Computational Capacity of the Universe

# Computational Capacity of the Universe -...

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Unformatted text preview: arXiv:quant-ph/0110141v1 24 Oct 2001 Computational capacity of the universe Seth Lloyd d’Arbeloff Laboratory for Information Systems and Technology MIT Department of Mechanical Engineering MIT 3-160, Cambridge, Mass. 02139 [email protected] Merely by existing, all physical systems register information. And by evolving dynamically in time, they transform and process that information. The laws of physics determine the amount of information that a physical system can register (number of bits) and the number of elementary logic operations that a system can perform (number of ops). The universe is a physical system. This paper quantifies the amount of information that the universe can register and the number of elementary operations that it can have performed over its history. The universe can have performed no more than 10 120 ops on 10 90 bits. ‘Information is physical’ 1 . This statement of Landauer has two complementary inter- pretations. First, information is registered and processed by physical systems. Second, all physical systems register and process information. The description of physical systems in terms of information and information processing is complementary to the conventional de- scription of physical system in terms of the laws of physics. A recent paper by the author 2 put bounds on the amount of information processing that can be performed by physical systems. The first limit is on speed. The Margolus/Levitin theorem 3 implies that the total number of elementary operations that a system can perform per second is limited by its energy: #ops / sec ≤ 2 E/π ¯ h , where E is the system’s average energy above the ground state and ¯ h = 1 . 0545 × 10- 34 joule-sec is Planck’s reduced constant. As the presence of Planck’s constant suggests, this speed limit is fundamentally quantum mechanical, 2- 4 and as shown in (2), is actually attained by quantum computers, 5- 27 including existing 1 devices. 15 , 16 , 21 , 23- 27 The second limit is on memory space. The total number of bits available for a system to process is limited by its entropy: #bits ≤ S/k B ln2, where S is the system’s thermody- namic entropy and k B = 1 . 38 × 10- 23 joule/K is Boltzmann’s constant. 2 This limit, too, is attained by existing quantum computers, which store one bit of information per nuclear spin or photon polarization state. The speed at which information can be moved from place to place is limited by the speed of light, c = 2 . 98 × 10 8 meters/sec. This limit can be combined with the first two to limit the input/output rate of computational systems. 2 The maximum rate at which information can be moved in and out of a system with size R is ≈ cS/k B R (attained by taking all the information S/k B ln2 in the system and moving it outward at the speed of light). The ratio between the maximum rate of information processing (ops/sec) and the input/output rate (bits in/out per second) is ≈ k B ER/ ¯ hcS , a quantity shown by Bekenstein to be...
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Computational Capacity of the Universe -...

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