Limits of Fault-Tolerance on Resource-Constrained Quantum Circuits for Classical Problems

Existing lower bounds on redundancy in fault-tolerant quantum circuits are applicable when both the input and the intended output are quantum states. These bounds may not necessarily hold, however, when the input and the intended output are classical bits, as in the Deutsch-Jozsa, Grover, or Shor algorithms. Here we show that indeed, noise thresholds obtained from existing bounds do not apply to a simple fault-tolerant implementation of the Deutsch-Jozsa algorithm. Then we obtain the first lower bound on the minimum required redundancy for fault-tolerant quantum circuits with classical inputs and outputs. Recent results show that due to physical resource constraints in quantum circuits, increasing redundancy can increase noise, which in turn may render many fault-tolerance schemes useless. So it is of both practical and theoretical interest to characterize the effect of resource constraints on the fundamental limits of fault-tolerant quantum circuits. Thus as an application of our lower bound, we characterize the fundamental limit of fault-tolerant quantum circuits with classical inputs and outputs under resource constraint-induced noise models.