Quantum Mechanincs

Civilisation has advanced as people discovered new ways of exploiting various physical resources such as materials, forces and energies. In the twentieth century information was added to the list when the invention of computers allowed complex information processing to be performed outside human brains. The history of computer technology has involved a sequence of changes from one type of physical realisation to another --- from gears to relays to valves to transistors to integrated circuits and so on. Today\'s advanced lithographic techniques can squeeze fraction of micron wide logic gates and wires onto the surface of silicon chips. Soon they will yield even smaller parts and inevitably reach a point where logic gates are so small that they are made out of only a handful of atoms. On the atomic scale matter obeys the rules of quantum mechanics, which are quite different from the classical rules that determine the properties of conventional logic gates. So if computers are to become smaller in the future, new, quantum technology must replace or supplement what we have now. The point is, however, that quantum technology can offer much more than cramming more and more bits to silicon and multiplying the clock-speed of microprocessors. It can support entirely new kind of computation with qualitatively new algorithms based on quantum principles!

To explain what makes quantum computers so different from their classical counterparts we begin by having a closer look at a basic chunk of information namely one bit. From a physical point of view a bit is a physical system which can be prepared in one of the two different states representing two logical values --- no or yes, false or true, or simply 0 or

1. For example, in digital computers, the voltage between the plates in a capacitor represents a bit of information: a charged capacitor denotes bit value 1 and an uncharged capacitor bit value 0. One bit of information can be also encoded using two different polarisations of light or two different electronic states of an atom. However, if we choose an atom as a physical bit then quantum mechanics tells us that apart from the two distinct electronic states the atom can be also prepared in a coherent superposition of the two states. This means that the atom is both in state 0 and state 1. To get used to the idea that a quantum object can be in `two states at once\' it is helpful to consider the following experiment (Fig.A and B)

Let us try to reflect a single photon off a half-silvered mirror i.e. a mirror which reflects exactly half of the light which impinges upon it, while the remaining half is transmitted directly through it (Fig. A). Where do you think the photon is after its encounter with the mirror --- is it in the reflected or in the transmitted beam? It seems that it would be sensible to say that the photon is either in the transmitted or in the reflected beam with the same probability. That is one might expect the photon to take one of the two paths choosing randomly which way to go. Indeed, if we place two photodetectors behind the half-silvered mirror in direct lines of the two beams, the photon will be registered with the same probability either in the detector 1 or in the detector 2. Does it really mean that after the half-silvered mirror the photon travels in either reflected or transmitted beam with the same probability 50%? No, it does not ! In fact the photon takes `two paths at once\'. This can be demonstrated by recombining the two beams with the help of two fully silvered mirrors and placing another half-silvered mirror at their meeting point, with two photodectors in direct lines of the two beams (Fig. B). With this set up we can observe a truly amazing quantum interference phenomenon.

If it were merely the case that there were a 50% chance that the photon followed one path and a 50% chance that it followed the other, then we should find a 50% probability that one of the detectors registers the photon and a 50% probability that the other one does. However, that is not what happens. If the two possible paths are exactly equal in length, then it turns out that there is a 100% probability that the photon reaches the detector 2 and 0% probability that it reaches the other detector 1. Thus the photon is certain to strike the detector 2! It seems inescapable that the photon must, in some sense, have actually travelled both routes at once for if an absorbing screen is placed in the way of either of the two routes, then it becomes equally probable that detector 1 or 2 is reached (Fig. 1c). Blocking off one of the paths actually allows detector 1 to be reached; with both routes open, the photon somehow knows that it is not permitted to reach detector 1, so it must have actually felt out both routes. It is therefore perfectly legitimate to say that between the two half-silvered mirrors the photon took both the transmitted and the reflected paths or, using more technical language, we can say that the photon is in a coherent superposition of being in the transmitted beam and in the reflected beam. By the same token an atom can be prepared in a superposition of two different electronic states, and in general a quantum two state system, called a quantum bit or a qubit, can be prepared in a superposition of its two logical states 0 and 1. Thus one qubit can encode at a given moment of time both 0 and 1.

Input Register Output Register

a1|000> Apply the Transformation F

==> a1F(|000>) = b1|000>

a2|001> a2F(|001>) b2|001>

a3|010> a3F(|010>) b3|010>

a4|011> a4F(|011>) b4|011>

a5|100> a5F(|100>) b5|100>

a6|101> a6F(|101>) b6|101>

a7|110> a7F(|110>) b7|110>

a8|111> a8F(|111>) b8|111>

Now we push the idea of superposition of numbers a bit further. Consider a register composed of three physical bits. Any classical register of that type can store in a given moment of time only one out of eight different numbers i.e the register can be in only one out of eight possible configurations such as 000, 001, 010, ... 111. A quantum register composed of three qubits can store in a given moment of time all eight numbers in a quantum superposition (Table 1 ). This is quite remarkable that all eight numbers are physically present in the register but it should be no more surprising than a qubit being both in state 0 and 1 at the same time. If we keep adding qubits to the register we increase its storage

...