Qubit

A qubit representation by a Bloch sphere.

A qubit is not to be confused with a cubit, which is an ancient measure of length.

A quantum bit, or qubit (sometimes qbit) ['kju.bɪt] or [k'bɪt] is a unit of quantum information. That information is described by a state vector in a two-level quantum mechanical system which is formally equivalent to a two-dimensional vector space over the complex numbers.

Benjamin Schumacher discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in a state, and storing the information on a smaller number of states. This is now known as Schumacher compression. In the acknowledgments of his paper (Phys. Rev. A 51, 2738), Schumacher states that the term qubit was invented in jest, during his conversations with Bill Wootters.

Bit versus qubit

A bit is the base of computer information. Regardless of its physical representation, it is always read as either a 0 or a 1. An analogy to this is a light switch - the down position can represent 0 (normally equated to off) and the up position can represent 1 (normally equated to on).

A qubit has some similarities to a classical bit, but is overall very different. Like a bit, a qubit can have only two possible values - normally a 0 or a 1. The difference is that whereas a bit must be either 0 or 1, a qubit can be 0, 1, or a superposition of both.

Representation

The states a qubit may be measured in are known as basis states (or vectors). As is the tradition with any sort of quantum states, Dirac, or bra-ket notation is used to represent them.

This means that the two computational basis states are conventionally written as $|0\rangle$ and $|1\rangle$ (pronounced: 'ket 0' and 'ket 1').

The french Commissariat à l'Énergie Atomique have created a representation of the superposition. [1]

Qubit states

A pure qubit state is a linear superposition of those two 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 that the outcome is $|1\rangle$ is $|\beta |^{2}$. Because the absolute squares of the amplitudes equate to probabilities, it follows that α and β must be constrained by the equation

$|\alpha |^{2}+|\beta |^{2}=1\,$

simply because this ensures you must measure either one state or the other.

The state space of a single qubit register can be represented geometrically by the Bloch sphere. This is a two dimensional space which has an underlying geometry of the surface of a sphere. This essentially means that the single qubit register space has two local degrees of freedom. Represented on such a sphere, a classical bit could only lie on one of the poles. An n-qubit register space has 2ⁿ⁺¹ − 2 degrees of freedom. This is much larger than 2n, which is what one would expect classically with no entanglement.

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

$|\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle ).$

(Note that in this state, there are equal probabilities of measuring either $|00\rangle$ or $|11\rangle$.)

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 the exact same measurement as Alice, i.e. if she measured 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 are 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 entangled qubits taken together is a qubit register. Quantum computers perform calculations by manipulating qubits within a register.

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 which 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.

This is an incomplete list of physical implementation of qubits:

Physical support	Name	Information support	"0"	"1"
Single photon (Fock states)	Polarization encoding	Polarization of light	Horizontal	Vertical
	Photon number	Photon number	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
	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

External links

• An update on qubits in the Jan 2005 issue of Scientific American
• An update on qubits in the Oct 2005 issue of Scientific American
• The organization cofounded by one of the pioneers in quantum computation, David Deutsch