What math is used in the theory of quantum computing?

Question

I'd like to know what rung of the math ladder one need be on to grasp how a quantum computer computes.

I realize this might not be a simple answer, so I'm just looking for an idea of the broad topics required.

Thanks.

Answer 1

For the foundation you need to understand linear algebra, projective geometry and how to build circuits out of AND, OR, NOT gates. For the algorithms themselves, you need to know a little about rational approximations and the Fourier transformation. You can start to learn about Quantum Computing from here but I also recommend working through the book he wrote.

Answer 2

Since all quantum algorithms I know, deal with finite dimensional system, knowledge of unitary groups $\text{U}(N)$ is important, because it governs the evolution of the finite quantum system without relaxation. For the QA to approximate the Jones Polynomial, it doesn't hurt to know something about knot theory.

Answer 3

I would say that the two most important topics would be linear algebra and complexity theory.

• Linear algebra is essential as it is the fundamental language of quantum mechanics (or at least finite-dimensional QM, which is how almost all of quantum computing and information is formulated). Understanding complex vector spaces, matrices (unitary and hermitian matrices especially), and eigenvalues is crucial, and then quantum mechanics is simply attaching labels, descriptions and interpretations to the maths. One can build on this and knowledge of group theory and analysis is also very helpful.

• Complexity theory (and classical algorithms from computer science) is important as it gives context to why we care about quantum computers. What is the fundamental idea of computation? Understanding asymptotic notation (e.g. 'big O'), decision problems, time and space complexities, upper and lower bounds, complexity classes, is essential to then understand how to approach and analyse quantum algorithms, such as Simons's, Grover's, and Shor's. Other topics relating to computer science, such as in information theory and data structures can be helpful too.

So linear algebra (for the maths of QM) and complexity theory (for an appreciation for how to analyse algorithms) is critical in my opinion. There are so many other mathematical topics that quantum computing intersects with: e.g. topology, group theory, analysis -- it's very broad and interdisciplinary!