To get an intuition for it, see the sample computation at: en.wikipedia.org/w/index.php?title=Ackermann_function&oldid=1170238965#TRS,_based_on_2-ary_function where in this context. From this, we immediately get the intuition that these functions are recursive somehow.
Quantum computer physical implementation by 
Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01Lists of the most promising implementations:
As of 2020, the hottest by far are:
How To Build A Quantum Computer by Lukas's Lab (2023)
Source. Super quick overview of the main types of quantum computer physical implementations, so doesn't any much to a quick Google.
He says he's going to make a series about it, so then something useful might actually come out. The first one was: Video "How to Turn Superconductors Into A Quantum Computer by Lukas's Lab (2023)", but it is still too basic.
The author's full name is Lukas Baker, www.linkedin.com/in/lukasbaker1331/, found with Google reverse image search, even though the LinkedIn image is very slightly different from the YouTube one.
Quantum Computation and Quantum Information by Nielsen and Chuang by Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01The last value we will likely every know for the busy beaver function! BB(6) is likely completely out of reach forever.
By 2023, it had basically been decided by the The Busy Beaver Challenge as mentioned at: discuss.bbchallenge.org/t/the-30-to-34-ctl-holdouts-from-bb-5/141, pending only further verification. It is going to be one of those highly computational proofs that will be needed to be formally verified for people to finally settle.
As that project beautifully puts it, as of 2023 prior to full resolution, this can be considered the:on the Busy beaver scale.
simplest open problem in mathematics
Aptitude test scene from the Snowden 2016 film
. Source. FISA Court Order The Guardian discussion scene from the Snowden 2016 film
. Source. How is this Possible? scene from the Snowden 2016 film
. Source. Fresh Brains for You scene from the Snowden 2016 film
. Source. The quantum NOT gate swaps the state of and , i.e. it maps: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
Quantum NOT gate symbol
. Source. 
Mathematics course of the University of Oxford by Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01
Ciro Santilli 35 Updated 2025-01-10 +Created 1970-01-01List 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
- Michaelmas Term
- Year 2:
- Mandatory big courses:
- Linear Algebra
- Differential Equations 1
- Metric Spaces and Complex Analysis
- 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
- Mandatory big courses:
- 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)
- Michaelmas
- 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)
- Michaelmas
The most common way to construct multi-qubit gates is to use single-qubit gates as part of a controlled quantum gate.
wormwideweb.org/
Browse freely moving whole-brain calcium imaging datasets
The first two that you should study are:
Controlled quantum gates are gates that have two types of input qubits:These gates can be understood as doing a certain unitary operation only if the control qubits are enabled or disabled.
- control qubits
- operand qubits (terminology made up by Ciro Santilli just now)
The first example to look at is the CNOT gate.
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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