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Unformatted text preview: arXiv:quantph/0004072v1 18 Apr 2000 Proceedings of Symposia in Applied Mathematics An Introduction to Quantum Error Correction Daniel Gottesman Abstract. Quantum states are very delicate, so it is likely some sort of quan tum error correction will be necessary to build reliable quantum computers. The theory of quantum errorcorrecting codes has some close ties to and some striking differences from the theory of classical errorcorrecting codes. Many quantum codes can be described in terms of the stabilizer of the codewords. The stabilizer is a finite Abelian group, and allows a straightforward character ization of the errorcorrecting properties of the code. The stabilizer formalism for quantum codes also illustrates the relationships to classical coding theory, particularly classical codes over GF(4), the finite field with four elements. 1. Background: the need for error correction Quantum computers have a great deal of potential, but to realize that potential, they need some sort of protection from noise. Classical computers dont use error correction. One reason for this is that classical computers use a large number of electrons, so when one goes wrong, it is not too serious. A single qubit in a quantum computer will probably be just one, or a small number, of particles, which already creates a need for some sort of error correction. Another reason is that classical computers are digital: after each step, they correct themselves to the closer of 0 or 1. Quantum computers have a continuum of states, so it would seem, at first glance, that they cannot do this. For instance, a likely source of error is overrotation: a state  ) +  1 ) might be supposed to become  ) + e i  1 ) , but instead becomes  ) + e i ( + )  1 ) . The actual state is very close to the correct state, but it is still wrong. If we dont do something about this, the small errors will build up over the course of the computation, and eventually will become a big error. Furthermore, quantum states are intrinsically delicate: looking at one collapses it.  ) +  1 ) becomes  ) with probability   2 and  1 ) with probability   2 . The environment is constantly trying to look at the state, a process called decoherence . One goal of quantum error correction will be to prevent the environment from looking at the data. There is a welldeveloped theory of classical errorcorrecting codes, but it doesnt apply here, at least not directly. For one thing, we need to keep the phase 1991 Mathematics Subject Classification. Primary 81P68; Secondary 94B60. c circlecopyrt 0000 (copyright holder) 1 2 DANIEL GOTTESMAN correct as well as correcting bit flips. There is another problem, too. Consider the simplest classical code, the repetition code: 000 (1) 1 111 (2) It will correct a state such as 010 to the majority value (becoming 000 in this case)....
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