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The full characterization of the possible transformations of quantumoperations is indispensable to developing algorithms in higher-order quantumcomputation, which is the quantum version of functional programming. Althoughuniversal transformations of unitary operations have been well investigated,their extensions to non-unitary operations are still missing, except for a fewexamples. Here we construct \emph{isometry adjointation} protocols,transforming an input isometry operation into its adjoint operation. This taskreduces to the transformation of unitary operation or that of quantum states inspecial cases. Parallel and sequential isometry adjointation protocols areconstructed by transforming unitary inversion protocols using the compositionof quantum combs. This construction achieves the optimal approximation error,which implies that the optimal performance does not depend on the outputdimension of the isometry operation. In particular, we explicitly obtain theasymptotically optimal parallel protocol achieving the approximation error$\epsilon = \Theta(d^2/n)$, where $d$ is the input dimension of the isometryoperation and $n$ is the number of calls of the isometry operation. We alsoconstruct the protocols for the related tasks called isometry inversion anduniversal error detection. We conduct semidefinite programming to investigatethe optimal performances of the tasks using general protocols includingindefinite causal order protocols. The numerical results show that the optimalperformances of general protocols do not depend on the output dimension of theisometry operation for isometry adjointation and universal error detection,which is shown analytically for parallel and sequential protocols. They alsoexhibit the advantage of indefinite causal order protocols over sequentialprotocols for isometry inversion and universal error detection.

Universal adjointation of isometry operations using transformation of quantum supermaps