Qubit
In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classical binary bit physically realized with a two-state device. A qubit is a two-state 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 which is fundamental to quantum mechanics and quantum computing.
Etymology
The coining of the term qubit is attributed to Benjamin Schumacher. 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 and 1, is used to represent information in classical computers.When averaged over both of its states, 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 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 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. 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 complex numbers.
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. These vectors are usually denoted asand
. 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 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
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. 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 has no physically observable consequences, so we can arbitrarily choose to be real, 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. 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. 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 pure qubit states
There are various kinds of physical operations that can be performed on pure qubit states.- Quantum logic gates, building blocks for a quantum circuit in a quantum computer, operate on one, two, or three qubits: mathematically, the qubits undergo a unitary transformation under the quantum gate. For a single qubit, unitary transformations correspond to rotations of the qubit vector on the Bloch sphere to specific superpositions. For two qubits, the Controlled NOT gate can be used to entangle or disentangle them.
- Standard basis measurement is an irreversible operation in which information is gained about the state of a single qubit. The result of the measurement will be either or . Measurement of the state of the qubit alters the magnitudes of α and β. For instance, if the result of the measurement is, α is changed to 0 and β is changed to the phase factor no longer experimentally accessible. When a qubit is measured, the superposition state collapses to a basis state and the relative phase is rendered inaccessible. Note that a measurement of a qubit state that is entangled with another quantum system transforms the qubit state, a pure state, into a mixed state as the relative phase of the qubit state is rendered inaccessible.
Quantum entanglement
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
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 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 and B 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 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. 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.
Quantum register
A number of qubits taken together is a qubit register. Quantum computers perform calculations by manipulating qubits within a register. A qubyte is a collection of eight qubits.Variations of the qubit
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. Note, however, that not all 3-level quantum systems are qutrits. The term "qu-d-it" denotes the unit of quantum information that can be realized in suitable d-level quantum systems. 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.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. 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 | ||
| Photon | Polarization encoding | Polarization of light | Horizontal | Vertical |
| Photon | Number of photons | Fock state | Vacuum | Single photon state |
| Photon | 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 | Charged superconducting island |
| Josephson junction | Superconducting flux qubit | Current | Clockwise current | Counterclockwise current |
| Josephson junction | 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 |
| Gapped topological system | Non-abelian anyons | Braiding of Excitations | Depends on specific topological system | Depends on specific topological system |
| van der Waals heterostructure | Electron localization | Charge | Electron on bottom sheet | Electron on top sheet |