Qubit

Unsolved problem in physics:

Is it possible to have three-dimensional, self-correcting, quantum memory?
(more unsolved problems in physics)

In quantum computing, a qubit (/ˈkjuːbɪt/) or quantum bit (sometimes qbit) is a unit of quantum information—the quantum analogue of the classical bit. A qubit is a two-state quantum-mechanical system, such as the polarization of a single photon: here the two states are vertical polarization and horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a superposition of both states at the same time, a property that is fundamental to quantum computing.

Origin of the concept and name

The concept of the qubit was unknowingly introduced by Stephen Wiesner in 1983, in his proposal for quantum money, which he had tried to publish for over a decade.^[1]^[2]

The coining of the term "qubit" is attributed to Benjamin Schumacher.^[3] In the acknowledgments of his paper, Schumacher states that the term qubit was invented in jest, during a conversation with William Wootters. The paper describes a way of compressing states emitted by a quantum source of information so that they require fewer physical resources to store. This procedure is now known as Schumacher compression.

Bit versus qubit

The bit is the basic unit of information. It is used to represent information by computers. Regardless of its physical realization, a bit has two possible states typically thought of as 0 and 1, but more generally—and according to applications—interpretable as true and false, or any other dichotomous choice. An analogy to this is a light switch—its OFF position can be thought of as 0 and its ON position as 1.

A qubit has a few similarities to a classical bit, but is overall very different. There are two possible outcomes for the measurement of a qubit—usually 0 and 1, like a bit. The difference is that whereas the state of a bit is either 0 or 1, the state of a qubit can also be a superposition of both.^[4] It is possible to fully encode one bit in one qubit. However, a qubit can hold even more information, e.g. up to two bits using superdense coding.

For a system of n components, a complete description of its state in classical physics requires only n bits, whereas in quantum physics it requires 2ⁿ−1 complex numbers.^[5]

Representation

The two states in which a qubit may be measured are known as basis states (or basis vectors). As is the tradition with any sort of quantum states, they are represented by Dirac—or "bra–ket"—notation. This means that the two computational basis states are conventionally written as $|0\rangle$ and $|1\rangle$ (pronounced "ket 0" and "ket 1").

Qubit states

Bloch sphere representation of a qubit. The probability amplitudes in the text are given by

\alpha =\cos \left({\frac {\theta }{2}}\right)

and

\beta =e^{i\phi }\sin \left({\frac {\theta }{2}}\right)

A pure qubit state is a linear superposition of the basis states. This means that the qubit can be represented as a linear combination of $|0\rangle$ and $|1\rangle$ :

|\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,\,

where α and β are probability amplitudes and can in general both be complex numbers.

When we measure this qubit in the standard basis, the probability of outcome $|0\rangle$ is $|\alpha |^{2}$ and the probability of outcome $|1\rangle$ is $|\beta |^{2}$ . Because the absolute squares of the amplitudes equate to probabilities, it follows that $\alpha$ and $\beta$ must be constrained by the equation

|\alpha |^{2}+|\beta |^{2}=1.

Bloch sphere

It might, at first sight, seem that there should be four degrees of freedom, as $\alpha$ and $\beta$ are complex numbers with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint $| α | 2 + | β | 2 = 1$ , which can be treated as the equation for a 3-sphere embedded in 4-dimensional space with a radius of $1$ (unit sphere). This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of Hopf coordinates:

{\begin{aligned}\alpha &=e^{i\psi }\cos {\frac {\theta }{2}},\\\beta &=e^{i(\psi +\phi )}\sin {\frac {\theta }{2}}.\end{aligned}}

Additionally, for a single qubit the overall phase of the state $e i ψ$ has no physically observable consequences, so we can arbitrarily choose $α$ to be real (or $β$ in the case that $α$ is zero), leaving just two degrees of freedom:

{\begin{aligned}\alpha &=\cos {\frac {\theta }{2}},\\\beta &=e^{i\phi }\sin {\frac {\theta }{2}}.\end{aligned}}

The possible states for a single qubit can be visualised using a Bloch sphere (see diagram). Represented on such a sphere, a classical bit could only be at the "North Pole" or the "South Pole", in the locations where $|0\rangle$ and $|1\rangle$ are respectively. The rest of the surface of the sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state ${{|0\rangle +i|1\rangle } \over {\sqrt {2}}}$ would lie on the equator of the sphere, on the positive y axis.

The surface of the sphere is a two-dimensional space, which represents the state space of the pure qubit states. This state space has two local degrees of freedom.

It is possible to put the qubit in a mixed state, a statistical combination of different pure states. Mixed states can be represented by points inside the Bloch sphere. A mixed qubit state has three degrees of freedom: the angles $\phi$ and $\theta$ , as well as the length $r$ of the vector that represents the mixed state.

Operations on pure qubit states

There are various kinds of physical operations that can be performed on pure qubit states.

A quantum logic gate can operate on a qubit: mathematically speaking, the qubit undergoes a unitary transformation. Unitary transformations correspond to rotations of the qubit vector in the Bloch sphere.
Standard basis measurement is an operation in which information is gained about the state of the qubit. The result of the measurement will be either $|0\rangle$ , with probability $|\alpha |^{2}$ , or $|1\rangle$ , with probability $|\beta |^{2}$ . Measurement of the state of the qubit alters the values of α and β. For instance, if the result of the measurement is $|0\rangle$ , α is changed to 1 (up to phase) and β is changed to 0. Note that a measurement of a qubit state entangled with another quantum system transforms a pure state into a mixed state.

Entanglement

An important distinguishing feature between a qubit and a classical bit is that multiple qubits can exhibit quantum entanglement. Entanglement is a nonlocal property that allows a set of qubits to express higher correlation than is possible in classical systems. Take, for example, two entangled qubits in the Bell state

{\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle ).

In this state, called an equal superposition, there are equal probabilities of measuring either $|00\rangle$ or $|11\rangle$ , as $|1/{\sqrt {2}}|^{2}=1/2$ .

Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining—with equal probabilities—either $|0\rangle$ or $|1\rangle$ . Because of the qubits' entanglement, Bob must now get exactly the same measurement as Alice; i.e., if she measures a $|0\rangle$ , Bob must measure the same, as $|00\rangle$ is the only state where Alice's qubit is a $|0\rangle$ . Entanglement also allows multiple states (such as the Bell state mentioned above) to be acted on simultaneously, unlike classical bits that can only have one value at a time. Entanglement is a necessary ingredient of any quantum computation that cannot be done efficiently on a classical computer. Many of the successes of quantum computation and communication, such as quantum teleportation and superdense coding, make use of entanglement, suggesting that entanglement is a resource that is unique to quantum computation.

Quantum register

A number of qubits taken together is a qubit register. Quantum computers perform calculations by manipulating qubits within a register. A qubyte (quantum byte) is a collection of eight qubits.^[6]

Variations of the qubit

Similar to the qubit, a qutrit is a unit of quantum information in a 3-level quantum system. This is analogous to the unit of classical information trit. The term "qudit" is used to denote a unit of quantum information in a d-level quantum system.

Physical representation

Any two-level system can be used as a qubit. Multilevel systems can be used as well, if they possess two states that can be effectively decoupled from the rest (e.g., ground state and first excited state of a nonlinear oscillator). There are various proposals. Several physical implementations that approximate two-level systems to various degrees were successfully realized. Similarly to a classical bit where the state of a transistor in a processor, the magnetization of a surface in a hard disk and the presence of current in a cable can all be used to represent bits in the same computer, an eventual quantum computer is likely to use various combinations of qubits in its design.

The following is an incomplete list of physical implementations of qubits, and the choices of basis are by convention only.

Physical support	Name	Information support	$\|0\rangle$	$\|1\rangle$
Photon	Polarization encoding	Polarization of light	Horizontal	Vertical
	Number of photons	Fock state	Vacuum	Single photon state
	Time-bin encoding	Time of arrival	Early	Late
Coherent state of light	Squeezed light	Quadrature	Amplitude-squeezed state	Phase-squeezed state
Electrons	Electronic spin	Spin	Up	Down
Electrons	Electron number	Charge	No electron	One electron
Nucleus	Nuclear spin addressed through NMR	Spin	Up	Down
Optical lattices	Atomic spin	Spin	Up	Down
Josephson junction	Superconducting charge qubit	Charge	Uncharged superconducting island (Q=0)	Charged superconducting island (Q=2e, one extra Cooper pair)
	Superconducting flux qubit	Current	Clockwise current	Counterclockwise current
	Superconducting phase qubit	Energy	Ground state	First excited state
Singly charged quantum dot pair	Electron localization	Charge	Electron on left dot	Electron on right dot
Quantum dot	Dot spin	Spin	Down	Up

Qubit storage

In a paper entitled: "Solid-state quantum memory using the ³¹P nuclear spin", published in the October 23, 2008 issue of the journal Nature,^[7] a team of scientists from the U.K. and U.S. reported the first relatively long (1.75 seconds) and coherent transfer of a superposition state in an electron spin "processing" qubit to a nuclear spin "memory" qubit. This event can be considered the first relatively consistent quantum data storage, a vital step towards the development of quantum computing. Recently, a modification of similar systems (using charged rather than neutral donors) has dramatically extended this time, to 3 hours at very low temperatures and 39 minutes at room temperature.^[8] Room temperature preparation of a qubit based on electron spins instead of nuclear spin was also demonstrated by a team of scientist from Switzerland and Australia.^[9]

References

↑ S. Weisner (1983). "Conjugate coding". Association for Computing Machinery, Special Interest Group in Algorithms and Computation Theory. 15: 78–88.
↑ A. Zeilinger, Dance of the Photons: From Einstein to Quantum Teleportation, Farrar, Straus & Giroux, New York, 2010, pp. 189, 192, ISBN 0374239665
↑ B. Schumacher (1995). "Quantum coding". Physical Review A. 51 (4): 2738–2747. Bibcode:1995PhRvA..51.2738S. doi:10.1103/PhysRevA.51.2738.
↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. p. 13. ISBN 978-1-107-00217-3.
↑ Shor, Peter (1996). "Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer∗". arXiv:quant-ph/9508027 .
↑ R. Tanburn; E. Okada; N. S. Dattani (2015). "Reducing multi-qubit interactions in adiabatic quantum computation without adding auxiliary qubits. Part 1: The "deduc-reduc" method and its application to quantum factorization of numbers". arXiv:1508.04816 .
↑ J. J. L. Morton; et al. (2008). "Solid-state quantum memory using the ³¹P nuclear spin". Nature. 455 (7216): 1085–1088. Bibcode:2008Natur.455.1085M. arXiv:0803.2021 . doi:10.1038/nature07295.
↑ Kamyar Saeedi; et al. (2013). "Room-Temperature Quantum Bit Storage Exceeding 39 Minutes Using Ionized Donors in Silicon-28". Science. 342 (6160): 830–833. Bibcode:2013Sci...342..830S. PMID 24233718. doi:10.1126/science.1239584.
↑ Náfrádi, Bálint; Choucair, Mohammad; Dinse, Klaus-Pete; Forró, László (July 18, 2016). "Room temperature manipulation of long lifetime spins in metallic-like carbon nanospheres". Nature Communications. 7: 12232. PMC 4960311 . PMID 27426851. doi:10.1038/ncomms12232.

External links

Qubit.org—cofounded by one of the pioneers in quantum computation, David Deutsch

Quantum information science

General

Quantum communication

Quantum algorithms

Quantum complexity theory

Quantum computing models

Decoherence prevention

Physical implementations

Quantum optics	Cavity QED Circuit QED Linear optical quantum computing KLM protocol Boson sampling
Ultracold atoms	Trapped ion quantum computer Optical lattice
Spin-based	Nuclear magnetic resonance QC Kane QC Loss–DiVincenzo QC Nitrogen-vacancy center
Superconducting quantum computing	Charge qubit Flux qubit Phase qubit Transmon

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