Efficient CMF protocols for tensor-network isometries

Determine systematically when an isometry specified as a tensor network admits an efficient circuit–measurement–feedforward protocol, with particular attention to protocols implemented using shallow, geometrically local circuits and simple feedforward.

Background

The paper establishes an exact correspondence between circuit–measurement–feedforward (CMF) protocols and quantum error-correcting codes: the premeasurement circuit functions as a code encoder, measurement projectors act as detectable errors, and unitary feedforward removes post-selection precisely when detectability holds. This framework turns CMF protocol design into a code-design problem and provides constructions for state preparation and unitary implementation, including protocols beyond the Clifford setting.

The Outlook identifies the converse direction as unresolved. Although constant-depth geometrically local CMF isometries with constant-depth feedforward admit exact tensor-network descriptions with constant bond dimension, it is not known how to characterize, in general, which tensor-network-specified isometries possess efficient CMF realizations.