austin j. adams

Survey of Qubit Representations

February 27, 2026

Because we humans are macroscopic beings who constantly interact with our environment, reasoning about the behavior of individual particles in a closed system — quantum behavior — can be difficult. Even the simplest quantum mechanical systems around, the spin-1/2 systems that people call quantum bits or qubits, aren’t much easier. Folks have designed many ways to represent qubit states. This post lists all the ones I have found.

Summary

Name Example 1Q State Example 2Q State Phase Refs.
Bra–ket/Dirac notation $\frac{1}{\sqrt{2}}\left(\vert 0\rangle - \vert 1\rangle \right)$ $\frac{1}{\sqrt{2}}\left(\vert 00\rangle - \vert 11\rangle \right)$ $\theta$ [1, Fig. 2.1], [2, App. A], [3, Sec. 2.2]
Matrices $\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\ -1\end{bmatrix}$ $\frac{1}{\sqrt{2}}\begin{bmatrix}1 \\ 0 \\ 0 \\ -1\end{bmatrix}$ $\theta$ [4, Chap. 3]
Coordinates $\begin{aligned}a(0) &= \frac{1}{\sqrt{2}} \\ a(1) &= -\frac{1}{\sqrt{2}}\end{aligned}$ $\begin{aligned}a(00) &= \frac{1}{\sqrt{2}} \\ a(11) &= -\frac{1}{\sqrt{2}}\end{aligned}$ $\theta$ [5, Chap. 3]
Bloch sphere
N/A $\theta$ [1, Sec. 1.2, Sec. 4.2], [3, Sec. 2.5.2], [4, Chap. 2], [5, Sec. 14.4]
2D plane/unit circle
$\pm$ [3, Fig. 2.4, Fig. 9.2], [1, Fig. 6.3], [2, Fig. 4.3]
Q is for Quantum
$\pm$ [6], [7]
Qwerty '0'-'1' '00'-'11' $\theta$ [8]
Knot N/A
❌ [9], [10]
Circle notation
$\theta$ [11]
Square notation
$\color{green}\theta$ [12]
Dimensional circle notation
$\theta$ [13]
Pie chart
❌ [14]
State-o-gram
$\theta$ [15]
Q-sphere
$\color{green}\theta$ [16]
BEADS
$\theta$ [16]

Phase column key:

Bibliography

  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition. Cambridge University Press, 2010.
  2. N. D. Mermin, Quantum Computer Science: An Introduction. Cambridge University Press, 2007.
  3. E. G. Rieffel and W. H. Polak, Quantum Computing: A Gentle Introduction. Cambridge, Massachusetts London, England: The MIT Press, 2014.
  4. B. Burd, Quantum Computing Algorithms: Discover how a little math goes a long way. Birmingham: Packt Publishing, 2023.
  5. R. J. Lipton and K. W. Regan, Introduction to Quantum Algorithms via Linear Algebra, second edition. Cambridge, Massachusetts: The MIT Press, 2021.
  6. T. Rudolph, Q is for Quantum. Wroclaw: Terence Rudolph, 2017.
  7. S. E. Economou, T. Rudolph, and E. Barnes, “Teaching quantum information science to high-school and early undergraduate students,” Aug. 08, 2020, arXiv: arXiv:2005.07874.
  8. A. J. Adams et al., “Qwerty: A Basis-Oriented Quantum Programming Language,” in Proceedings of the 2025 IEEE International Conference on Quantum Computing and Engineering (QCE ‘25), Aug. 2025, pp. 804–815.
  9. P. K. Aravind, “Borromean Entanglement of the GHZ State,” in Potentiality, Entanglement and Passion-at-a-Distance: Quantum Mechanical Studies for Abner Shimony Volume Two, R. S. Cohen, M. Horne, and J. Stachel, Eds., in Boston Studies in the Philosophy of Science. Dordrecht: Springer Netherlands, 1997, pp. 53–59.
  10. A. Sugita, “Borromean Entanglement Revisited,” In Proceedings of International Workshop on Knot Theory for Scientic Objects, Mar. 2006, Osaka, Japan.
  11. E. R. Johnston, N. Harrigan, and M. Gimeno-Segovia, Programming Quantum Computers: Essential Algorithms and Code Samples. O’Reilly Media, Inc., 2019.
  12. B. Just, Quantum Computing Compact: Spooky Action at a Distance and Teleportation Easy to Understand. Berlin, Heidelberg: Springer, 2022.
  13. J. Bley et al., “Visualizing entanglement in multiqubit systems,” Phys. Rev. Res., vol. 6, no. 2, p. 023077, Apr. 2024.
  14. K. Yeung, Quantum Computing & Some Physics: The Quantum Computing Comics Notebook. 2020.
  15. F. Schinkel, “state-o-gram – A Novel 2D Visualization for Quantum States,” Aug. 25, 2025, arXiv: arXiv:2508.18390.
  16. IBM Quantum Composer. 2025.