Ciro Santilli @cirosantilli 37

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Pauli-X gate by

Ciro Santilli 37 Updated 2025-07-16

The quantum NOT gate swaps the state of

∣ 0 ⟩

and

∣ 1 ⟩

, i.e. it maps:

x ∣ 0 ⟩ + y ∣ y ⟩ \to y ∣ 0 ⟩ + x ∣ y ⟩

As a result, this gate also inverts the probability of measuring 0 or 1, e.g.

if the old probability of 0 was 0, then it becomes 1
if the old probability of 0 was 0.2, then it becomes 0.8

[0110]

Quantum NOT gate matrix

.

Figure 1.
Quantum NOT gate symbol
. Source.

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Phase shift gate by

Ciro Santilli 37 Updated 2025-07-16

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Mathematics course of the University of Oxford by

Ciro Santilli 37 Updated 2025-07-16

www.ox.ac.uk/admissions/undergraduate/courses/course-listing/mathematics

List of handbooks open as of 2022 at: www.maths.ox.ac.uk/members/students/undergraduate-courses/teaching-and-learning/handbooks-synopses Kudos, e.g. unlike the physics course of the University of Oxford which paywalled them. 2022 one: www.maths.ox.ac.uk/system/files/attachments/UG%20Handbook%202022.pdf

The Oxford mathematics Moodle has detailed course listings, and most PDFs are not paywalled.

E.g. the 2024 course:

Year 1: everything seems mandatory:
- Michaelmas Term
  - Introduction to University Mathematics
  - Introduction to Complex Numbers
  - Linear Algebra I
  - Analysis I
  - Introductory Calculus
  - Probability
  - Geometry
- Hilary Term
  - Linear Algebra II
  - Groups and Group Actions
  - Analysis II
  - Dynamics
  - Fourier Series and Partial Differential Equations
  - Multivariable Calculus
- Trinity Term
  - Groups and Group Actions
  - Analysis III
  - Statistics and Data Analysis
  - Constructive Mathematics
Year 2:
- Mandatory big courses:
- long options:
  - Rings and Modules
  - Integration
  - Topology
  - Differential Equations 2
  - Numerical Analysis
  - Probability
  - Statistics
  - Fluids and Waves
  - Quantum Theory
- short options
  - Number Theory
  - Group Theory
  - Projective Geometry
  - Integral Transforms
  - Calculus of Variations
  - Graph Theory
  - Mathematical Modelling in Biology
Year 3: pick any 8 courses. Does not say which courses exist in PDF but we can get them from courses.maths.ox.ac.uk/course/index.php?categoryid=814 of the Oxford mathematics Moodle:
- Michaelmas
  - B1.1 Logic (2024-25)
  - B2.1 Introduction to Representation Theory (2024-25)
  - B3.2 Geometry of Surfaces (2024-25)
  - B3.5 Topology and Groups (2024-25)
  - B4.1 Functional Analysis I (2024-25)
  - B5.2 Applied Partial Differential Equations (2024-25)
  - B5.3 Viscous Flow (2024-25)
  - B5.5 Further Mathematical Biology (2024-25)
  - B6.1 Numerical Solution of Partial Differential Equations (2024-25)
  - B6.3 Integer Programming (2024-25)
  - B7.1 Classical Mechanics (2024-25)
  - B8.1 Probability, Measure and Martingales (2024-25)
  - B8.4 Information Theory (2024-25)
  - B8.5 Graph Theory (2024-25)
  - BO1.1 History of Mathematics (2024-25)
  - BOE Other Mathematical Extended Essay (2024-25)
  - BSP Structured Projects (2024-25)
- Hilary
  - B1.2 Set Theory (2024-25)
  - B2.2 Commutative Algebra (2024-25)
  - B2.3 Lie Algebras (2024-25)
  - B3.1 Galois Theory (2024-25)
  - B3.3 Algebraic Curves (2024-25)
  - B3.4 Algebraic Number Theory (2024-25)
  - B4.3 Distribution Theory (2024-25)
  - B4.2 Functional Analysis II (2024-25)
  - B5.1 Stochastic Modelling of Biological Processes (2024-25)
  - B5.4 Waves and Compressible Flow (2024-25)
  - B5.6 Nonlinear Dynamics, Bifurcations and Chaos (2024-25)
  - B6.2 Optimisation for Data Science (2024-25)
  - B7.2 Electromagnetism (2024-25)
  - B7.3 Further Quantum Theory (2024-25)
  - B8.2 Continuous Martingales and Stochastic Calculus (2024-25)
  - B8.3 Mathematical Models of Financial Derivatives (2024-25)
  - B8.6 High Dimensional Probability (2024-25)
  - SB3.1 Applied Probability (2024-25)
  - BO1.1 History of Mathematics (2024-25)
  - BOE Other Mathematical Extended Essay (2024-25)
  - BSP Structured Projects (2024-25)
Year 4: pick any 8 courses (up to 10 if you're crazy). Does not say which courses exist in PDF but we can get them from courses.maths.ox.ac.uk/course/index.php?categoryid=814 of the Oxford mathematics Moodle:
- Michaelmas
  - C1.1 Model Theory (2024-25)
  - C1.4 Axiomatic Set Theory (2024-25)
  - C2.2 Homological Algebra (2024-25)
  - C2.4 Infinite Groups (2024-25)
  - C2.7 Category Theory (2024-25)
  - C3.1 Algebraic Topology (2024-25)
  - C3.3 Differentiable Manifolds (2024-25)
  - C3.4 Algebraic Geometry (2024-25)
  - C3.7 Elliptic Curves (2024-25)
  - C3.8 Analytic Number Theory (2024-25)
  - C4.1 Further Functional Analysis (2024-25)
  - C4.3 Functional Analytic Methods for PDEs (2024-25)
  - C5.2 Elasticity and Plasticity (2024-25)
  - C5.5 Perturbation Methods (2024-25)
  - C5.7 Topics in Fluid Mechanics (2024-25)
  - C5.11 Mathematical Geoscience (2024-25)
  - C5.12 Mathematical Physiology (2024-25)
  - C6.1 Numerical Linear Algebra (2024-25)
  - C6.5 Theories of Deep Learning (2024-25)
  - C7.1 Theoretical Physics (C6) (2024-25)
  - C7.5 General Relativity I (2024-25)
  - C8.1 Stochastic Differential Equations (2024-25)
  - C8.3 Combinatorics (2024-25)
  - CCD Dissertations on a Mathematical Topic (2024-25)
  - COD Dissertations on the History of Mathematics (2024-25)
- Hilary
  - C1.2 Gödel's Incompleteness Theorems (2024-25)
  - C1.3 Analytic Topology (2024-25)
  - C2.3 Representation Theory of Semisimple Lie Algebras (2024-25)
  - C2.5 Non-Commutative Rings (2024-25)
  - C2.6 Introduction to Schemes (2024-25)
  - C3.2 Geometric Group Theory (2024-25)
  - C3.5 Lie Groups (2024-25)
  - C3.6 Modular Forms (2024-25)
  - C3.9 Computational Algebraic Topology (2024-25)
  - C3.10 Additive Combinatorics (2024-25)
  - C3.11 Riemannian Geometry (2024-25)
  - C3.12 Low-Dimensional Topology and Knot Theory (2024-25)
  - C4.6 Fixed Point Methods for Nonlinear PDEs (2024-25)
  - C4.9 Optimal Transport & Partial Differential Equations (2024-25)
  - C5.1 Solid Mechanics (2024-25)
  - C5.4 Networks (2024-25)
  - C5.6 Applied Complex Variables (2024-25)
  - C6.2 Continuous Optimisation (2024-25)
  - C6.4 Finite Element Method for PDEs (2024-25)
  - C7.1 Theoretical Physics (C6) (2024-25)
  - C7.4 Introduction to Quantum Information (2024-25)
  - C7.6 General Relativity II (2024-25)
  - C7.7 Random Matrix Theory (2024-25)
  - C8.2 Stochastic Analysis and PDEs (2024-25)
  - C8.4 Probabilistic Combinatorics (2024-25)
  - C8.7 Optimal Control (2024-25)
  - CCD Dissertations on a Mathematical Topic (2024-25)
  - COD Dissertations on the History of Mathematics (2024-25)

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Multi-qubit gate by

Ciro Santilli 37 Updated 2025-07-16

The most common way to construct multi-qubit gates is to use single-qubit gates as part of a controlled quantum gate.

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Ciro Santilli 37 Updated 2025-07-16

wormwideweb.org/

Browse freely moving whole-brain calcium imaging datasets

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Single-qubit gate by

Ciro Santilli 37 Updated 2025-07-16

The first two that you should study are:

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Geography of Europe by

Ciro Santilli 37 Updated 2025-07-16

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Ciro Santilli 37 Updated 2025-07-16

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Controlled quantum gate by

Ciro Santilli 37 Updated 2025-07-16

Controlled quantum gates are gates that have two types of input qubits:

control qubits
operand qubits (terminology made up by Ciro Santilli just now)

These gates can be understood as doing a certain unitary operation only if the control qubits are enabled or disabled.

The first example to look at is the CNOT gate.

Figure 1.
Generic controlled quantum gate symbol
. Source.
The black dot means "control qubit", and "U" means an arbitrary Unitary operation.
When the operand has a conventional symbol, e.g. the Figure "Quantum NOT gate symbol" for the quantum NOT gate to form the CNOT gate, that symbol is used in the operand instead.

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Evening raga by

Ciro Santilli 37 Updated 2025-07-16

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Ciro Santilli 37 Updated 2025-07-16

Raag Puriya Dhanashri performed on the Sitar by Shahid Parvez

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Ungulate subclade by

Ciro Santilli 37 Updated 2025-07-16

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Cow by

Ciro Santilli 37 Updated 2025-07-16

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Information Engineering group of the University of Oxford by

Ciro Santilli 37 Updated 2025-07-16

www.robots.ox.ac.uk

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Hadamard gate by

Ciro Santilli 37 Updated 2025-07-16

The Hadamard gate takes

∣ 0 ⟩

or

∣ 1 ⟩

(quantum states with probability 1.0 of measuring either 0 or 1), and produces states that have equal probability of 0 or 1.

H = \frac{1}{2} [11 1 - 1]

Hadamard gate matrix

.

Figure 1.
Hadamard gate symbol
. Source.

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Empty circle control qubit notation by

Ciro Santilli 37 Updated 2025-07-16

Some authors use the convention of:

filled black circle: conventional controlled quantum gate, i.e. operate if control qubit is active
empty (White) circle: operate if control qubit is inactive

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Alps by

Ciro Santilli 37 Updated 2025-07-16

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Raga by

Ciro Santilli 37 Updated 2025-07-16

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Raga by time of the day by

Ciro Santilli 37 Updated 2025-07-16

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Ustad by

Ciro Santilli 37 Updated 2025-07-16

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