t b a b a

very limited. To see the spectacular speedups we have to concentrate on problems not in the
classical computational class P(although quantum algorithms may be faster). Many believe
that quantum algorithms solving NP-complete problems in polynomial time do not exist,
even though no proof of this assertion is available at this time. Shor argues that if we assume
that no polynomial time quantum algorithms exist for solving NP-complete problems, then
the class of problems we have to search for is neither NP-complete, nor P, and the population
of this class is relatively small.
2.6 Classical vs Quantum Error Correction
Classical error correction operates by the judicious use of redundancy, that is, making the
system larger in order to make it more resilient to perturbations. A good encoding should
1. make transmission resilient to errors,
2. reduce the amount of data transmitted through a channel, and
3. ensure information confidentiality.
The type of redundancy, or encoding, employed must be carefully chosen according to
the type of noise it wants to fight. The simplest example is the Repetition Code which is
given by the map 0 → 000, 1 → 111. The information can be decoded with majority logic:
If the majority of the three bits is 0, output 0, otherwise output 1.
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