# Bloch Sphere

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## 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

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
• 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?
• 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.