Qubit

Of course you need an article on the basic unit of quantum information. Let's get this over with. Don't confuse it with a Cubit, the ancient unit of length. One measures the mundane, the other measures the fabric of reality. Try to keep up.

Units of information

Information-theoretic

Data storage

Quantum information

qubit (binary)
qutrit (ternary)
qudit ( d -dimensional)

In the strange and often counter-intuitive world of quantum computing, the qubit (or quantum bit, if you insist on being formal) serves as the fundamental unit of quantum information. It is a binary qudit, the quantum-mechanical successor to the classical binary bit, physically brought to life by a two-state device. At its core, a qubit is a two-state (or two-level) quantum-mechanical system—one of the simplest constructs that still manages to exhibit the full, baffling peculiarity of quantum mechanics.

Consider, for instance, the spin of an electron, where the two levels can be designated as "spin up" and "spin down." Or think of the polarization of a single photon, where its two spin states (left-handed and right-handed circular polarization) can also be measured as horizontal and vertical linear polarization. In a classical system, a bit is tragically confined to one state or the other. It's either a zero or a one; there is no in-between. Quantum mechanics, however, is far more generous. It permits the qubit to exist in a coherent superposition of multiple states at once, a property that is not merely a novelty but the very foundation of quantum mechanics and, by extension, quantum computing.

Etymology

The term qubit is attributed to Benjamin Schumacher, who, in the acknowledgments of his 1995 paper, confessed that the name was conceived in jest during a conversation with William Wootters. It seems even the nomenclature of reality's building blocks isn't immune to the dry humor of physicists.

Comparison with classical bits

A binary digit, the familiar 0 or 1, is the bedrock of classical computing. It's how your computer represents information. When averaged over its two possible states, a single binary digit can convey up to one bit of information content, with the bit being the fundamental unit of information. For the purposes of this discussion, let's treat "bit" and "binary digit" as synonymous, because life is too short for pedantry.

In the machinery of classical computers, a bit is physically manifested by one of two distinct levels of low direct current voltage. To switch from one state to the other, the system must traverse a "forbidden zone" that separates the two logic levels. This transition must be executed as swiftly as possible, a brute-force maneuver necessitated by the physical reality that electrical amplitude cannot change instantaneously.

A qubit, when measured, also yields one of two outcomes, which we conveniently label "0" and "1" to maintain some semblance of continuity with the classical world. But that's where the similarity ends. While a classical bit is always definitively either 0 or 1, the general state of a qubit, as dictated by quantum mechanics, can be an arbitrary coherent superposition of every computable state it could possibly be, all at the same time. Furthermore, measuring a classical bit is a benign act; it doesn't alter its state. Measuring a qubit, on the other hand, is an act of violence. It shatters the delicate coherence and irrevocably collapses the superposition.

While it's possible to encode one classical bit within a single qubit, to say a qubit "holds more information" is a dangerously simplistic metaphor. It can, for example, carry up to two bits of classical information using protocols like superdense coding, but its true power lies not in storage capacity but in its computational potential.

A classical bit is always fully committed to one of its two states. A collection of n bits, such as a processor register or a bit array, can only occupy one of its 2ⁿ possible configurations at any given moment. A quantum state, however, can be in a superposition, meaning the qubit possesses a non-zero probability amplitude for both of its states simultaneously. This is popularly, if imprecisely, described as being "in both states at once." To describe this state, a qubit requires two complex numbers—its probability amplitudes. These can be visualized as a 2-dimensional complex vector, known as a quantum state vector. Equivalently, the qubit's value can be seen as a single point within a 2-dimensional complex coordinate space.

The difference becomes exponentially more dramatic with scale. A set of n classical bits is described by n binary digits. A set of n qubits, forming a quantum register, requires 2ⁿ complex numbers to fully describe its superposition state vector. This exponential growth in descriptive complexity is the source of a quantum computer's theoretical power.

Standard representation

In the language of quantum mechanics, a qubit's general quantum state is expressed as a linear superposition of its two orthonormal basis states, or basis vectors. These foundational vectors are typically denoted as:

$|0\rangle ={\bigl [}{\begin{smallmatrix}1\\0\end{smallmatrix}}{\bigr ]}$ and $|1\rangle ={\bigl [}{\begin{smallmatrix}0\\1\end{smallmatrix}}{\bigr ]}$ .

This is the conventional Dirac notation, or "bra–ket" notation. The symbols $|0\rangle$ and $|1\rangle$ are pronounced "ket 0" and "ket 1," respectively. These two orthonormal basis states, collectively referred to as the computational basis, are said to span the two-dimensional linear vector (Hilbert) space that the qubit inhabits.

Qubit basis states can be combined to form product basis states. A group of qubits treated as a single system is called a quantum register. For instance, two qubits can be represented within a four-dimensional linear vector space spanned by these product basis states:

$|00\rangle ={\biggl [}{\begin{smallmatrix}1\\0\\0\\0\end{smallmatrix}}{\biggr ]}$ , $|01\rangle ={\biggl [}{\begin{smallmatrix}0\\1\\0\\0\end{smallmatrix}}{\biggr ]}$ , $|10\rangle ={\biggl [}{\begin{smallmatrix}0\\0\\1\\0\end{smallmatrix}}{\biggr ]}$ , and $|11\rangle ={\biggl [}{\begin{smallmatrix}0\\0\\0\\1\end{smallmatrix}}{\biggr ]}$ .

As a general rule, a system of n qubits is described by a superposition state vector existing in a 2ⁿ-dimensional Hilbert space.

Qubit states

The polarization of light provides a remarkably clear way to demonstrate orthogonal states. Using a standard mapping of ${|H\rangle }={|0\rangle }$ (horizontal polarization) and ${|V\rangle }={|1\rangle }$ (vertical polarization), quantum states like $({|0\rangle }\pm {|1\rangle })/{\sqrt {2}}$ have a direct, physical representation. These diagonal polarization states are easily produced and measured in a lab with linear polarizers and, for real coefficients $\alpha$ and $\beta$ , align with the high-school geometry definition of orthogonality.

A pure qubit state is a coherent superposition of its basis states. This means a single qubit, denoted as $\psi$ , can be described by a linear combination of ${|0\rangle }$ and ${|1\rangle }$ :

${|\psi \rangle }=\alpha {|0\rangle }+\beta {|1\rangle }$

Here, α and β are the probability amplitudes, and they are both complex numbers. When this qubit is measured in the standard basis, the Born rule dictates that the probability of the outcome being ${|0\rangle }$ (a value of "0") is ${|\alpha |}^{2}$ , and the probability of the outcome being ${|1\rangle }$ (a value of "1") is ${|\beta |}^{2}$ . Since the absolute squares of these amplitudes correspond to probabilities, it logically follows that $\alpha$ and $\beta$ must adhere to the second axiom of probability theory, constrained by the equation:

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

The probability amplitudes $\alpha$ and $\beta$ hold more information than just probabilities. The relative phase between them is responsible for phenomena like quantum interference, famously demonstrated in the universe's oldest party trick, the double-slit experiment.

Bloch sphere representation

It might seem that the state ${|\psi \rangle }=\alpha {|0\rangle }+\beta {|1\rangle }\,$ should have four degrees of freedom, since $\alpha$ and $\beta$ are complex numbers, each with two degrees of freedom. However, the normalization constraint |α|² + |β|² = 1 removes one degree of freedom. This allows for a change of coordinates, such as Hopf coordinates, to simplify the representation:

${\begin{aligned}\alpha &=e^{i\delta }\cos {\frac {\theta }{2}},\\\beta &=e^{i(\delta +\varphi )}\sin {\frac {\theta }{2}}.\end{aligned}}$

Furthermore, for a single qubit, the global phase of the state, $e^{i\delta }$ , has no physically observable consequences.^[a] We can therefore arbitrarily choose α to be real (or β, if α is zero), which leaves us with just two meaningful degrees of freedom:

${\begin{aligned}\alpha &=\cos {\frac {\theta }{2}},\\\beta &=e^{i\varphi }\sin {\frac {\theta }{2}},\end{aligned}}$

where $e^{i\varphi }$ represents the physically significant relative phase.^[b]

The landscape of possible quantum states for a single qubit can be visualized on a Bloch sphere. On this 2-sphere, a classical bit would be confined to either the "North Pole" (representing ${|0\rangle }$ ) or the "South Pole" (representing ${|1\rangle }$ ). The choice of this axis is arbitrary, but the confinement is not. The rest of the sphere's surface is inaccessible to a classical bit. A pure qubit state, however, can be represented by any point on the surface. For example, the pure state $({|0\rangle }+{|1\rangle })/{\sqrt {2}}$ resides on the equator, along the positive X-axis. In the classical limit, the qubit, with its infinite possibilities on the Bloch sphere, collapses back into the classical bit, which can only occupy one of the two poles.

The surface of the Bloch sphere is a two-dimensional space that represents the observable state space of pure qubit states. This space is defined by two local degrees of freedom, represented by the angles $\varphi$ and $\theta$ .

Mixed state

A pure state is perfectly defined by a single ket, ${|\psi \rangle }=\alpha {|0\rangle }+\beta {|1\rangle },\,$ , representing a coherent superposition as a point on the surface of the Bloch sphere. This coherence is the fragile heart of a qubit's quantum nature. Through interactions with its environment—quantum noise and decoherence—a qubit can devolve into a mixed state. This is a statistical, "incoherent mixture" of different pure states. Mixed states are represented by points inside the Bloch sphere (or in the Bloch ball). A mixed qubit state is described by three degrees of freedom: the angles $\varphi$ and $\theta$ , and the length $r$ of the vector representing the state, where a shorter vector signifies a more mixed, less pure state.

The Sisyphean task of quantum error correction is dedicated to fighting this decay and maintaining the purity of qubits.

Operations on qubits

A variety of physical operations can be performed on qubits, governed by the stringent rules of quantum mechanics.

Quantum logic gates: These are the building blocks of a quantum circuit within a quantum computer. They operate on a set of qubits (a register). Mathematically, the qubits undergo a reversible unitary transformation, described by multiplying the gate's unitary matrix with the quantum state vector. The result is a new, transformed quantum state.
Quantum measurement: This is a clumsy, irreversible operation where information about a qubit's state is extracted, but its coherence is destroyed in the process. Measuring a single qubit in the state ${|\psi \rangle }=\alpha {|0\rangle }+\beta {|1\rangle }$ will yield either ${|0\rangle }$ with probability ${|\alpha |}^{2}$ or ${|1\rangle }$ with probability ${|\beta |}^{2}$ . The act of measurement alters the amplitudes α and β. For instance, if the measurement result is ${|1\rangle }$ , α is forced to 0 and β is forced to 1 (up to a phase factor, $e^{i\phi }$ , which is no longer experimentally accessible). If the measured qubit is entangled with others, this measurement can instantly collapse the state of the other entangled qubits as well.
Initialization: This operation resets a qubit to a known value, typically ${|0\rangle }$ . Like measurement, this collapses the quantum state. Initialization can be implemented logically (by measuring and then applying a Pauli-X gate if the result was ${|1\rangle }$ ) or physically (for a superconducting phase qubit, this could involve lowering the system's energy to its ground state).
Transmission: A qubit can be sent through a quantum channel to a remote system, a fundamental I/O operation for any potential quantum network.

Quantum entanglement

A crucial feature distinguishing qubits from their classical counterparts is their capacity for quantum entanglement. Indeed, a single qubit in a superposition is itself an exhibition of entanglement with its own basis states. More commonly, entanglement refers to a local or nonlocal property of two or more qubits, allowing them to exhibit correlations far stronger than anything possible in classical systems.

The most straightforward system for demonstrating entanglement involves two qubits. Consider two qubits entangled in the ${|\Phi ^{+}\rangle }$ Bell state:

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

In this state, known as an equal superposition, there is an equal probability of measuring the product state ${|00\rangle }$ or ${|11\rangle }$ , since ${|1/{\sqrt {2}}|}^{2}=1/2$ . Before measurement, there is no way to know if the first qubit is "0" or "1," and the same is true for the second.

Now, imagine these two entangled qubits are separated and given to two individuals, Alice and Bob. Alice measures her qubit. She obtains—with equal probability—either ${|0\rangle }$ or ${|1\rangle }$ . Her qubit's state is now known. Due to the entanglement, Bob's qubit must instantly assume a state that perfectly correlates with Alice's measurement. If Alice measures a ${|0\rangle }$ , Bob must also measure a ${|0\rangle }$ , because ${|00\rangle }$ is the only component of the original state where Alice's qubit is ${|0\rangle }$ . In essence, whatever Alice measures, Bob measures too, with perfect correlation, regardless of the basis they choose or the distance separating them. This is a profoundly strange circumstance with no classical explanation.

Controlled gate to construct the Bell state

Controlled gates operate on two or more qubits, where one qubit acts as a control for an operation on another. The controlled NOT gate (CNOT or CX) acts on two qubits. It performs a NOT operation on the second (target) qubit only if the first (control) qubit is in the state ${|1\rangle }$ ; otherwise, it does nothing. With respect to the unentangled product basis $\{{|00\rangle }$ , ${|01\rangle }$ , ${|10\rangle }$ , ${|11\rangle }}\}$ , its action is as follows:

${|00\rangle }\mapsto {|00\rangle }$
${|01\rangle }\mapsto {|01\rangle }$
${|10\rangle }\mapsto {|11\rangle }$
${|11\rangle }\mapsto {|10\rangle }$

A primary application of the CNOT gate is to generate the maximally entangled ${|\Phi ^{+}\rangle }$ Bell state. To do this, the inputs A (control) and B (target) are prepared as:

${\frac {1}{\sqrt {2}}}({|0\rangle }+{|1\rangle })_{A}$ $\otimes$ ${|0\rangle }_{B}$ = ${\frac {1}{\sqrt {2}}}$ $({|00\rangle }+{|10\rangle })$ .

After applying the CNOT gate, the output is the ${|\Phi ^{+}\rangle }$ Bell State: ${\frac {1}{\sqrt {2}}}({|00\rangle }+{|11\rangle })$ .

Applications

The ${|\Phi ^{+}\rangle }$ Bell state is a foundational component in algorithms for superdense coding, quantum teleportation, and entangled quantum cryptography.

Quantum entanglement allows multiple states to be acted upon simultaneously, a parallelism impossible for classical bits. It is an indispensable ingredient for any quantum computation that aims to outperform a classical computer. The successes of quantum communication and computation heavily rely on entanglement, suggesting it is a unique resource intrinsic to the quantum world. A formidable obstacle facing quantum computing in its quest to surpass classical machines is the pervasive noise in quantum gates, which severely limits the size and reliability of quantum circuits that can be executed.

Quantum register

A collection of qubits treated as a single entity is a qubit register. Quantum computers execute calculations by manipulating the qubits within such a register.

Qudits and qutrits

The term qudit denotes a unit of quantum information realized in a d-level quantum system. Qudits are the quantum analog of integer types in classical computing and can sometimes be mapped to arrays of qubits. However, when d is not a power of 2, a qudit cannot be represented by a simple array of qubits. It is entirely possible, for instance, to have 5-level qudits.

In 2017, scientists at the National Institute of Scientific Research created a pair of qudits each with 10 distinct states, yielding more computational power than 6 qubits.
In 2022, researchers at the University of Innsbruck successfully developed a universal qudit quantum processor using trapped ions. That same year, researchers at Tsinghua University's Center for Quantum Information implemented a dual-type qubit scheme in trapped ion computers using the same ion species.
In 2025, the Innsbruck team managed to simulate two-dimensional lattice gauge theories on their qudit quantum computer.
Also in 2022, researchers at the University of California, Berkeley developed a method to dynamically control cross-Kerr interactions between fixed-frequency qutrits, achieving high two-qutrit gate fidelities. This was followed in 2024 by a demonstration of extensible control of superconducting qudits up to $d=4$ .

The qutrit, analogous to the classical trit of ternary computers, is the unit of quantum information in a 3-level quantum system. Beyond the enlarged computational space, the third level of a qutrit can be exploited for more efficient compilation of multi-qubit gates.

Physical implementation

Any two-level quantum-mechanical system can, in principle, be used as a qubit. Multilevel systems are also viable if two of their states can be effectively isolated from the others (for example, the ground state and first excited state of a nonlinear oscillator). Numerous proposals exist, and several physical implementations that approximate two-level systems have been realized with varying degrees of success. Just as a classical computer uses transistors, magnetic domains on a hard disk, and electrical currents to represent bits, a future quantum computer will likely employ a hybrid design with different physical qubits for different tasks.

All physical implementations are plagued by environmental noise. The so-called T₁ lifetime (energy relaxation time) and T₂ dephasing time are critical metrics that characterize a qubit's sensitivity to this noise. A longer time is generally better, but it is not the only factor; gate operation times and fidelities are equally important for practical quantum computing. Different applications such as quantum sensing, quantum computing, and quantum communication demand different types of qubit implementations optimized for their specific needs.

The following is an incomplete list of physical implementations. The choices of basis states are purely by convention.

| Physical support | Name | Information support | ${|0\rangle }$ | ${|1\rangle }$ | | :--- | :--- | :--- | :--- | :--- | | 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 | two electron | | nucleus | nuclear spin addressed through NMR | spin | up | down | | neutral atom | atomic energy level | spin | up | down | | trapped ion | atomic energy level | spin | up | down | | Josephson junction | superconducting charge qubit | charge | uncharged superconducting island ( Q = 0) | charged superconducting island ( Q = 2 e , 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 | | gapped topological system | non-abelian anyons | braiding of excitations | depends on specific topological system | depends on specific topological system | | vibrational qubit | vibrational states | phonon/vibron | ${|01\rangle }$ superposition | ${|10\rangle }$ superposition | | van der Waals heterostructure | electron localization | charge | electron on bottom sheet | electron on top sheet |

Qubit storage

In 2008, a team of UK and US scientists reported the first relatively long and coherent transfer of a superposition state from an electron spin "processing" qubit to a nuclear spin "memory" qubit, lasting 1.75 seconds. This was a landmark event, representing the first relatively stable quantum data storage—a critical step toward functional quantum computing. By 2013, a modification of this system (using charged instead of neutral donors) extended this storage time dramatically: to 3 hours at cryogenic temperatures and 39 minutes at room temperature. The preparation of a room-temperature qubit based on electron spins was also demonstrated by scientists from Switzerland and Australia. Researchers continue to explore increased coherence times, testing the limits of structures like a Germanium hole spin-orbit qubit.

Notes

a. ^{^} This is a consequence of the Born rule. The probability of observing an outcome upon measurement is the modulus squared of the probability amplitude for that outcome. The global phase factor $e^{i\delta }$ is not measurable because it applies to both basis states and, being on the complex unit circle, its modulus squared is always 1: ${|e^{i\delta }|}^{2}=1.$ Note that removing $e^{i\delta }$ means that quantum states that differ only by a global phase cannot be distinguished as distinct points on the surface of the Bloch sphere.
b. ^{^} While the relative phase has no effect on measurements in the computational basis (the Pauli Z basis), it is crucial when measuring in other bases, like the X or Y Pauli basis. For example, the state $({|0\rangle }+e^{i\pi /2}{|1\rangle })/{\sqrt {2}}$ (which lies on the positive pole of the Y-axis) will always be measured to the same value in the Y-basis. In the Z-basis, however, it has an equal probability of being measured as ${|0\rangle }$ or ${|1\rangle }$ . Because measurement collapses the quantum state, measuring in one basis obscures information that would have been accessible in another, a direct manifestation of the uncertainty principle.