Quantum Computers, Factoring, and Decoherence

It is known that quantum computers can dramatically speed up the task of finding factors of large numbers, a problem of practical significance for cryptographic applications. Factors of an L -digit number can be found in ∼ L 2 time compared to ∼exp( L 1/3 ) time by a quantum computer, which simultaneously follows all paths corresponding to distinct classical inputs, obtaining the solution from the coherent quantum interference of the alternatives. Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm. For a quantum computer performing logical operations, an exponential decay of quantum coherence is inevitable. However, even in the presence of exponential decoherence, quantum computation can be useful as long as a sufficiently low decoherence rate can be achieved to allow meaningful results to be extracted from the calculation.