Single-Qubit Quantum Gates on a Bloch Sphere

Using the Bloch sphere, a cubit can be represented as a unit vector (shown in red) from the origin to the point on the unit sphere with spherical coordinates . A single-qubit quantum gate operating on produces a rotated qubit , represented by the green vector. Check the box for "add gate 2?" to perform a second operation using gate . This produces another qubit , which is represented by the blue vector. You can choose from the gates H, X, Y, Z, S and T as defined in the Details.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The action of the single-qubit quantum gates can be represented by unitary matrices acting on the qubit
Hadamard gate: ,
Pauli X gate: ,
Pauli Y gate: ,
Pauli Z gate: ,
phase (or gate: ,
gate: .
In general, the operation gives a qubit of the form
with an overall phase factor of no physical significance. Multiplication by removes this factor and reduces the qubit to the canonical form
[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition, Cambridge: Cambridge University Press, 2010.
[2] G. Fano and S. M. Blinder, Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers, Berlin: Springer, 2017.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2017 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+