Glossary

Computational Basis

The computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.

by Frank Zickert

November 13, 2025

All Quantum States Are Equal, Selected Few Are More Equal

When you walk into Quantum ComputingQuantum Computing is a different kind of computation that builds upon the phenomena of Quantum Mechanics.
Learn more about Quantum Computing, it's easy to assume that the states labeled is a basis state.
Learn more about and is a basis state.
Learn more about are the true, physical states of a Qubit.A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit Everything else feels like decoration. This belief is understandable because Dirac's famous text The Principles of Quantum MechanicsQuantum mechanics is the branch of physics that describes the behavior of matter and energy at atomic and subatomic scales.
Learn more about Quantum Mechanics presents these KetsA ket (written as |ψ⟩) represents a vector in a complex Hilbert space, describing the state of a quantum system. It encodes all measurable information about that system. In linear algebra terms, it’s a column vector, while the corresponding bra (⟨ψ|) is its conjugate transpose (a row vector).
Learn more about Ket early as anchor points, and modern software like QiskitQiskit is an open-source Python framework for programming and simulating quantum computers. It lets users create quantum circuits, run them on real quantum hardware or simulators, and analyze the results. Essentially, it bridges high-level quantum algorithms with low-level hardware execution.
Learn more about Qiskit initializes every QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in is a basis state.
Learn more about by default. The result is a mental model that quietly elevates is a basis state.
Learn more about and is a basis state.
Learn more about to special status.

This belief creates a specific kind of confusion. A simple equation such as the Quantum SuperpositionSuperposition in quantum computing means a quantum bit (qubit) can exist in multiple states (0 and 1) at the same time, rather than being limited to one like a classical bit. Mathematically, it’s a linear combination of basis states with complex probability amplitudes. This allows quantum computers to process many possible inputs simultaneously, enabling exponential speedups for certain problems.
Learn more about Superposition becomes a mysterious blend of possibilities. A tiny QiskitQiskit is an open-source Python framework for programming and simulating quantum computers. It lets users create quantum circuits, run them on real quantum hardware or simulators, and analyze the results. Essentially, it bridges high-level quantum algorithms with low-level hardware execution.
Learn more about Qiskit Quantum CircuitA quantum circuit is a sequence of quantum gates applied to qubits, representing the operations in a quantum computation. Each gate changes the qubits’ state using quantum mechanics principles like superposition and entanglement. The final qubit states, when measured, yield the circuit’s computational result probabilistically.
Learn more about Quantum Circuit as depicted in ? looks like sorcery.

from qiskit import QuantumCircuit

qc = QuantumCircuit(1)

qc.h(0)

qc.measure_all()

Listing 1 A simple quantum circuit

Instead of ordinary Linear AlgebraLinear algebra is the branch of mathematics that studies vectors, vector spaces, and linear transformations between them. It provides tools for solving systems of linear equations and understanding geometric operations like rotations, projections, and scaling. Its core objects—matrices and vectors—form the basis for much of modern computation, physics, and machine learning.
Learn more about Linear Algebra in a chosen coordinate system, it suggests a supernatural fusion of and . But if you mistake the notation for the physics, you misread what is actually happeningNielsen, M.A., 2010, Cambridge university press, .

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There is a cleaner way to understand all of this. One shift in perspective removes the mystery without changing the math, the code or the predictions. It turns the narrative from spooky paradox into normal geometry.

But the problem is not the Qubit. The problem is the coordinate system we impose on it.

A Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State is a VectorA vector is a mathematical object that has both magnitude (size) and direction. It’s often represented as an arrow or as an ordered list of numbers (components) that describe its position in space, such as . Vectors are used to represent quantities like velocity, force, or displacement.
Learn more about Vector in a complexA complex number is a number that has two parts: a real part and an imaginary part, written as ( a + bi ), where ( i = \sqrt ). The real part ( a ) behaves like ordinary numbers, while the imaginary part ( bi ) represents a direction perpendicular to the real axis on the complex plane. Complex numbers let us represent and calculate quantities involving square roots of negative numbers.
Learn more about Complex Number two dimensional space. The KetsA ket (written as |ψ⟩) represents a vector in a complex Hilbert space, describing the state of a quantum system. It encodes all measurable information about that system. In linear algebra terms, it’s a column vector, while the corresponding bra (⟨ψ|) is its conjugate transpose (a row vector).
Learn more about Ket is a basis state.
Learn more about and is a basis state.
Learn more about are simply the coordinate axes in that space. Nothing more. Dirac's Bra KetBra–ket notation is a compact way to describe quantum states and their inner products. A **ket** (|ψ⟩) represents a column vector describing a quantum state, while a **bra** (⟨φ|) represents its conjugate transpose (a row vector). The inner product ⟨φ|ψ⟩ gives a complex number (like an overlap or probability amplitude), and the outer product |ψ⟩⟨φ| gives an operator.
Learn more about Bra Ket notation was introduced precisely to clarify this. And the Bloch SphereThe Bloch sphere is a geometric representation of a single qubit’s quantum state as a point on or inside a unit sphere. The north and south poles represent the classical states |0⟩ and |1⟩, while any other point corresponds to a superposition of them. Its position encodes the qubit’s relative phase and probability amplitudes, making it a visual tool for understanding quantum state evolution.
Learn more about Bloch Sphere depicted in ? visualizes it.

Figure 1 The Bloch Sphere

Once you see this, the Quantum SuperpositionSuperposition in quantum computing means a quantum bit (qubit) can exist in multiple states (0 and 1) at the same time, rather than being limited to one like a classical bit. Mathematically, it’s a linear combination of basis states with complex probability amplitudes. This allows quantum computers to process many possible inputs simultaneously, enabling exponential speedups for certain problems.
Learn more about Superposition stops looking mystical. It becomes a coordinates statement. It tells you that the Quantum State VectorA quantum state vector is a mathematical object (usually denoted |ψ⟩) that fully describes the state of a quantum system. Its components give the probability amplitudes for finding the system in each possible basis state. The squared magnitude of each component gives the probability of measuring that corresponding outcome.
Learn more about Quantum State Vector has a coefficient of along the is a basis state.
Learn more about axis and a coefficient of along the is a basis state.
Learn more about axis. That is all.

In our code example, the line QuantumCircuit(1) means that the circuit has one Qubit,A qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit which QiskitQiskit is an open-source Python framework for programming and simulating quantum computers. It lets users create quantum circuits, run them on real quantum hardware or simulators, and analyze the results. Essentially, it bridges high-level quantum algorithms with low-level hardware execution.
Learn more about Qiskit initializes as is a basis state.
Learn more about . That is not a law of nature. It is a design choice.

The line QCA quantum circuit is a sequence of quantum gates applied to qubits, representing the operations in a quantum computation. Each gate changes the qubits’ state using quantum mechanics principles like superposition and entanglement. The final qubit states, when measured, yield the circuit’s computational result probabilistically. Learn more about Quantum Circuit.h(0) applies a Hadamard OperatorThe Hadamard operator, often denoted H, is a single-qubit quantum gate that creates an equal superposition of and . In other words, It turns the states of the computational basis and into the states of the Fourier basis and .
Learn more about Hadamard Operator to the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit. It turns the Quantum State VectorA quantum state vector is a mathematical object (usually denoted |ψ⟩) that fully describes the state of a quantum system. Its components give the probability amplitudes for finding the system in each possible basis state. The squared magnitude of each component gives the probability of measuring that corresponding outcome.
Learn more about Quantum State Vector from is a basis state.
Learn more about to the state with coordinates and . In other words, after this line the VectorA vector is a mathematical object that has both magnitude (size) and direction. It’s often represented as an arrow or as an ordered list of numbers (components) that describe its position in space, such as . Vectors are used to represent quantities like velocity, force, or displacement.
Learn more about Vector has been rotated to match the equation above. However, we could also describe it on a different basis, which we actually often do. Then it simply reads .

Nothing mysterious happens. This is only a change in coordinates.

If is a basis state.
Learn more about and is a basis state.
Learn more about were truly special, would it be allowed to replace them?

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The answer is no. You can choose any pair of OrthonormalOrthonormal states are perfectly distinct and normalized, which makes them reliable building blocks for quantum algorithms. Two quantum states are orthonormal if they’re at right angles in Hilbert space (their inner product is zero) and each has length 1. This guarantees the states don’t overlap and can be cleanly distinguished when measuring a quantum system.
Learn more about Orthonormal states as your axes.

So, our example state looks like a SuperpositionSuperposition in quantum computing means a quantum bit (qubit) can exist in multiple states (0 and 1) at the same time, rather than being limited to one like a classical bit. Mathematically, it’s a linear combination of basis states with complex probability amplitudes. This allows quantum computers to process many possible inputs simultaneously, enabling exponential speedups for certain problems.
Learn more about Superposition in the Computational Basis.The computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis But in another perfectly valid basis, namely the Fourier BasisThe Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
Learn more about Fourier Basis whose Basis StatesA **basis state** in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are (|0⟩) and (|1⟩); any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of 0s and 1s (e.g., (|00⟩, |01⟩, |10⟩, |11⟩)), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State are and , the very same looks like a pure basis Vector.A vector is a mathematical object that has both magnitude (size) and direction. It’s often represented as an arrow or as an ordered list of numbers (components) that describe its position in space, such as . Vectors are used to represent quantities like velocity, force, or displacement.
Learn more about Vector In that basis, is simply . There is no mixing at all. Therefore, SuperpositionSuperposition in quantum computing means a quantum bit (qubit) can exist in multiple states (0 and 1) at the same time, rather than being limited to one like a classical bit. Mathematically, it’s a linear combination of basis states with complex probability amplitudes. This allows quantum computers to process many possible inputs simultaneously, enabling exponential speedups for certain problems.
Learn more about Superposition is not a physical feature of a state but a feature of how you choose to describe it.

So, the notion that a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in state is in SuperpositionSuperposition in quantum computing means a quantum bit (qubit) can exist in multiple states (0 and 1) at the same time, rather than being limited to one like a classical bit. Mathematically, it’s a linear combination of basis states with complex probability amplitudes. This allows quantum computers to process many possible inputs simultaneously, enabling exponential speedups for certain problems.
Learn more about Superposition, whereas a QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in state is a basis state.
Learn more about is not, is simply wrong. Every Quantum SystemA quantum system is any physical system that is subject to the laws of quantum mechanics, whereby quantities such as energy or spin can only assume discrete (quantized) values. Its behavior is described by a wave function that encodes the probabilities of possible measurement results.
Learn more about Quantum System is in a Quantum StateA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State of SuperpositionSuperposition in quantum computing means a quantum bit (qubit) can exist in multiple states (0 and 1) at the same time, rather than being limited to one like a classical bit. Mathematically, it’s a linear combination of basis states with complex probability amplitudes. This allows quantum computers to process many possible inputs simultaneously, enabling exponential speedups for certain problems.
Learn more about Superposition. Always.

For instance, we can formulate is a basis state.
Learn more about as a superposition of the Fourier Basis'The Fourier basis is a set of sine and cosine functions that can represent any periodic signal as a weighted sum of these functions. Each basis function corresponds to a specific frequency, capturing how much of that frequency is present in the signal. In essence, it’s the coordinate system for expressing signals in terms of their frequency components instead of time.
Learn more about Fourier Basis Basis StatesA **basis state** in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are (|0⟩) and (|1⟩); any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of 0s and 1s (e.g., (|00⟩, |01⟩, |10⟩, |11⟩)), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State like this .

But what about MeasurementIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement? It forces you to return to the Computational Basis.The computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis That's right. Yet, it is not because nature prefers it. It happens because engineers built the Quantum HardwareQuantum hardware is the physical technology that builds and runs quantum computers, using quantum bits (qubits) instead of classical bits. These qubits exploit quantum properties like superposition and entanglement to process information in fundamentally different ways. The hardware can be based on systems such as superconducting circuits, trapped ions, or photons, each requiring extreme control and isolation to maintain quantum coherence.
Learn more about Quantum Hardware that way.

If the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis were fundamentally special, you would expect nature to insist on it. Instead, we find the opposite. The outcome probabilities depend entirely on the basis in which you choose to measure.

So, in our example, , we can compute the probabilities of measuring the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in either Basis StateA **basis state** in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are (|0⟩) and (|1⟩); any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of 0s and 1s (e.g., (|00⟩, |01⟩, |10⟩, |11⟩)), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State:

These numbers describe the outcome of projecting onto the is a basis state.
Learn more about and is a basis state.
Learn more about axes. If we had chosen a different MeasurementIn quantum computing, measurement is the process of extracting classical information from a quantum state. It collapses a qubit’s superposition into one of its basis states (usually or ), with probabilities determined by the amplitudes of those states. After measurement, the qubit’s state becomes definite, destroying the original superposition.
Learn more about Measurement basis, we would get different numerical probabilities.

Once you see this, the idea that is a basis state.
Learn more about and is a basis state.
Learn more about are special falls apart. So, the truth is that your QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit does not live inside the Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis. The Computational BasisThe computational basis is the standard set of basis states used to describe qubits in quantum computing. These are typically and for a single qubit. For multi-qubit systems all possible combinations of basis states denote the computational basis, like , , , and . These states correspond to classical bit strings and form an orthonormal basis for the system's Hilbert space. Any quantum state can be expressed as a superposition of these computational basis states.
Learn more about Computational Basis is not a feature of the QubitA qubit is the basic unit of quantum information, representing a superposition of 0 and 1 states.
Learn more about Quantum Bit in general.

However, it is a feature of the Quantum HardwareQuantum hardware is the physical technology that builds and runs quantum computers, using quantum bits (qubits) instead of classical bits. These qubits exploit quantum properties like superposition and entanglement to process information in fundamentally different ways. The hardware can be based on systems such as superconducting circuits, trapped ions, or photons, each requiring extreme control and isolation to maintain quantum coherence.
Learn more about Quantum Hardware we use. It is a coordinate system we standardize on. Therefore, even if not special, the Basis StatesA **basis state** in quantum computing is one of the fundamental states that form the building blocks of a quantum system’s state space. For a single qubit, the basis states are (|0⟩) and (|1⟩); any other qubit state is a superposition of these. In systems with multiple qubits, basis states are all possible combinations of 0s and 1s (e.g., (|00⟩, |01⟩, |10⟩, |11⟩)), forming an orthonormal basis for the system’s Hilbert space.
Learn more about Basis State of the computational basis are more equal than the other Quantum StatesA quantum state is the complete mathematical description of a quantum system, containing all the information needed to predict measurement outcomes. It’s usually represented by a wavefunction or a state vector in a Hilbert space. The state defines probabilities, not certainties, for observable quantities like position, momentum, or spin.
Learn more about Quantum State after all.