Bloch Sphere

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  • Name and department of each person.
  • Student or staff partner?
  • How is/was each person involved?
  • What rough dates did they contribute?

Aims & Learning Outcomes

  • Explain the motivation for your visualisation.

Being able to visualize quantum states and the effects of quantum gates on them gives the learner a visual and intuitive understanding of quantum logic gates (rather than just as mathematical equations) - this is important to then be able to build more complex algorithms and quantum circuits which are the fundamental building blocks of quantum computing

  • Introduce the subject of your visualisation.

Bloch sphere

  1. What is a qubit? Bit vs qubit - bits are the building blocks of classical computers (only 0s and 1s) whereas the building blocks of quantum computers are qubits (superpositions of the |0> and |1> quantum states)
  2. What is a Bloch sphere? A way of geometrically representing these quantum states as vectors of a 3d unit sphere, where |0> can be represented by the 0 vector  and |1> can be represented by the 1 vector    using bra-ket notation
  3. The general representation of these pure quantum states as α|0>+β|1>
  4. Pure vs mixed states

where α and β are complex numbers representing the probability amplitudes i.e. the probability of getting |0> is |α|^2.

  • How can we represent this general form of the quantum state geometrically? - We need to introduce coordinate parameters such as φ and θ
    • Rewrite the general quantum state form in terms of polar coordinates (using Euler's identity)
    • Global phase - two quantum states which differ only by a factor of exp(i theta) are considered to be the same
    • Normalization constraint (probabilities must sum to 1)
    • To get: cos(θ) |0> + e^(iφ)sin(θ) |1>
    • θ and φ restrictions means we get cos(θ/2) |0> + e^(iφ)sin(θ/2) |1>
    • Pure vs mixed states (pure states on the surface, mixed states within sphere)

Quantum gates

  • A way of manipulating qubits which is useful for creating algorithms/quantum circuits
  • Pauli-X gate:
    • Matrix representation
    • Flipping the state -> corresponds to 180 degree rotation about the x axis on the Bloch sphere
  • Likewise for Pauli-Y
  • Likewise for Pauli-Z
  • Hadamard
    • Matrix representation
    • 90 degree rotation around Y-axis followed by 180 degree around X axis
  • Which module and year is it intended for and which setting (lecture or self study)?

Useful for the Quantum Information course (PHYS97080)

Can also be used for self-study

  • List learning outcomes. E.g.: "After using this visualisation, students should be able to explain that..."

After using this visualisation, students should be able to:

  • Represent states of qubits (pure and mixed) and their dynamics on the Bloch sphere
  • Understand the dynamics of single qubits by quantum gates (Pauli X, Pauli Y, Pauli Z and Hadamard)

Design Overview

  • Once the design is agreed, describe the final outcome, how it looks, how it functions etc.
  • Include graphics.

Design Justification

Optionally describe any notable decisions made for the design, e.g.

  • Educational design: breaking down of concepts (scaffolding)
  • How were accessibility issues considered?
  • How was space used effectively?
  • How is the design intuitive?
  • Choice of interactive element(s) that fit in organically with the visualisation [inspiration of choice might be from lecture/in-class activity or other sources] - Sliders/Buttons/Cursor (hover/click).

Progress and Future Work

  • Is the design finalised (i.e. agreed by all partners)?
  • If applicable, which pages have been uploaded to website?
  • Any ideas for future improvements.


  • Link to GitHub repository for code in development:
  • Link to visualisation on ImpVis website (when uploaded):
  • Link to Collection on ImpVis website (when created):
  • Any other links to resources (Miro boards / notes pages / Google Docs etc):