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From what I have seen so far, there is a lot of linear algebra. Curious what other kinds of maths are used in QC & the specific fields in which they are most predominately invoked.

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From what I have seen so far, there is a lot of linear algebra. Curious what other kinds of maths are used in QC & the specific fields in which they are most predominately invoked.

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- Calculus (e.g. $\int |\psi(x)|^2dx = 1$ )
- Differential Equations (e.g. Schroedinger equation)
- Complex analysis
- Statistics/Probability theory
- Stochastics (especially in studying open quantum systems)
- Information theory
- Topology (e.g. topological quantum computing)
- Group theory (e.g. in stabilizer codes)
- Representation theory (e.g. in stabilizer codes)
- Graph theory (e.g. graph state quantum computing)
- Functional analysis (e.g. Quantum states are unit vectors... with respect to which norm?)
- Algebraic geometry (e.g. for factoring numbers using quantum annealing)
- Discrete optimization (e.g. for factoring numbers using quantum annealing and also.)
- Optimal control theory (here's a review on quantum optimal control theory)
- Game theory (for quantum games)
- Boolean algebra (see this book on Boolean functions and quantum mechanics)
- Coding theory (e.g. Quantum Error Correcting codes)
- Number theory (e.g. Shor's algorithm)
- Category theory (e.g. What is the use of Categorical quantum mechanics?)
- Differential geometry (for quantum information theorists working on quantum gravity)
- Formal Language Theory (in studying QC as a computation model)
**Amazingly**, I've even seen fractional calculus come up for sub-Ohmic baths (s=1/2 in this)- Basically any area of mathematics because we would like to make quantum algorithms to do things more efficiently, regardless of the mathematical field. For example "semi-definite programming" could be added to the list, because quantum algorithms for semi-definite programming is a very active current area of research.
^{A better question might be "is there any area of mathematics for which you cannot see any possible connection to quantum computing?"}

1is there any area of mathematics for which *you* cannot see any possible connection to qc? – user820789 – 2018-07-15T08:07:42.087

2I've been thinking about it since seeing your question! – user1271772 – 2018-07-15T08:08:09.307

2I can't think of a lot of examples where vector calculus comes up. But they're the heart of Maxwell's equations, which you'd use if you're implementing an optical quantum computer. – user1271772 – 2018-07-15T08:27:14.400

1This question is really, really broad. As the current answer shows, pretty much any and all maths can be used in QC. – heather – 2018-07-16T18:18:13.410

@heather I did not realize that when asking. I think it serves as a good reference for the future tho. – user820789 – 2018-07-16T18:37:23.043

4https://abstrusegoose.com/105 – DaftWullie – 2018-07-19T11:47:13.260

2How can you edit an existing question such as to completely reverse its meaning? This makes all answers void! – Norbert Schuch – 2018-07-19T12:23:26.713

1As @NorbertScuch's comment above, it's not a great idea to edit a question to such an extent after it has a valid answer as this invalidates the answer, so I'm reverting the edit. Normally, I'd say to ask a separate question but in this case, it would still be too broad/off-topic by virtue of essentially being a 'shopping list question', so you'd be better off asking that sort of thing in [chat], if there's someone there willing to answer – Mithrandir24601 – 2018-07-19T12:45:28.383