Summary: Limitations of Noisy Reversible Computation
Dorit Aharonov \Lambda Michael BenOr y Russell Impagliazzo z Noam Nisan x
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 noiseresistant) circuit can be converted into a reversible one that is
noiseresistant 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 1 .
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 high
light the fact that current errorcorrection 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 quasipolynomial noisy quantum circuits are at least powerful as quantum
circuits with logarithmic depth (or QNC 1 ). Making these bounds tight is left open in the quantum