Qubit

A qubit, short for quantum bit, is the basic unit of quantum information. It is described by a two-dimensional quantum state space and can be implemented using two distinguishable states of a physical system. Unlike a classical bit, whose value is either 0 or 1, a qubit can occupy a coherent superposition of its two basis states. Its behavior follows quantum mechanics, making it a fundamental component of a quantum computer. The term denotes both an abstract information unit and a physical system that carries it. (learning.quantum.ibm.com)

Mathematical description

A qubit’s pure state is a normalized vector in a two-dimensional complex Hilbert space. Choosing an orthonormal basis called the computational basis, denoted $|0\rangle$ and $|1\rangle$, gives

The coefficients $\alpha$ and $\beta$ are complex numbers called probability amplitudes. Their magnitudes determine measurement probabilities, while their relative phase affects how amplitudes combine during subsequent operations. Thus, a superposition is not simply uncertainty about a pre-existing classical value. Multiplying both amplitudes by the same global phase leaves the physical state unchanged. (learning.quantum.ibm.com)

Ignoring global phase, a pure state can be written as

The angles specify a point on the Bloch sphere, a geometric representation of single-qubit states. The computational basis states occupy opposite poles; equal-magnitude superpositions lie on the equator. (quantum.cloud.ibm.com)

More generally, a qubit is described by a density matrix, which accommodates both pure and mixed states. Pure states lie on the sphere’s surface, while mixed states lie inside the associated Bloch ball. The center represents the maximally mixed state, $\rho=I/2$, which gives equal probabilities for either outcome in every orthonormal measurement basis. (web.mit.edu)

Measurement and information

For an ideal computational-basis measurement, the Born rule assigns probabilities $|\alpha|^2$ and $|\beta|^2$ to outcomes 0 and 1. After this projective measurement, the qubit is in the corresponding basis state. A measurement in a different basis can reveal different aspects of the same preparation. (learning.quantum.ibm.com)

For example,

both produce equal probabilities for 0 and 1 in the computational basis. Nevertheless, they are distinct, orthogonal states that can be distinguished perfectly by measuring in the $\{|+\rangle,|-\rangle\}$ basis. Their difference illustrates the operational importance of relative phase and interference. (quantum.cloud.ibm.com)

The continuous range of qubit states does not make one qubit an unlimited readable classical memory: a single measurement cannot determine an arbitrary unknown state. Moreover, the no-cloning theorem forbids a universal operation that perfectly copies an arbitrary unknown quantum state. This restriction does not prevent copying known, mutually orthogonal basis states. (learning.quantum.ibm.com)

Operations and multiple qubits

An ideal reversible quantum gate transforms a qubit through a unitary matrix, preserving normalization. The $X$ gate exchanges $|0\rangle$ and $|1\rangle$; the Hadamard gate maps $|0\rangle$ to $|+\rangle$. Gates can change relative phases as well as outcome probabilities. Initialization and measurement are additional operations, not generally reversible unitary transformations. (learning.quantum.ibm.com)

The state space of $n$ qubits is the tensor product of their individual spaces and has dimension $2^n$. A general pure state therefore has $2^n$ amplitudes associated with binary basis strings. Some states exhibit quantum entanglement: their joint state cannot be expressed as a product of individual pure states. For instance, the Bell state

produces perfectly correlated computational-basis outcomes, although each individual outcome is random. (learning.quantum.ibm.com)

This exponentially growing state space is not equivalent to simultaneously reading exponentially many answers. Measurement yields limited classical outcomes. Quantum computational advantages depend on carefully structured operations that exploit interference and correlations to extract useful information, rather than on unrestricted access to every amplitude. (tsapps.nist.gov)

Physical implementations

A physical qubit requires two controllable states, reliable preparation and readout, and sufficiently isolated quantum behavior. Different technologies realize these requirements in different ways:

• Trapped-ion qubits encode information in internal states of an ion, confined by electromagnetic fields.
• Neutral-atom qubits use selected internal states of an atom held in an array.
• Superconducting qubits use selected energy levels of engineered circuits.
• Spin qubits encode information in spin states, for example those of an electron.
• Photonic qubits use two optical modes, such as orthogonal polarizations of a photon. (nist.gov)

A qubit need not be a system with exactly two available physical levels. It may instead use a chosen two-dimensional subspace of a larger system. Unwanted transitions outside that subspace are called leakage and require separate attention in control and error correction. (tsapps.nist.gov)

Noise and logical qubits

Environmental interactions cause quantum decoherence, while imperfect controls and measurements introduce further errors. Performance consequently depends on coherence, gate accuracy, readout, and available interactions—not simply the number of physical qubits. Noise can affect stored information even when no gates are being applied. (learning.quantum.ibm.com)

Quantum error correction encodes a logical qubit across multiple physical qubits. Measurements extract error syndromes without directly measuring the encoded logical information. This encoding distributes information through collective states rather than creating independent copies of an unknown qubit. Fault-tolerant computation additionally arranges operations so that errors introduced during processing and correction remain controllable. The required overhead depends on the code, hardware noise, and desired reliability. (learning.quantum.ibm.com)