The physics of classical information storage. Why is it that your hard drive works? A modern miracle, I tell you! Part III of my attempt to explain one of my main research interests in quantum computing: “self-correcting quantum computers.” Prior parts: Part I, Part II
The Physics of Classical Information Storage
Despite the fact that Shannon and von Neumann showed that, a least in theory, a reliable, fault-tolerant computer could be built out of faulty, probabilistic components, if we look at our classical computing devices it is not obvious that these ideas matter much. I mean really, once you see how reliable modern transistors or magnetic media work, it is easy to forget that deep down they are highly analog and probabilistic devices. So this leads one to the question of how to reconcile the observations of Shannon and von Neumann with our everyday computing devices. The answer to this seeming conundrum is that systems like transistors and magnetic media actually enact the ideas of the classical theory of error correction, but they do this via the physics of the devices out of which the computer is built.
|A toy model for how information is stored in you hard drive|
Let’s examine a concrete example of how this works. This will be a cartoon model but it will capture a lot of why you hard drive is able to robustly store information. Our model is made up of a large number of “spins” which are arranged in a regular lattice. For our purposes, let’s just make this a two dimensional lattice. These “spins” are simple binary systems which have two configurations, pointing up or pointing down. Now associate with a configuration of the spins (i.e. an assignment of values of pointing up or pointing down) an energy. This energy is calculated in a straightforward way: for each two neighbors on the lattice, add to the energy a value of +J if the spins are pointing in different directions or subtract from the energy a value of -J if the spins are pointing in the same directions. From this description it is easy to see that the lowest energy configuration of spins is when they all point up or when they all point up, since in this configuration all links in the lattice contribute an energy -J in this configuration. Now for any particular configurations of the spins, we can count the number of spins which are pointing up and the number which are pointing down. This is roughly how “information” can be encoded into a device. If the majority of the spins are pointing up, we call this configuration “0”. If a majority of the spins are pointing down, we call this configuration “1”.
|Equilibrium distribution for the two dimensional model of a hard drive|
Okay, so how can we view a magnetic media in terms of classical error correction. Well first of all note that we have encoded “0” and “1” by copying their values across a large number of spins. This is what is called a redundancy code in classical error correction. But, now, here is the important point: the system also performs error correction. In particular consider staring the system with all the spins pointing up. If you now flip a single spin, this will cost you some energy because now your spin will be unaligned with the four neighboring spin. Further if you want to flip another spin, this will cost you even more energy. In general one can see that it requires an amount of energy proportional to the perimeter of a domain to flip all of the spins in this domain. Now at a given temperature, the environment of the hard drive is exchanging energy with the system and is constantly fouling up the information of the value of spins. Sometimes energy goes from the environment to the system and the system is driven from one of the two encoded states of all up or all down. Sometimes energy goes the other way from the system to it’s environment. This latter process “fixes” the information by driving back towards the encoded state it came from. At a given temperature, the ratio of the rate of these two processes is related to the temperature of the system/envornment. At low enough temperature, if you store information into the all up or all down configuration, then this information will remain there, essentially forever, as the process of cooling beats out the process of heating. At high enough temperature this fails, and the information which you try to store into such a system will be rapidly destroyed. The figure on the right is the way that physicists would talk about these effects. At low temperature, most of the spins are pointing along one of two different directions. As one raises the temperature most of the spins remain pointing mostly up and mostly down, until one nears a critical temperature. At this critical temperature, these two configurations merge into one and information can no longer be reliably store in the equilibrium configurations of this system. Thus we see that one can encode information into the majority vote of these spins, and, at low enough temperature, the rate of errors occurring on the system is dominated by the processes of the errors being fixed. Thus physics does error correction for you in this setup.
|Incorrect preparation is fixed below the critical temperature|
Another important property of the model just described is that it is also robust to imperfect manipulations. For example, suppose that you attempt to prepare the system into a mostly up state, but you don’t do such a good job and a large (but not majority) number of the spins are accidentally prepared in the down state. Then, the above system will “fix” this problem and will take the system prepared imperfectly to one that is closer to a mostly up state. Similarly if you try to flip the value of the spins, if you don’t correctly flip all of them, then the system will naturally relax back to its equilibrium value which will have a higher proportion of spins in the correct distribution. Such a system is “self-correcting” in the sense that the error correction is being done not by an external agent, but instead by the natural dynamics of the system.
So, in this simple example, we have seen that physical systems which are currently used to store classical information actually do embody the ideas of Shannon and error correction: it’s just that the physics of these devices enacts these ideas behind the scenes. So, a natural question to ask for quantum computers is whether you can build similar systems for quantum information. Of course, before answering this question you’d better know whether it is possible to mimic the classical ideas of error correction and fault-tolerance for a quantum computer.
If classical computation is possible because of the physics of the devices out of which they are made, can quantum computers take a similar route?