Qubit

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This article is about the quantum computing unit. For other uses, see Qubit (disambiguation).

In quantum computing, a qubit (/ˈkjuːbɪt/) or quantum bit 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 which 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 unforgeable 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 due to its phonological resemblance with an ancient unit of length called cubit, 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, night and day, 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.

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 and (pronounced "ket 0" and "ket 1").

Qubit states

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

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 and  :

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 is and the probability of outcome is . Because the absolute squares of the amplitudes equate to probabilities, it follows that α and β must be constrained by the equation

simply because this ensures you must measure either one state or the other (the total probability of all possible outcomes must be 1).

Bloch sphere

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 and 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 would lie on the equator of the sphere, on the positive y axis.

The surface of the sphere is two-dimensional space, which represents the state space of the pure qubit states. This state space has two local degrees of freedom. It might at first sight seem that there should be four degrees of freedom, as α and β are complex numbers with two degrees of freedom each. However, one degree of freedom is removed by the constraint . Another, the overall phase of the state, has no physically observable consequences, so we can arbitrarily choose α to be real, leaving just two 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 and , 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.^{[citation needed]}

• 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 , with probability , or , with probability . Measurement of the state of the qubit alters the values of α and β. For instance, if the result of the measurement is , 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

In this state, called an equal superposition, there are equal probabilities of measuring either or , as .

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 or . Because of the qubits' entanglement, Bob must now get exactly the same measurement as Alice; i.e., if she measures a , Bob must measure the same, as is the only state where Alice's qubit is a . 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 entangled qubits. It was first demonstrated by a team at the Institute of Quantum Optics and Quantum Information at the University of Innsbruck in Austria in December 2005.^[5]

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.

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
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
	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,^[6] 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.^[7]

Quantum state

From Wikipedia, the free encyclopedia

In quantum physics, quantum state refers to the state of a quantum system.

A quantum state can be either pure or mixed. A pure quantum state is represented by a vector, called a state vector, in a Hilbert space. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number, written . For a more complicated case, consider Bohm's formulation of the EPR experiment, where the state vector

involves superposition of joint spin states for two particles. Mathematically, a pure quantum state is represented by a state vector in a Hilbert space over complex numbers, which is a generalization of our more usual three-dimensional space.^[1] If this Hilbert space is represented as a function space, then its elements are called wave functions.

A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. Mixed states are described by so-called density matrices. A pure state can also be recast as a density matrix; in this way, pure states can be represented as a subset of the more general mixed states.

For example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional. A pure state here is represented by a two-dimensional complex vector , with a length of one; that is, with

where and are the absolute values of and . A mixed state, in this case, is a matrix that is Hermitian, positive-definite, and has trace 1.

Before a particular measurement is performed on a quantum system, the theory usually gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the observable describing the measurement. These probability distributions arise for both mixed states and pure states: it is impossible in quantum mechanics (unlike classical mechanics) to prepare a state in which all properties of the system are fixed and certain. This is exemplified by the uncertainty principle, and reflects a core difference between classical and quantum physics. Even in quantum theory, however, for every observable there are some states that have an exact and determined value for that observable.^[2][3]

Conceptual description

Pure states

Probability densities for the electron of a hydrogen atom in different quantum states.

In the mathematical formulation of quantum mechanics, pure quantum states correspond to vectors in a Hilbert space, while each observable quantity (such as the energy or momentum of a particle) is associated with a mathematical operator. The operator serves as a linear function which acts on the states of the system. The eigenvalues of the operator correspond to the possible values of the observable, i.e. it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s.
The corresponding eigenvector (which physicists call an eigenstate) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s.

On the other hand, a system in a linear combination of multiple different eigenstates does in general have quantum uncertainty for the given observable. We can represent this linear combination of eigenstates as:

.

The coefficient which corresponds to a particular state in the linear combination is complex thus allowing interference effects between states. The coefficients are time dependent. How a quantum system changes in time is governed by the time evolution operator. The symbols "|" and "⟩"^[4] surrounding the are part of bra–ket notation.

Statistical mixtures of states are different from a linear combination. A statistical mixture of states is a statistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of different states . A number represents the probability of a randomly selected system being in the state . Unlike the linear combination case each system is in a definite eigenstate.^[5][6]

The expectation value of an observable A is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories.

There is no state which is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) are known exactly; at least one of them will have a range of possible values.^[a] This is the content of the Heisenberg uncertainty relation.

Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state^{[clarification needed]} More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive^{[clarification needed]} in time, then they will produce the same results. This has some strange consequences, however, as follows.

Consider two observables, A and B, where A corresponds to a measurement earlier in time than B.^[7] Suppose that the system is in an eigenstate of B at the experiment's begin. If we measure only B, we will not notice statistical^{[clarification needed]} behaviour. If we measure first A and then B in the same run of the experiment, the system will transfer to an eigenstate of A after the first measurement, and we will generally notice that the results of B are statistical. Thus: Quantum mechanical measurements influence one another, and it is important in which order they are performed.

Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see entanglement. These entangled states lead to experimentally testable properties (Bell's theorem) that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.

Schrödinger picture vs. Heisenberg picture

One can take the observables to be dependent on time, while the state σ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observables P(t), Q(t).) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger picture. (This approach was taken in the earlier part of the discussion above, with a time-varying state .) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention.

Both viewpoints are used in quantum theory. While non-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with Dirac picture.^[8]

Formalism in quantum physics

Pure states as rays in a Hilbert space

Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit sphere in the Hilbert space.

Multiplying a pure state by a scalar is physically inconsequential (as long as the state is considered by itself). If one vector is obtained from the other by multiplying by a scalar of unit magnitude, the two vectors are said to correspond to the same "ray" in Hilbert space^[9] and also to the same point in the projective Hilbert space.

Bra–ket notation

Calculations in quantum mechanics make frequent use of linear operators, inner products, dual spaces and Hermitian conjugation. In order to make such calculations flow smoothly, and to obviate the need (in some contexts) to fully understand the underlying linear algebra, Paul Dirac invented a notation to describe quantum states, known as bra-ket notation. Although the details of this are beyond the scope of this article (see the article bra–ket notation), some consequences of this are:

• The expression used to denote a state vector (which corresponds to a pure quantum state) takes the form (where the "" can be replaced by any other symbols, letters, numbers, or even words). This can be contrasted with the usual mathematical notation, where vectors are usually bold, lower-case letters, or letters with arrows on top.
• Instead of vector, the term ket is used synonymously.
• Each ket is uniquely associated with a so-called bra, denoted , which corresponds to the same physical quantum state. Technically, the bra is the adjoint of the ket. It is an element of the dual space, and related to the ket by the Riesz representation theorem. In a finite-dimensional space with a chosen basis, writing as a column vector, is a row vector; to obtain it just take the transpose and entry-wise complex conjugate of .
• Inner products (also called brackets) are written so as to look like a bra and ket next to each other: . (The phrase "bra-ket" is supposed to resemble "bracket".)

Spin

The angular momentum has the same dimension as the Planck constant and, at quantum scale, behaves as a discrete degree of freedom. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described with spinors. In non-relativistic quantum mechanics the group representations of the Lie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative number S that, in units of Planck's reduced constant ħ, is either an integer (0, 1, 2 ...) or a half-integer (1/2, 3/2, 5/2 ...). For a massive particle with spin S, its spin quantum number m always assumes one of the 2S + 1 possible values in the set

As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C^2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).

Many-body states and particle statistics

The quantum state of a system of N particles, each potentially with spin, is described by a complex-valued function with four variables per particle, e.g.

Here, the spin variables m_ν assume values from the set

where is the spin of νth particle. for a particle that does not exhibit spin.

The treatment of identical particles is very different for bosons (particles with integer spin) versus fermions (particles with half-integer spin). The above N-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not all N particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic).

Electrons are fermions with S = 1/2, photons (quanta of light) are bosons with S = 1 (although in the vacuum they are massless and can't be described with Schrödingerian mechanics).

When symmetrization or anti-symmetrization is unnecessary, N-particle spaces of states can be obtained simply by tensor products of one-particle spaces, to which we will return later.

Basis states of one-particle systems

As with any Hilbert space, if a basis is chosen for the Hilbert space of a system, then any ket can be expanded as a linear combination of those basis elements. Symbolically, given basis kets , any ket can be written

where c_i are complex numbers. In physical terms, this is described by saying that has been expressed as a quantum superposition of the states . If the basis kets are chosen to be orthonormal (as is often the case), then .

One property worth noting is that the normalized states are characterized by

Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the are eigenstates (with eigenvalues k_i) of an observable, and that observable is measured on the normalized state , then the probability that the result of the measurement is k_i is |c_i|². (The normalization condition above mandates that the total sum of probabilities is equal to one.)

A particularly important example is the position basis, which is the basis consisting of eigenstates of the observable which corresponds to measuring position. If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket is associated with a complex-valued function of three-dimensional space:^{[clarification needed]}

This function is called the wavefunction corresponding to .

Superposition of pure states

One aspect of quantum states, mentioned above, is that superpositions of them can be formed. If and are two kets corresponding to quantum states, the ket

is a different quantum state (possibly not normalized). Note that which quantum state it is depends on both the amplitudes and phases (arguments) of and . In other words, for example, even though and (for real θ) correspond to the same physical quantum state, they are not interchangeable, since for example and do not (in general) correspond to the same physical state. However, and do correspond to the same physical state. This is sometimes described by saying that "global" phase factors are unphysical, but "relative" phase factors are physical and important.

One example of a quantum interference phenomenon that arises from superposition is the double-slit experiment. The photon state is a superposition of two different states, one of which corresponds to the photon having passed through the left slit, and the other corresponding to passage through the right slit. The relative phase of those two states has a value which depends on the distance from each of the two slits. Depending on what that phase is, the interference is constructive at some locations and destructive in others, creating the interference pattern. By the analogy with coherence in other wave phenomena, a superposed state can be referred to as a coherent superposition.

Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.

Mixed states

A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics). Mixed states inevitably arise from pure states when, for a composite quantum system with an entangled state on it, the part is inaccessible to the observer. The state of the part is expressed then as the partial trace over .
A mixed state cannot be described as a ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Note that density matrices can describe both mixed and pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by a Hilbert space can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system for a sufficiently large Hilbert space .

The density matrix describing a mixed state is defined to be an operator of the form

where is the fraction of the ensemble in each pure state The density matrix can be thought of as a way of using the one-particle formalism to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in.

A simple criterion for checking whether a density matrix is describing a pure or mixed state is that the trace of ρ² is equal to 1 if the state is pure, and less than 1 if the state is mixed.^[10] Another, equivalent, criterion is that the von Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.

The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observable A is given by

where are eigenkets and eigenvalues, respectively, for the operator A, and "tr" denotes trace. It is important to note that two types of averaging are occurring, one being a weighted quantum superposition over the basis kets of the pure states, and the other being a statistical (said incoherent) average with the probabilities p_s of those states.

According to Wigner,^[11] the concept of mixture was put forward by Landau.^[12][13]

Interpretation

Although theoretically, for a given quantum system, a state vector provides the full information about its evolution, it is not easy to understand what information about the "real world" it carries. Due to the uncertainty principle, a state, even if it has the value of one observable exactly defined (i.e. the observable has this state as an eigenstate), cannot exactly define values of all observables.
For state vectors (pure states), probability amplitudes offer a probabilistic interpretation. It can be generalized for all states (including mixed), for instance, as expectation values mentioned above.

Mathematical generalizations

States can be formulated in terms of observables, rather than as vectors in a vector space. These are positive normalized linear functionals on a C*-algebra, or sometimes other classes of algebras of observables. See State on a C*-algebra and Gelfand–Naimark–Segal construction for more details.