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We explore the possibility of adding complex absorbing potential at theboundaries when solving the one-dimensional real-time Schr\"odinger evolutionon a grid using a quantum computer with a fully quantum algorithm described ona $n$ qubit register. Due to the complex potential, the evolution mixes real-and imaginary-time propagation and the wave function can potentially becontinuously absorbed during the time propagation. We use the dilation quantumalgorithm to treat the imaginary-time evolution in parallel to the real-timepropagation. This method has the advantage of using only one reservoir qubit ata time, that is measured with a certain success probability to implement thedesired imaginary-time evolution. We propose a specific prescription for thedilation method where the success probability is directly linked to thephysical norm of the continuously absorbed state evolving on the mesh. Weexpect that the proposed prescription will have the advantage of keeping a highprobability of success in most physical situations. Applications of the methodare made on one-dimensional wave functions evolving on a mesh. Results obtainedon a quantum computer identify with those obtained on a classical computer. Wefinally give a detailed discussion on the complexity of implementing thedilation matrix. Due to the local nature of the potential, for $n$ qubits, thedilation matrix only requires $2^n$ CNOT and $2^n$ unitary rotation for eachtime step, whereas it would require of the order of $4^{n+1}$ C-NOT gates toimplement it using the best-known algorithm for general unitary matrices.

Efficient solution of the non-unitary time-dependent Schrodinger equation on a quantum computer with complex absorbing potential