Glossary

Basis (quantum basis / basis state)

A basis is a set of linearly independent states that span the space of all possible quantum states.

For a single qubit, the basis states are and , meaning the quantum state of a qubit can be written as a superpossition of these basis states.

Further reading: https://www.quantum-inspire.com/kbase/qubit-basis-states/

Circuit (quantum circuit)

A quantum circuit is a series of quantum gates applied to qubits to perform computation.

It is analogous to electical circuits, with unitary operations acting on qubits, rather than gates acting on wires.

Further reading: https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/quantum-circuits/introduction

Eigenvalue

An eigenvalue is scalar for a given matrix , such that:

Eigenvalues are fundamentally a concept from linear algebra, however in quantum computing they often correspond to measurement outcomes.

Further reading: https://mathforquantum.quantumtinkerer.tudelft.nl/6_eigenvectors_QM/

Eigenvector

An eigenvector is a non-zero vector that only scales (does not rotate) when a transformation matrix A is applied:

Eigenvectors are fundamentally a concept from linear algebra, however in quantum computing they often correspond to definite measurement results.

Further reading: https://mathforquantum.quantumtinkerer.tudelft.nl/6_eigenvectors_QM/

Entanglement (entangled state / entangled bit)

Entanglement is a phenomenon where the states of multiple qubits become correlated, and therefore cannot be described independently of each other.

Measuring one qubit instantly determined all qubits entangled with it, even at distance.

Further reading: https://scienceexchange.caltech.edu/topics/quantum-science-explained/entanglement

Gate (quantum gate)

A quantum gate is a reversible operation applied to qubits, represented by a unitary matrix.

Quantum gates can involve a single or multiple qubit, and can create superposition and entanglement.

Further reading: https://quantum.cloud.ibm.com/learning/en/courses/utility-scale-quantum-computing/bits-gates-and-circuits

Hamiltonian

Hermitian (hermitian matrix)

Inverse

An inverse of a matrix is a matrix that undoes its transformation:

In quantum computing, as all valid gates are unitary, they must have inverses.

Further reading: https://www.mathcentre.ac.uk/resources/Engineering%20maths%20first%20aid%20kit/latexsource%20and%20diagrams/5_5.pdf

Measurement

Measurement collapses a quantum state into a basis state, providing a classical outcome.

Further reading: https://quantumatlas.umd.edu/entry/measurement/

Operation

A quantum operation is any valid transformation on quantum states.

This includes gate application (unitary operations) and measurements.

Further reading:

Qubit

A qubit is the fundamental unit of quantum information.

Analogous to a bit in traditional computing, a qubit represents a single piece of information.

A qubit can exist in a superposition of two basis states:

such that

Register

A quantum register is a system comprising multiple qubits, used to store and process information.

n qubits can be used to store an exponential amount of information in superposition.

Further reading: https://www.sciencedirect.com/topics/mathematics/quantum-register

State (quantum state)

A quantum state () describes the complete information about a quantum system, it contains probabilities for all possible measurement outcomes.

Superposition

|Superposition is a fundamental principle of quantum mechanics by which a physical system exists in all of its possible states until measured.|

In quantum computing, a qubit can be in superposition, existing in all of its possible states until a measurement is taken.

Further reading: https://quantum.microsoft.com/en-us/insights/education/concepts/superposition

Unitary (unitary matrix)

A unitary matrix is a complex square matrix whose inverse is equal to its conjugate transpose.

In quantum computing, these matrices represent valid, reversible quantum gates.