Difference between revisions of "Bloch Sphere"
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== Contributors == | == Contributors == | ||
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* ''Explain the motivation for your visualisation.'' | * ''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. | ||
*''Which module and year is it intended for and which setting (lecture or self study)?'' | |||
* 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..."'' | *''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 == | == Design Overview == | ||
Revision as of 00:00, 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):