Parameterized quantum comb and simpler circuits for reversing unknown qubit-unitary operations

In quantum computing, we are capable not only of transforming states but also of transforming processes. Designing quantum circuits to transform input operations has a wide range of applications in quantum computing, quantum information processing, and quantum machine learning. The networks that perform such transformations are known as super-channels [1, 2], which take processes as inputs and output the corresponding transformed process. In general, all these super-channels can be realized with the quantum comb architecture [1, 2]. Figure 1 illustrates an example where a quantum comb takes m quantum operations as input and outputs a target new operation. Quantum comb is widely applied in solving process transformation problems and optimizing the ultimate achievable performance, including transformations of unitary operations such as inversion [3, 4], complex conjugation, control-U analysis [5], as well as learning tasks [6, 7]. It can also be used for analyzing more general processes [8] and has also inspired structures like the indefinite causal network [9, 10]. However, obtaining the explicit quantum circuit required for the desired transformation is a challenging problem. A major problem of the semidefinite programming (SDP) approach based on the Choi-Jamiołkowski isomorphism is that the dimension of the Choi operator of the quantum comb, i.e., the dimension of the variable in such SDP problems, grows exponentially fast with the increase in the number of comb slots. Another issue is that the SDP ultimately returns the Choi operator of the quantum comb; however, finding a physical implementation of this network, such as converting it into a standard circuit model, is not straightforward.