I know only little thing about Lie theory but I would like to learn more about its link to quantum computing.
Has someone got some references explaining it well ?
1 Answer
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Howard Georgi's book 'Lie Algebras in Particle Physics' is a wonderful and surprisingly accessible introduction to the subject and its use in quantum mechanics. You don't need to know or even care about particle physics to find it useful. There may be an even better book out there that covers how Lie theory is used specifically in quantum computation, but that might be a bit too niche. An example that such a hypothetical book should cover is Solovay–Kitaev theorem, so you could use that to search more.
As for the connections, in quantum mechanics the main objects we deal with are Lie algebras and Lie groups.* The set of unitaries acting on a quantum system forms a Lie group, and this group is generated by a set of operators in a Lie algebra. For example, in the time-evolution operator, $U=e^{-iHt}$, $U$ is an element of a Lie group and $H$ is an element of a Lie algebra. Lie groups are closed under multiplication, while Lie algebras are closed under commutation (in addition to the closure properties inherited from being a vector space), which is a reason why commutators show up so much. An important example is the Baker-Campbell-Hausdorff formula in all of its forms, such as $\exp(X)\exp(Y) = \exp(X+Y+\frac{1}{2}[X,Y]+...)$, which in a sense converts between multiplication of Lie groups and commutation of Lie algebras. This is a basic result from Lie theory.
Actually strictly speaking, we mostly work with representations of the above objects, which is why representation theory often gets mentioned in the same sentence as Lie theory. Lots of important concepts in quantum mechanics like Clebsch-Gordon coefficients and addition of angular momentum are basically just using representation theory to cleanly understand the symmetries of a system.
*However note that discrete groups like the Clifford group are not Lie groups. Lie groups are continuous.
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$\begingroup$ +1; I think you are right that $H$ is an element of the algebra and $U$ is analogous to the group. But, I think you mean “Lie algebras are closed under addition”, right? $\endgroup$ Mar 26 at 17:06
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$\begingroup$ Actually I meant "Lie algebras are closed under commutation!" Fixed. $\endgroup$ Mar 26 at 17:52