$\mathcal{PT}$-symmetric mapping of three states and its implementation
  on a cloud quantum processor
We develop a new $\mathcal{PT}$-symmetric approach for mapping three pure
qubit states, implement it by the dilation method, and demonstrate it with a
superconducting quantum processor provided by the IBM Quantum Experience. We
derive exact formulas for the population of the post-selected
$\mathcal{PT}$-symmetric subspace and show consistency with the Hermitian case,
conservation of average projections on reference vectors, and Quantum Fisher
Information. When used for discrimination of $N = 2$ pure states, our algorithm
gives an equivalent result to the conventional unambiguous quantum state
discrimination. For $N = 3$ states, our approach provides novel properties
unavailable in the conventional Hermitian case and can transform an arbitrary
set of three quantum states into another arbitrary set of three states at the
cost of introducing an inconclusive result. For the QKD three-state protocol,
our algorithm has the same error rate as the conventional minimum error,
maximum confidence, and maximum mutual information strategies. The proposed
method surpasses its Hermitian counterparts in quantum sensing using non-MSE
metrics, providing an advantage for precise estimations within specific data
space regions and improved robustness to outliers. Applied to quantum database
search, our approach yields a notable decrease in circuit depth in comparison
to traditional Grover's search algorithm while maintaining the same average
number of oracle calls, thereby offering significant advantages for NISQ
computers. Additionally, the versatility of our method can be valuable for the
discrimination of highly non-symmetric quantum states, and quantum error
correction. Our work unlocks new doors for applying $\mathcal{PT}$-symmetry in
quantum communication, computing, and cryptography.
Yaroslav Balytskyi Yevgen Kotukh Gennady Khalimov Sang-Yoon Chang