I am writing a book on QC so - try this for starters:

Silicon chips consist of relatively simple components, namely resistors, transistors, capacitors and inductors. Just add a battery and you can make a phone, a personal computer or a supercomputer.
Initially these components were quite large and over the years scientists have perpetually been miniaturising them. These components are very useful as together they can store and manipulate zeroes and ones. We now have integrated circuits which contain these components in tiny form. It is now possible to fit billions of components into the space that used to occupy just one of them in the early days.
The integrated circuits are developed onto silicon using a technique similar to photography. The components are not made individually any more, but stamped and printed onto the silicon. The engineers use all sorts of tricks and techniques to project smaller and smaller images onto the silicon chips.
However, a couple of awkward problems occur when these silicon chips grow in size to accommodate more and more components. Firstly, there is a problem of overheating and secondly and more interestingly, different laws of physics apply as the components get smaller and smaller.
The overheating problem occurs as each of the tiny transistors has a resistance, and as you put power through a resistance, the power is dissipated. As you pack more and more transistors onto a silicon chip, more and more heat is produced, resulting in a requirement for more and more advances in the dissipation of heat with such things as heat syncs.
This overheating applies to a two dimensional silicon chip, so there is little hope of producing three dimensional silicon blocks which would theoretically be much faster. The problem of overheating would be exacerbated many-fold; to such an extent they would overheat and melt.
However, we still want to develop faster and faster processors to perform more and more demanding tasks.
There are materials called superconductors which have no electrical resistance below a certain temperature. A normal metal will maintain some resistance down to very low temperatures, whereas a superconductor will suddenly possess a totally zero resistance once it reaches a certain temperature.
Superconductors can be made of metals or ceramic based materials. The temperature at which the metal superconductors start operating is quite low, but the temperature at which the ceramic superconductors can operate is much higher. The higher the temperature the superconductor can operate at with zero resistance, the more cost effective the processing capability will become - simply because cooling materials to low temperatures is a fairly costly exercise.
One of the best superconductors for this job is referred to as YBCO, its full name is Yttrium barium copper oxide, which is a crystalline chemical compound.
As a superconductor has zero resistance, so if we were able to build computer chips from superconductors then when tightly packing billions of components together, the zero resistance would produce zero heat. Therefore we would no longer experience any overheating problems.
The superconducting material could now be stacked into three dimensional blocks which would be extremely efficient for processing.
Another property the superconducting materials possess is the ability to operate at very small scales where the different laws of physics apply; namely quantum physics. As mentioned earlier, these different laws of physics are prevalent as the components get smaller and smaller. Superconductors seem to be able to obey this weird world of quantum physics - this conformity appears to be inherent within their normal composition and operation.
One problem presented to the designers of very small circuit boards is the fact that an electron is not actually a point-like particle when viewed on a very small scale; it is actually more spread out like waves. Within the electronic circuits the electron waves are still very small, but they are actually big enough to start causing a problem. If an electron is spreading out, then when it comes in contact with a small component such as a transistor, you are not sure where the electron is; it could be on one side of the transistor, or on another side of the transistor. What you ideally want to be able to do is control exactly where the electron should be.
This is currently a major problem with the design of very small printed circuit boards. Because the components are so tiny, the electrons behave as waves, and this causes operational problems.
Is there a way we can utilise these quantum waves to our advantage?
What is interesting is that if we pass an electric current through a superconducting material such that the electrons have two paths they could take, when converging them together again they act in a similar fashion to the light in the double slit experiment. What is especially useful about quantum waves is that they possess some very strange properties.
Because the wave-like behaviour allows the electrons to spread out, they can be in two places at the same time. If you were to attribute a state to each of these two positions of the electron wave, not only would it be in two places at the same time, the electron would also be in two states at the same time!
Whereas a normal Personal Computer works by storing information as bits in the form of ones and zeroes, a quantum computer will allow you to store a state of zero and one at the same time using the quantum waves. This is effectively a superposition of zero and one called a QUBIT, coined from the terms quantum and bit. It can take a value of zero, one, or a mixture of the two. This is what gives a quantum computer the potential for enormous processing power.
The power of a quantum computer comes from the fact that it can put these zeroes and ones into a superposition state. Because a QUBIT can be in two states at the same time they can explore several calculation possibilities at the same time. A classical register on a classical computer will require four computational steps to upload to four register values. However, a quantum computer register can have both states at the same time, so you can generate all the answers in a single computational step. This is effectively a very powerful parallel computation capability.
A quantum computing QUBIT is created inside what is termed a SQUID, a Superconducting Quantum Interface Device. The SQUID is made from little loops of superconductor that allow the electron waves to pass in two directions. The superconductor used is normally niobium as it allows the electron waves to get larger; this means they can be controlled better. These components can be made as small as one micrometre, bearing in mind that a grain of pollen is about five micrometres wide, shows how small they are.
The quantum computer benefits from the superposition state of the electron wave, this means that other methods or systems can be utilised to gain the same effect rather than just using superconductors. Ions trapped in electrical fields, nuclear spins in the centre of atoms, and photons from lasers can also be used to encode these zeroes and ones in their special states. No one knows which of all these different systems will work best yet – scientists are still exploring.
QUBITS are simply placed together in large clusters; so long as they are designed correctly using superconductors they can be positioned onto an integrated circuit and operated as a quantum computer. The quantum computer circuits look very similar to what the conventional silicon circuits look like – so it is easy to build quantum computers using existing technology.
You may ask; so what benefits will a quantum computer bring?
It so happens that due to the nature of a quantum computer it can use clever algorithms. A little like a classical computer may solve a jigsaw puzzle by trying every piece in every other piece to see if it fits, a quantum computer can use clever algorithms to start with the edge pieces and separate out different coloured pieces.
To put this in perspective let us compare how a classical computer can cope with factoring numbers compared with a quantum computer. A classical computer runs into problems very quickly with very large numbers – to such an extent that to find the factors of very large numbers would take longer than the lifetime of the Universe. With quantum computers there is an algorithm that allows these factors to be found much more quickly.
The true beauty of the quantum computer is that it is similar to the way the human brain works; a vast number of neurons all connected in a parallel way. It is interesting to think how quantum computing will help us in the future – perhaps one day we could create additional processing packs for our brains!