Efficient solution of the non-unitary time-dependent Schrodinger equation on a quantum computer with complex absorbing potential
Quantum(2023)
摘要
We explore the possibility of adding complex absorbing potential at the
boundaries when solving the one-dimensional real-time Schrödinger evolution
on a grid using a quantum computer with a fully quantum algorithm described on
a n qubit register. Due to the complex potential, the evolution mixes real-
and imaginary-time propagation and the wave function can potentially be
continuously absorbed during the time propagation. We use the dilation quantum
algorithm to treat the imaginary-time evolution in parallel to the real-time
propagation. This method has the advantage of using only one reservoir qubit at
a time, that is measured with a certain success probability to implement the
desired imaginary-time evolution. We propose a specific prescription for the
dilation method where the success probability is directly linked to the
physical norm of the continuously absorbed state evolving on the mesh. We
expect that the proposed prescription will have the advantage of keeping a high
probability of success in most physical situations. Applications of the method
are made on one-dimensional wave functions evolving on a mesh. Results obtained
on a quantum computer identify with those obtained on a classical computer. We
finally give a detailed discussion on the complexity of implementing the
dilation matrix. Due to the local nature of the potential, for n qubits, the
dilation matrix only requires 2^n CNOT and 2^n unitary rotation for each
time step, whereas it would require of the order of 4^n+1 C-NOT gates to
implement it using the best-known algorithm for general unitary matrices.
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