Super-recursive algorithm
The famous Church-Turing Thesis claims that recursive algorithms are the most general. All later mathematical models of algorithms were either equivalent to or even weaker than Turing machines (or equivalently, partial recursive functions).
However, if you abandon the requirement that an algorithm must halt then you get something more general, called a super-recursive algorithm.
Several decades ago the theory of super-recursive algorithms appeared. This theory refutes the Church-Turing Thesis because the computing power of super-recursive algorithms is much greater than that of the conventional, or recursive, algorithms.
Super-recursive algorithms could theoretically provide even greater efficiency gains than using quantum algorithms.