Discriminating Quantum States in the Presence of a Deutschian CTC: A Simulation Analysis

In an article published in 2009, Brun et al. proved that in the presence of a 'Deutschian' closed timelike curve, one can map K distinct nonorthogonal states (hereafter, input set) to the standard orthonormal basis of a K-dimensional state space. To implement this result, the authors proposed a quantum circuit that includes, among SWAP gates, a fixed set of controlled operators (boxes) and an algorithm for determining the unitary transformations carried out by such boxes. To our knowledge, what is still missing to complete the picture is an analysis evaluating the performance of the aforementioned circuit from an engineering perspective. The objective of this article is, therefore, to address this gap through an in-depth simulation analysis, which exploits the approach proposed by Brun et al. in 2017. This approach relies on multiple copies of an input state, multiple iterations of the circuit until a fixed point is (almost) reached. The performance analysis led us to a number of findings. First, the number of iterations is significantly high even if the number of states to be discriminated against is small, such as 2 or 3. Second, we envision that such a number may be shortened as there is plenty of room to improve the unitary transformation acting in the aforementioned controlled boxes. Third, we also revealed a relationship between the number of iterations required to get close to the fixed point and the Chernoff limit of the input set used: the higher the Chernoff bound, the smaller the number of iterations. A comparison, although partial, with another quantum circuit discriminating the nonorthogonal states, proposed by Nareddula et al. in 2018, is carried out and differences are highlighted.