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Abstract We present the first detailed polarimetric studies of Cygnus A at 230 GHz with the Submillimeter Array (SMA) to constrain the mass accretion rate onto its supermassive black hole. We detected the polarized emission associated with the core at a fractional polarization of . This low fractional polarization suggests that the polarized emission is highly depolarized. One of the possible explanations is due to a significant variance in the Faraday rotation measure within the synthesized beam. By assuming the Faraday depolarization caused by inhomogeneous column density of the magnetized plasma associated with the surrounding radiatively-inefficient accretion flow within the SMA beam, we derived the constraint on the mass accretion rate to be larger than 0.15 yr −1 at the Bondi radius. The derived constraint indicates that an adiabatic inflow–outflow solution or an advection-dominated accretion flow should be preferable as the accretion flow model in order to explain the jet power of Cygnus A.