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BQP is the set of problems solvable in polynomial time for a given error tolerance, and it is suspected to be larger than P (and BPP, which is probably equal to P). However, inability for the gates to act perfectly, etc would require error-checking overhead. What is the overhead cost in the algorithm? In particular, does either the time or the number of q-bits overhead grow more than polynomially in the problem size (if it did then BQP would be altered)?

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The threshold theorem says that if the error rate is below the threshold, a quantum algorithm with T locations (breadth times depth) can be made fault-tolerant with a blow-up (in both number of qubits and circuit size) by a factor which is a polynomial in the log of T. This is not enough to change BQP.

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Reference:Level Reduction and the Quantum Threshold Theorem. Also it also requires "weak correlation", personally I think that physical systems in practice will have this property. –  Kevin Kostlan Nov 5 '12 at 22:38

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