Qubit

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In quantum computing, a qubit ( /ˈkjuːbɪt/ ) or quantum bit (sometimes qbit[ citation needed ]) is the basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the 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 coherent superposition  of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing.

Contents

Etymology

The coining of the term qubit is attributed to Benjamin Schumacher. [1] In the acknowledgments of his 1995 paper, Schumacher states that the term qubit was created 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

A binary digit, characterized as 0 or 1, is used to represent information in classical computers. When averaged over both of its states (0,1), a binary digit can represent up to one bit of Shannon information, where a bit is the basic unit of information. However, in this article, the word bit is synonymous with a binary digit.

In classical computer technologies, a processed bit is implemented by one of two levels of low DC voltage, and whilst switching from one of these two levels to the other, a so-called "forbidden zone" between two logic levels must be passed as fast as possible, as electrical voltage cannot change from one level to another instantaneously.

There are two possible outcomes for the measurement of a qubit—usually taken to have the value "0" and "1", like a bit or binary digit. However, whereas the state of a bit can only be either 0 or 1, the general state of a qubit according to quantum mechanics can be a coherent superposition  of both. [2] Moreover, whereas a measurement of a classical bit would not disturb its state, a measurement of a qubit would destroy its coherence and irrevocably disturb the superposition state. It is possible to fully encode one bit in one qubit. However, a qubit can hold 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 (or a single point in a 2n-dimensional vector space). [3]

Standard representation

In quantum mechanics, the general quantum state of a qubit can be represented by a linear superposition of its two orthonormal basis states (or basis vectors). These vectors are usually denoted as and . They are written in the conventional Dirac—or "bra–ket"—notation; the and are pronounced "ket 0" and "ket 1", respectively. These two orthonormal basis states, , together called the computational basis, are said to span the two-dimensional linear vector (Hilbert) space of the qubit.

Qubit basis states can also be combined to form product basis states. For example, two qubits could be represented in a four-dimensional linear vector space spanned by the following product basis states: , , , and .

In general, n qubits are represented by a superposition state vector in 2n dimensional Hilbert space.

Qubit states

A pure qubit state is a coherent superposition of the basis states. This means that a single qubit can be described by 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, according to the Born rule, the probability of outcome with value "0" is and the probability of outcome with value "1" is . Because the absolute squares of the amplitudes equate to probabilities, it follows that and must be constrained by the equation

Note that a qubit in this superposition state does not have a value in between "0" and "1"; rather, when measured, the qubit has a probability of the value “0” and a probability of the value "1". In other words, superposition means that there is no way, even in principle, to tell which of the two possible states forming the superposition state actually pertains. Furthermore, the probability amplitudes, and , encode more than just the probabilities of the outcomes of a measurement; the relative phase of and is responsible for quantum interference, e.g., as seen in the two-slit experiment.

Bloch sphere representation

Bloch sphere representation of a qubit. The probability amplitudes for the superposition state,
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Bloch sphere representation of a qubit. The probability amplitudes for the superposition state, are given by and .

It might, at first sight, seem that there should be four degrees of freedom in , as and 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. 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:

Additionally, for a single qubit the overall phase of the state ei 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:

where is the physically significant relative phase.

The possible quantum states for a single qubit can be visualised using a Bloch sphere (see diagram). Represented on such a 2-sphere, a classical bit could only be at the "North Pole" or the "South Pole", in the locations where and are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch 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 at the positive y-axis. In the classical limit, a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles.

The surface of the Bloch 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, which can be represented by the two angles and .

Mixed state

A pure state is one fully specified by a single ket, a coherent superposition as described above. Coherence is essential for a qubit to be in a superposition state. With interactions and decoherence, it is possible to put the qubit in a mixed state, a statistical combination or incoherent mixture of different pure states. Mixed states can be represented by points inside the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles and , as well as the length of the vector that represents the mixed state.

Operations on qubits

There are various kinds of physical operations that can be performed on qubits.

Quantum entanglement

An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit quantum entanglement. Quantum entanglement is a nonlocal property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.

The simplest system to display quantum entanglement is the system of two qubits. Consider, for example, two entangled qubits in the Bell state:

In this state, called an equal superposition, there are equal probabilities of measuring either product state or , as . In other words, there is no way to tell if the first qubit has value “0” or “1” and likewise for the second qubit.

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 , i.e., she can now tell if her qubit has value “0” or “1”. Because of the qubits' entanglement, Bob must now get exactly the same measurement as Alice. For example, if she measures a , Bob must measure the same, as is the only state where Alice's qubit is a . In short, for these two entangled qubits, whatever Alice measures, so would Bob, with perfect correlation, in any basis, however far apart they may be and even though both can not tell if their qubit has value “0” or “1” — a most surprising circumstance that can not be explained by classical physics.

Controlled gate to construct the Bell state

Controlled gates act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, the controlled NOT gate (or CNOT or cX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is , and otherwise leaves it unchanged. With respect to the unentangled product basis , , , , it maps the basis states as follows:

.

A common application of the CNOT gate is to maximally entangle two qubits into the Bell state. To construct , the inputs A (control) and B (target) to the CNOT gate are:

and

After applying CNOT, the output is the Bell State: .

Applications

The Bell state forms part of the setup of the superdense coding, quantum teleportation, and entangled quantum cryptography algorithms.

Quantum 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. [4] A major hurdle facing quantum computing, as of 2018, in its quest to surpass classical digital computing, is noise in quantum gates that limits the size of quantum circuits that can be executed reliably. [5]

Quantum register

A number of qubits taken together is a qubit register. Quantum computers perform calculations by manipulating qubits within a register.

Qudits and qutrits

The term "qu-d-it" (quantumd-git) denotes the unit of quantum information that can be realized in suitable d-level quantum systems. [6] A quantum register that can be measured to N states is identical to an N-level qudit. This is similar to how an integer type in classical computing is mapped to an array of bits. Qudits where the d-level system is not an exponent of 2 can however not be mapped to arrays of qubits. It is for example possible to have 5-level qudits.

In 2017, scientists at the National Institute of Scientific Research constructed a pair of qudits with 10 different states each, giving more computational power than 6 qubits. [7]

Similar to the qubit, the qutrit is the unit of quantum information that can be realized in suitable 3-level quantum systems. This is analogous to the unit of classical information trit of ternary computers.

Physical implementations

Any two-level quantum-mechanical 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 supportNameInformation support
Photon Polarization encoding Polarization of light HorizontalVertical
Number of photons Fock state VacuumSingle photon state
Time-bin encoding Time of arrivalEarlyLate
Coherent state of light Squeezed light Quadrature Amplitude-squeezed statePhase-squeezed state
Electrons Electronic spin Spin UpDown
Electron number Charge No electronOne electron
Nucleus Nuclear spin addressed through NMR SpinUpDown
Optical lattices Atomic spinSpinUpDown
Josephson junction Superconducting charge qubit ChargeUncharged superconducting island (Q=0)Charged superconducting island (Q=2e, one extra Cooper pair)
Superconducting flux qubit CurrentClockwise currentCounterclockwise current
Superconducting phase qubit EnergyGround stateFirst excited state
Singly charged quantum dot pairElectron localizationChargeElectron on left dotElectron on right dot
Quantum dot Dot spinSpinDownUp
Gapped topological system Non-abelian anyons Braiding of Excitations Depends on specific topological systemDepends on specific topological system
van der Waals heterostructure [8] Electron localizationChargeElectron on bottom sheetElectron on top sheet

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 , [9] 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. [10] Room temperature preparation of a qubit based on electron spins instead of nuclear spin was also demonstrated by a team of scientists from Switzerland and Australia. [11] An increased coherence of qubits is being explored by researchers who are testing the limitations of a Ge hole spin-orbit qubit structure. [12]

See also

Further reading

Related Research Articles

In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist. The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem.

Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. Moreover, the sender may not know the location of the recipient, and does not know which particular quantum state will be transferred.

Quantum decoherence Loss of quantum coherence

Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

Probability amplitude Complex number whose squared absolute value is a probability

In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability density.

Quantum logic gate Basic circuit in quantum computing

In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.

The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1: . Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will collapse it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will assign one of two possible values to the other qubit instantly, where the value assigned depends on which Bell state the two qubits are in. Bell states can be generalized to represent specific quantum states of multi-qubit systems, such as the GHZ state for 3 or more subsystems.

A qutrit is a unit of quantum information that is realized by a quantum system described by a superposition of three mutually orthogonal quantum states.

Superdense coding Two-bit quantum communication protocol

In quantum information theory, superdense coding is a quantum communication protocol to communicate a number of classical bits of information by only transmitting a smaller number of qubits, under the assumption of sender and received pre-sharing an entangled resource. In its simplest form, the protocol involves two parties, often referred to as Alice and Bob in this context, which share a pair of maximally entangled qubits, and allows Alice to transmit two bits to Bob by sending only one qubit. This protocol was first proposed by Bennett and Wiesner in 1992 and experimentally actualized in 1996 by Mattle, Weinfurter, Kwiat and Zeilinger using entangled photon pairs. Superdense coding can be thought of as the opposite of quantum teleportation, in which one transfers one qubit from Alice to Bob by communicating two classical bits, as long as Alice and Bob have a pre-shared Bell pair.

Greenberger–Horne–Zeilinger state "Highly entangled" quantum state of 3 or more qubits

In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state that involves at least three subsystems. It was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989. Extremely non-classical properties of the state have been observed.

In computational complexity theory, PostBQP is a complexity class consisting of all of the computational problems solvable in polynomial time on a quantum Turing machine with postselection and bounded error.

The time-evolving block decimation (TEBD) algorithm is a numerical scheme used to simulate one-dimensional quantum many-body systems, characterized by at most nearest-neighbour interactions. It is dubbed Time-evolving Block Decimation because it dynamically identifies the relevant low-dimensional Hilbert subspaces of an exponentially larger original Hilbert space. The algorithm, based on the Matrix Product States formalism, is highly efficient when the amount of entanglement in the system is limited, a requirement fulfilled by a large class of quantum many-body systems in one dimension.

Time-bin encoding is a technique used in quantum information science to encode a qubit of information on a photon. Quantum information science makes use of qubits as a basic resource similar to bits in classical computing. Qubits are any two-level quantum mechanical system; there are many different physical implementations of qubits, one of which is time-bin encoding.

In physics, the no-deleting theorem of quantum information theory is a no-go theorem which states that, in general, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. It is a time-reversed dual to the no-cloning theorem, which states that arbitrary states cannot be copied. This theorem seems remarkable, because, in many senses, quantum states are fragile; the theorem asserts that, in a particular case, they are also robust. Physicist Arun K. Pati along with Samuel L. Braunstein proved this theorem.

Entanglement distillation is the transformation of N copies of an arbitrary entangled state into some number of approximately pure Bell pairs, using only local operations and classical communication (LOCC).

In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces.

In the theory of quantum communication, an amplitude damping channel is a quantum channel that models physical processes such as spontaneous emission. A natural process by which this channel can occur is a spin chain through which a number of spin states, coupled by a time independent Hamiltonian, can be used to send a quantum state from one location to another. The resulting quantum channel ends up being identical to an amplitude damping channel, for which the quantum capacity, the classical capacity and the entanglement assisted classical capacity of the quantum channel can be evaluated.

Linear Optical Quantum Computing or Linear Optics Quantum Computation (LOQC) is a paradigm of quantum computation, allowing universal quantum computation. LOQC uses photons as information carriers, mainly uses linear optical elements, or optical instruments to process quantum information, and uses photon detectors and quantum memories to detect and store quantum information.

Quantum complex network Notion in network science of quantum information networks

Being a component part of network science the study of quantum complex networks aims to explore the impact of complexity science and network architectures in quantum systems. According to quantum information theory it is possible to improve communication security and data transfer rates by taking advantage of quantum mechanics. In this context the study of quantum complex networks is motivated by the possibility of quantum communications being used on a massive scale in the future. In such case it is likely that quantum communication networks will acquire non trivial features as is common in existing communication networks today.

Optical cluster states are a proposed tool to achieve quantum computational universality in linear optical quantum computing (LOQC). As direct entangling operations with photons often require nonlinear effects, probabilistic generation of entangled resource states has been proposed as an alternative path to the direct approach.

In quantum information science, monogamy is a fundamental property of quantum entanglement that describes the fact that entanglement cannot be freely shared between arbitrarily many parties.

References

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