I've lately been taking further coursework in abstract algebra, and it has struck me as fairly reminiscent of quantum computing. Of course, Pauli matrices, etc. have relevant roots within abstract algebra -- however, at a higher level (applications of quantum computing such as Grover's algorithm, decoherence and quantum information, etc.), how can quantum computing be understood through an algebraic framework?
7
-
1$\begingroup$ What level of abstraction are you interested in? Linear algebra or operator algebras? $\endgroup$– RammusCommented Jan 8 at 9:23
-
$\begingroup$ with "how can quantum computing be understood through [...]" do you mean to ask how is quantum computing formalised in general?, or whether quantum computing can be formalised algebraically?, or how is it the case that algebra is sufficient/apt to describe quantum computing?, or something else? $\endgroup$– glS ♦Commented Jan 8 at 9:49
-
$\begingroup$ @Rammus I have more experience with the linear algebra side, so operator algebras. $\endgroup$– NurdickCommented Jan 8 at 16:12
-
$\begingroup$ @glS How is quantum computing formalized, and how are techniques from abstract algebra applied in quantum computing. $\endgroup$– NurdickCommented Jan 8 at 16:14
-
$\begingroup$ @Nurdick the problem with that is that it's highly unspecific. It seems like you're just asking how quantum computing (and thus quantum mechanics in general) is described mathematically, which is extremely broad, and the topic of basic courses on quantum mechanics. $\endgroup$– glS ♦Commented Jan 9 at 8:48
|
Show 2 more comments
1 Answer
$\begingroup$
$\endgroup$
2
You are correct that quantum computing is algebraic at a theoretical level. The primary mathematical framework uses finite-dimensional inner product spaces over the field of complex numbers.
This is in contrast to actual quantum physics, which mostly uses calculus-based methods. That being said, these calculus-based methods are used when studying physical models of quantum computing.
-
$\begingroup$ What function/s is/are represented by the inner product operation? And how are such functions used? $\endgroup$– NurdickCommented Jan 9 at 4:22
-
$\begingroup$ A good starting place for learning about inner products in the context of quantum computing is the course Basics of quantum information. $\endgroup$ Commented Jan 10 at 10:55