# Difference between revisions of "Bloch Sphere"

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*Link to Collection on ImpVis website (when created): | *Link to Collection on ImpVis website (when created): | ||

*Any other links to resources (Miro boards / notes pages / Google Docs etc): | *Any other links to resources (Miro boards / notes pages / Google Docs etc): | ||

[[Category: | [[Category:Project pages]] |

## Latest revision as of 12:31, 16 February 2022

## Contributors

*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**

- 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)
- 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
- The general representation of these pure quantum states as α|0>+β|1>
- 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.*

## Links

- 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):