is only possible using a density matrix, which in turn indicates the ignorance (or non-consideration) of information about the qubit: In this case, the missing information is precisely the entanglement with other qubits. However, the complete state cannot be described by specifying the density matrices for each individual qubit. The entanglement is rather a non-local property that is expressed in the correlations between the interlaced qubits.
Complementary observables of the qubit
Two observables are complementary when complete knowledge of the value of one observable implies complete ignorance of the other. Since complete ignorance of the value is synonymous with projection onto the center of the Bloch sphere in the description of the measurement given above, it immediately follows that observables that are complementary to one another are described by mutually orthogonal directions in the Bloch sphere. Accordingly, for a single qubit one always finds exactly three pairwise complementary observables, corresponding to the three spatial directions.
If one has many similarly prepared copies of a qubit, the state can be determined by measuring the probabilities of a set of three pairwise complementary observables (each measurement having to be made on a new copy, since the measurement destroyed the original state). The coordinates of the point on the Bloch sphere describing the state and thus the state result directly from the probabilities.
Bug fix
As with classic bits, external influences cannot be completely eliminated with qubits either. Therefore, error correction codes are also required here. In contrast to classic error correction codes, however, there are important limitations for correcting errors in qubits:
- The collapse of the wave function ensures that any measurement that provides information about the state of a qubit destroys that state.
- The no-cloning theorem prohibits copying the state of a qubit.
- Since qubits, unlike classic bits, allow a continuum of states , errors can also be continuous.
Despite these restrictions, it is possible to correct an error because you don't really need the result to correct an error, you just need to know which error has occurred. Is z. If, for example, a so-called bit flip has occurred which interchanges and with one another, it is clear that the problem is eliminated by performing another bit flip; 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 a qubit can still be represented by two states of a system made up of several qubits. Only then one generally has no copies, but a set of entangled states.
The problem of continuous errors is solved by the superposition principle: The result of a small disturbance caused by a certain type of error can be understood quantum mechanically as the superposition of two states: One in which this error did not occur at all and one in which this error occurred at its maximum . If you now measure whether the error has occurred, the collapse of the wave function ensures that exactly one of these two cases is found; one therefore only has to deal with a limited number of discrete errors.
Types of errors that can occur with a single qubit are
- Not a mistake: the qubit is not changed. Represented by the unit operator.
- Bit flip: swapping and . The state becomes . Represented by the operator
- Phase: The sign is reversed for the state . The state becomes . Represented by the operator .
- Bit phase: Combination of the two errors above. The state becomes . Represented by the operator .
General 1-qubit errors can be described by linear combinations of these errors.
The elementary qubit errors are combinations of these error types for each individual qubit (e.g. qubit 1 is without error, but qubit 2 has made a bit flip). Again, a general error is described by a linear combination; In this way, it is also possible to describe such complicated types of errors as “qubit 1 had a phase error, provided qubit was 2 ”.
A simple example is the repetition code. The information is simply distributed symmetrically over several qubits. For example, if there are three qubits, the value is encoded by. With this coding, it is already possible to reliably correct bit flip errors with three qubits. If you use two Bell states as a basis instead, phase errors can be corrected. The combination of both mechanisms leads to the so-called Shor code developed by Peter Shor, in which all three elementary types of errors can be corrected using nine qubits. However, error correction is also possible with fewer qubits; Andrew Steane has developed an error correction code that manages with just seven qubits per stored qubit.
literature
- M. Homeister: Understanding Quantum Computing. Springer Vieweg, Wiesbaden 2018, fourth edition, ISBN 978-3-658-22883-5 .
- AJ Leggett : Quantum computing and quantum bits in mesoscopic systems. Kluwer Academic, New York 2004, ISBN 978-0-306-47904-5 .
- B. Lenze: Mathematik und Quantum Computing Logos Verlag, Berlin 2020, second edition, ISBN 978-3-832-54716-5 .
- RJ Lipton , KW Regan: Quantum Algorithms via Linear Algebra: A Primer MIT Press, Cambridge MA 2014, ISBN 978-0-262-02839-4 .
- MA Nielsen, IL Chuang: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge 2010, ISBN 978-1-107-00217-3 .
- O. Morsch: Quantum bits and quantum secrets - how quantum physics is revolutionizing codes and computers. Wiley-VCH, Weinheim 2008, ISBN 978-3-527-40710-1 .
- J. Rink: Superconductivity quantum computer. In: c't. 2009, issue 16, ISSN 0724-8679 , p. 52.
- W. Scherer: Mathematics of Quanteninformatik Springer-Verlag, Berlin-Heidelberg 2016, ISBN 978-3-662-49079-2 .
- CP Williams: Explorations in Quantum Computing Springer-Verlag, London 2011, second edition, ISBN 978-1-846-28886-9 .
Web links
- Oxford Quantum portal page about quantum computers of the Center for Quantum Computation
- Physics 219 Course Information - Lecture notes on quantum information and quantum computers
Individual evidence
- ^ David P. DiVincenzo: The Physical Implementation of Quantum Computation . February 25, 2000, arxiv : quant-ph / 0002077 (English).
- ↑ Wolfgang Hänsel: A look at the cards of quantum mechanics: quantum bits in the ion trap . In: Physics in Our Time . tape 37 , no. 2 , February 21, 2006, p. 64 , doi : 10.1002 / piuz.200501093 .
- ↑ N. Friis, O. Marty, C. Maier, C. Hempel, M. Holzäpfel, P. Jurcevic, MB Plenio, M. Huber, C. Roos, R. Blatt and B. Lanyon: Observation of Entangled States of a Fully Controlled 20-Qubit System . In: Phys. Rev. X . tape 8 , 2018, p. 021012 , doi : 10.1103 / PhysRevX.8.021012 (English).
- ↑ C. Kloeffel and D. Loss: Prospects for Spin-Based Quantum Computing in Quantum Dots . In: Annu. Rev. Condens. Matter Phys. tape 4 , 2013, p. 51 , doi : 10.1146 / annurev-conmatphys-030212-184248 , arxiv : 1204.5917 (English).
- ↑ C. Song, K. Xu, W. Liu, C.-p. Yang, S.-B. Zheng, H. Deng, Q. Xie, K. Huang, Q. Guo, L. Zhang, P. Zhang, D. Xu, D. Zheng, X. Zhu, H. Wang, Y.-A. Chen, C.-Y. Lu, S. Han and J.-W. Pan: 10-Qubit Entanglement and Parallel Logic Operations with a Superconducting Circuit . In: Phys. Rev. Lett. tape 119 , 2017, pp. 180511 (English).
- ^ Emily Conover: Google moves toward quantum supremacy with 72-qubit computer. In: ScienceNews. March 5, 2018, accessed August 31, 2018 .
- ↑ Bruce E. Kane: A silicon-based nuclear spin quantum computer . In: Nature . tape 393 , 1998, pp. 133 , doi : 10.1038 / 30156 (English).
- ↑ Fedor Jelezko: Defects with Effect . In: Physics Journal . tape 7 , no. 8/9 , 2008, p. 63 ( pro-physik.de [PDF]).
- ^ TC Ralph, A. Gilchrist, GJ Milburn, WJ Munro and S. Glancy: Quantum computation with optical coherent states . In: Phys. Rev. A . tape 68 , 2003, p. 042319 , doi : 10.1103 / PhysRevA.68.042319 , arxiv : quant-ph / 0306004 (In this case, the two-dimensional Hilbert space of the qubit is spanned by the two states , but these are not orthogonal to one another and can therefore only be identified approximately with the base states . The approximation gets exponentially better with increasing .).
- ↑ E. Knill, R. Laflamme and GJ Milburn: A scheme for efficient quantum computation with linear optics . In: Nature . tape 409 , 2001, p. 46 , doi : 10.1038 / 35051009 , arxiv : quant-ph / 0006088 (English).
- ^ Benjamin W. Schumacher: Quantum Coding. In: Physical Review. A, 51 (4), 1995, pp. 2738–2747, doi: 10.1103 / PhysRevA.51.2738 (after Jozef Gruska, Quantum Computing, 1999)