Quantum computers exploit some special properties of quantum mechanics (the most accurate and complete description of the world known) to solve in an incredibly more efficient way some computational tasks (for example: factoring large integer numbers into primes, searching through unstructured data or molecular simulation), that also on a classical supercomputers, require an exponentially amount of time and resources.
The unbelievable power of a quantum computer is due to the massive parallelism obtained by exploiting three fundamental properties of quantum mechanics: quantum superposition, entanglement and interference.
A brief explanation: the classical bit can only take one of two values, it is either 0 or 1, while the quantum bit (or qubit) can also be found in a superposition of both states 0 and 1; essentially it can be 0 and 1 at the same time until a measurement is made. One important aspect of the measurement process is that it alters the state of the qubit: the effect of the measurement is that the new state is exactly the outcome of the measurement. Moreover, in quantum mechanics we can exploit the special property of entanglement. One way to produce the entanglement, it is by bringing two qubits close together, perform an operation to entangle them and then move them apart again. When they are entangled, you can move them arbitrarily far apart from each other and they will remain entangled. This entanglement will manifest itself in the outcomes of measurements on these qubits. When measured these qubits will always yield zero or one perfectly at random, but no matter how far away they are from each other,they will always yield the same outcome.
A fundamental idea in quantum computing is to control the probability a system of qubits collapses into particular measurement states. Quantum interference, a byproduct of superposition, is what allows us to bias the measurement of a qubit toward a desired state or set of states.
Interference is a typical property of wave nature of particles (like electrons, photons, etc.). A wave is characterized by ridges (peak: highest point) and by bellies (valley: lowest point) and the distance between two consecutive ridges or between two consecutive ventri is called wavelength.
When two waves meet each other, they can interfere in different ways: if a crest of a wave meets a crest of another wave of the same frequency at the same point, then the amplitude is the sum of the individual amplitudes — this is constructive interference. If a crest of one wave meets a trough of another wave, then the amplitude is equal to the difference in the individual amplitudes — this is known as destructive interference.
Thanks to these bizarre properties of quantum mechanics, quantum computers will be able to help us to solve problems currently considered unsolvable. We will have a big revolution in pharmaceutical industry, because researchers will be able to simulate the behavior of thousands of molecules in just a few days or weeks, thus obtaining predictive results before performing the classic clinical trials.
The chemical industry has always exploited catalysts to do reactions as near to ambient temperature as is practical, thus keeping energy usage and costs down. Today industry faces additional pressure to be cleaner and greener, which will require the development of new catalysts. Quantum Computers can help the chemical industry to solve these issues.
Quantum Computing will be able to solve complex AI (Artificial Intelligence) problems and obtain multiple solutions to complex problems simultaneously. This will result in artificial intelligence more efficiently performing complex tasks in human-like ways.
And last but not least, the laws of quantum physics can guarantee unconditional security in emerging quantum communication and cryptography systems.
These are just some of the possible applications of this fascinating computational paradigm, others, totally new, probably will be invented as soon as we will have a fault-tolerant scalable quantum computer.