Friday, December 28, 2007

The Margolus-Levitin Theorem, the Beckenstein Bound, and Computronium

Computronium is a proposed form of matter that is optimized for computation. The speed of the fastest possible computer can be calculated by the Margolus-Levitin theorem, which states that that rate is (E * 4) / h, where E is the amount of energy available to the system in joules (the maximum value is E = mc^2, where m is mass in kilograms and c is the speed of light, 3 *10^8 m/s) and h is Planck's constant, which is approximately 6.626 * 10^-34 J * s. The units of the result are operations per second. The maximum possible memory capacity of a computer is related to the Beckenstein bound, which is the maximum amount of information that a given region of space can contain. This equals one bit per 4 square Planck lengths (a Planck length is approximately 1.616 * 10^-35 meters. This means that one kilogram of computronium taking up a volume of one liter could perform 10^51 operations per second and store 10^31 bits. For comparison, the world's fastest computer can currently perform 10^15 operations per second and a typical large hard drive can store one terabyte (approximately 10^13 bits). Processing speeds and memory capacities are currently doubling about every 12 and 15 months, respectively. If these trends continue, the maximum possible storage density will be reached around the year 2080 and the fastest possible computing speed will be reached around 2200. The Margolus-Levitin theorem also implies that computation speed is inversly proportional to entropy. On a tangentially related note, there is an interesting post on the generally obstructive effect of disorder on intelligent processes at Tom McCabe's blog.

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