Testing non-isometry is QMA-complete

Abstract

Determining the worst-case uncertainty added by a quantum circuit is shown to be computationally intractable. This is the problem of detecting when a quantum channel implemented as a circuit is close to a linear isometry, and it is shown to be complete for the complexity class QMA of verifiable quantum computation. This is done by relating the problem of detecting when a channel is close to an isometry to the problem of determining how mixed the output of the channel can be when the input is a pure state. How mixed the output of the channel is can be detected by a protocol making use of the swap test: this follows from the fact that an isometry applied twice in parallel does not affect the symmetry of the input state under the swap operation.