  It is generally considered that the signal output by a quantum circuit is
attenuated exponentially fast in the number of gates. This letter explores how
algorithms using mid-circuit measurements and classical conditioning as
computational tools (and not as error mitigation or correction subroutines) can
be naturally resilient to complete decoherence, and maintain quantum states
with useful properties even for infinitely deep noisy circuits. Specifically,
we introduce a quantum channel built out of mid-circuit measurements and
feed-forward, that can be used to compute dynamical correlations at infinite
temperature and canonical ensemble expectation values for any Hamiltonian. The
unusual property of this algorithm is that in the presence of a depolarizing
channel it still displays a meaningful, non-zero signal in the large depth
limit. We showcase the noise resilience of this quantum channel on Quantinuum's
H1-1 ion-trap quantum computer.
