Polarization requirements for ensemble implementations of quantum algorithms with a single-bit output

We compare the failure probabilities of ensemble implementations of quantum algorithms which use pseudopure initial states, quantified by their polarization, to those of competing classical probabilistic algorithms. Specifically we consider a class algorithms which require only one bit to output the solution to problems. For large ensemble sizes, we present a general scheme to determine a critical polarization beneath which the quantum algorithm fails with greater probability than its classical competitor. We apply this to the Deutsch-Jozsa algorithm and show that the critical polarization is 86.6%.