Limitations of Noisy Reversible Computation

In this paper we study noisy reversible circuits. Noisy computation and reversible computation have been studied separately, and it is known that they are equivalent in power to unrestricted computation. We study the case where both noise and reversibility are combined and show that the combined model is weaker than unrestricted computation. We consider the model of reversible computation with noise, where the value of each wire in the circuit is flipped with some fixed probability 1/2 > p > 0 each time step, and all the inputs to the circuit are present in time 0. We prove that any noisy reversible circuit must have size exponential in its depth in order to compute a function with high probability. This is tight as we show that any (not necessarily reversible or noise-resistant) circuit can be converted into a reversible one that is noise-resistant with a blow up in size which is exponential in the depth. This establishes that noisy reversible computation has the power of the complexity class NC. We extend the upper bound to quantum circuits, and prove that any noisy quantum circuit must have size exponential in its depth in order to compute a function with high probability. This highlight the fact that current error-correction schemes for quantum computation require constant inputs throughout the computation (and not just at time 0), and shows that this is unavoidable. As for the lower bound, we show that quasi-polynomial noisy quantum circuits are at least powerful as quantum circuits with logarithmic depth (or QNC). Making these bounds tight is left open in the quantum case.