Hi, welcome to my first serious attempt at wrapping my head around quantum computing.
This glossary is my attempt to organize the mass of terms and concepts I'm encountering as I dive into the field of quantum computing.
To be honest, I'm grasping in the dark with most of this material. While some concepts feel tangentially familiar from my high school informatics classes, the field has leaped forward dramatically since the days when I was using a ball mouse with Windows 98.
In short, this glossary will be a reflection of my familiarity with quantum concepts - a kind of a live document that I'll keep expanding and refining as I go along.
If you're also beginning to learn, maybe we can do it together.
Note
This glossary is a work in progress. I'll be adding new terms and deepening definitions as I continue learning. Some definitions may be simplified or incomplete - that's part of the learning process! I will keep a list of sources.
Core Math & Notation
Vector
An object you can add to other vectors and multiply by scalars, obeying the vector-space rules (closure, associativity, distributivity, etc.). In coordinates it's a column of numbers; its length is given by a norm.
Complex Vector Space
A vector space where the scalars are complex numbers (ℂ) rather than reals; the usual linear-algebra operations apply. [1]
Ket
Dirac's notation |ψ⟩ for a (column) vector in a complex inner-product space; the corresponding bra ⟨ψ| is the conjugate transpose (row) in the dual space. Inner products are written ⟨φ|ψ⟩. [2]
Unit Vector / Normalized
A vector whose length (norm) is 1. "Normalizing" a state means scaling it so its norm is 1.
Quantum Basics
Bit
Classical information unit with two values 1. Shannon popularized the term "binary digit" → "bit" in his 1948 communication theory paper. [3]
Qubit
A two-level quantum system. A pure qubit state can be written as |ψ⟩ = α|0⟩ + β|1⟩ with complex amplitudes α, β satisfying |α|² + |β|² = 1 (i.e., a unit vector in ℂ²).
Superposition (Linear Combination)
Any quantum state can be expressed as a linear combination of basis kets with complex coefficients (amplitudes). For a qubit: |ψ⟩ = α|0⟩ + β|1⟩.
Amplitude
The complex coefficient (e.g., α or β) in a superposition. Measurement in the computational basis yields outcome probabilities given by the squared magnitudes, e.g., Pr(0)=|α|², Pr(1)=|β|² (Born rule).
Normalization Constraint
For a closed system's state vector, amplitudes satisfy a unit-length condition; for a qubit |α|² + |β|² = 1.
Quantum State
Mathematically, a state is a ray in a complex Hilbert space (vectors that differ only by a nonzero complex factor—global phase—represent the same physical state). [4]
Greek Letter ψ to Denote an Arbitrary Quantum State
Using |ψ⟩ for a generic state is standard Dirac notation in quantum mechanics texts and lecture notes. [2], [4]
The Quantum State of a Qubit
The quantum state of a qubit is a vector of unit length in a two-dimensional complex vector space (state space). This is the usual formal statement: a qubit's state lives as a unit vector in ℂ² (a 2-dimensional Hilbert space).
Circuits & Gates
X Gate (aka NOT Gate)
The single-qubit Pauli-X operation that flips computational-basis states: X|0⟩=|1⟩, X|1⟩=|0⟩. Matrix X = [[0,1],[1,0]]; equivalently a π-rotation about the x-axis of the Bloch sphere. [5]
Quantum Wire (in Circuit Diagrams)
Each horizontal wire represents a single qubit; gates placed on that wire act on that qubit, with time flowing left→right. [6]
People
Steven Weinberg
Theoretical physicist; Nobel Prize in Physics (1979) for electroweak unification (with Glashow and Salam). [7]
Wolfgang Pauli
Theoretical physicist; Nobel Prize in Physics (1945) for the exclusion principle; namesake of the Pauli matrices (including X). [8]
References
- Lecture Notes for Ph219/CS219: Quantum Information - Preskill, Caltech
- Lecture 9: Dirac's Bra and Ket Notation - MIT OpenCourseWare
- A Mathematical Theory of Communication - Shannon, Harvard
- Chapter 2 Foundations I: States and Ensembles - Preskill, Caltech
- XGate Documentation - IBM Quantum
- Quantum Circuit Diagram Conventions - Microsoft Azure Quantum
- The Nobel Prize in Physics 1979 - NobelPrize.org
- The Nobel Prize in Physics 1945 - NobelPrize.org
