Efficient Solution Of The Non-unitary Time-dependent Schrodinger Equation On A Quantum Computer With Complex Absorbing Potential

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Mariane Mangin-Brinet¹, Jing Zhang², Denis Lacroix², and Edgar Andres Ruiz Guzman²

¹Laboratoire de Physique Subatomique et de Cosmologie, CNRS/IN2P3, 38026 Grenoble, France
²Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

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Abstract

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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Featured image: Schematic illustration of one time-step evolution including the implementation of the Complex Absorbing Potential using the reservoir qubit (schematic example with 3 qubits only).

► BibTeX data

@article{ManginBrinet2024efficientsolutionof, doi = {10.22331/q-2024-04-08-1311}, url = {https://doi.org/10.22331/q-2024-04-08-1311}, title = {Efficient solution of the non-unitary time-dependent {S}chrodinger equation on a quantum computer with complex absorbing potential}, author = {Mangin-Brinet, Mariane and Zhang, Jing and Lacroix, Denis and Ruiz Guzman, Edgar Andres}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1311}, month = apr, year = {2024} }

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Source: https://quantum-journal.org/papers/q-2024-04-08-1311/

Time Stamp: April 8, 2024

Time Stamp: Mar 16, 2023

Quantum simulation in the semi-classical regime

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Quantum Journal

Source Node: 1577385

Time Stamp: Jun 17, 2022

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