The mathematical proof that quantum computers can outperform classical machines in certain computational tasks is based on a field of study called quantum complexity theory. The most famous result in this field is Peter Shor's algorithm for factoring large numbers, which demonstrated that a quantum computer can efficiently factorize numbers, a task that is believed to be intractable for classical computers.
Shor's algorithm is based on the principles of quantum superposition and quantum interference, which allow a quantum computer to explore a vast number of possibilities simultaneously. By leveraging these quantum properties, Shor's algorithm can find the prime factors of a large number significantly faster than any known classical algorithm.
The classical counterpart of Shor's algorithm, known as the best-known classical factoring algorithm, is the General Number Field Sieve (GNFS). The GNFS has a sub-exponential time complexity, meaning that the time it takes to factorize a number using the GNFS grows exponentially, but at a slower rate than the brute-force method. In contrast, Shor's algorithm has polynomial time complexity, which means its runtime grows as a polynomial function of the input size.
The advantage of quantum computers over classical computers for factoring large numbers has not been proven rigorously using complexity theory, mainly due to the unproven conjecture that factoring is indeed a hard problem for classical computers. However, Shor's algorithm demonstrates a significant speedup over the best-known classical algorithm for factoring, indicating the potential superiority of quantum computers for certain tasks.
It's worth noting that Shor's algorithm is just one example where quantum computers can offer exponential speedup over classical computers. There are other quantum algorithms, such as Grover's algorithm for unstructured search, which provide a quadratic speedup compared to classical algorithms.
Overall, the mathematical proofs for quantum computers' advantage lie in the development of quantum algorithms that demonstrate substantial computational speedup compared to classical counterparts for specific problems. These proofs are supported by the principles of quantum mechanics and complexity theory.