Introduction to Quantum Computation

A computation is a physical process. It may be performed by a piece of electronics or on an abacus, or in your brain, but it is a process that takes place in nature and as such it is subject to the laws of physics. Quantum computers are machines that rely on characteristically quantum phenomena, such as quantum interference and quantum entanglement in order to perform computation. In this series of lectures I want to elaborate on the computational power of such machines. 1 Physical Representation of Information Suppose you have n physical objects and each object has k distinguishable states. If you can access each object separately and put it into any of the k states then with very little effort you can prepare any of the N = k different configurations of the combined systems. Let us put k = 2 and refer to each object of this type as a physical bit. We label the two states of a physical bit as 0 and 1. Any collection of n physical bits can be prepared in N = 2 different configurations which can be used to store N messages, N binary strings or N different numbers. Suppose the two states in the physical bit are separated by the energy difference E0 then a preparation of any particular configuration will cost not more than E = E0n = E0 logN units of energy (the log is taken to the base 2). If you choose to encode N configurations into one chunk of matter, say a single harmonic oscillator with the interstate energy separation E0, then, in the worst case, one has to use E = E0N units of energy (e.g. to go from the ground state labelled as 0 to the most excited state labelled as N ). For large N this gives an exponential gap in the energy expenditure between the binary encoding using physical bits and the so-called unary encoding using a harmonic oscillators. One can, of course, try to switch from harmonic oscillators to objects which have a finite spread in the energy spectrum. For example, if one wants to use the energy states of the hydrogen atom to encode any number of configurations then one is guaranteed not to spend more than E0 = 13.6eV (otherwise the atoms is ionised). The snag is that in this case some of the electronic states will be separated by the energy difference of the order of E0/N and to drive the system selectively from one state to another one has to tune into the frequency E0/ N which requires a sufficiently long wavepacket (so that the frequency is well defined) and consequently the interaction time of the order N( /E0). Thus we have to trade energy for time. It turns out that whichever way we try to map N configurations into a single chunk of matter we end up depleting our physical W. Dieter Heiss (Ed.): LNP 587, pp. 47–76, 2002. c © Springer-Verlag Berlin Heidelberg 2002