What kind of mathematics is common in quantum computing?

Question

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.

Answer 1

• 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)

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• 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.

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• ^{A better question might be "is there any area of mathematics for which you cannot see any possible connection to quantum computing?"}