Digital quantum simulation of non-perturbative dynamics of open systems with orthogonal polynomials

José D. Guimarães1,2,3, Mikhail I. Vasilevskiy3,4,5, and Luís S. Barbosa3,6

1Centro de Física das Universidades do Minho e do Porto, Braga 4710-057, Portugal
2Institute of Theoretical Physics and IQST, Ulm University, Albert-Einstein-Allee 11, Ulm 89081, Germany
3International Iberian Nanotechnology Laboratory, Av. Mestre José Veiga s/n, Braga 4715-330, Portugal
4Laboratório de Física para Materiais e Tecnologias Emergentes (LaPMET), Universidade do Minho, Braga 4710-057, Portugal
5Departamento de Física, Universidade do Minho, Braga 4710-057, Portugal
6INESC TEC, Departamento de Informática, Universidade do Minho, Braga 4710-057, Portugal

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Classical non-perturbative simulations of open quantum systems' dynamics face several scalability problems, namely, exponential scaling of the computational effort as a function of either the time length of the simulation or the size of the open system. In this work, we propose the use of the Time Evolving Density operator with Orthogonal Polynomials Algorithm (TEDOPA) on a quantum computer, which we term as Quantum TEDOPA (Q-TEDOPA), to simulate non-perturbative dynamics of open quantum systems linearly coupled to a bosonic environment (continuous phonon bath). By performing a change of basis of the Hamiltonian, the TEDOPA yields a chain of harmonic oscillators with only local nearest-neighbour interactions, making this algorithm suitable for implementation on quantum devices with limited qubit connectivity such as superconducting quantum processors. We analyse in detail the implementation of the TEDOPA on a quantum device and show that exponential scalings of computational resources can potentially be avoided for time-evolution simulations of the systems considered in this work. We applied the proposed method to the simulation of the exciton transport between two light-harvesting molecules in the regime of moderate coupling strength to a non-Markovian harmonic oscillator environment on an IBMQ device. Applications of the Q-TEDOPA span problems which can not be solved by perturbation techniques belonging to different areas, such as the dynamics of quantum biological systems and strongly correlated condensed matter systems.

The paper introduces Quantum Time Evolving Density operator with Orthogonal Polynomials algorithm (Q-TEDOPA), an adaptation of the classical TEDOPA method for quantum computation, where non-perturbative dynamics of open quantum systems linearly coupled with bosonic environments are simulated. Designed for quantum computers with restricted qubit connectivity, such as superconducting quantum processors, Q-TEDOPA only requires local nearest-neighbor interactions. We analyze the complexity of the method and suggest that Q-TEDOPA may achieve exponential speedups relatively to its classical counterpart (TEDOPA). We demonstrate its utility by simulating the exciton transport between light-harvesting molecules on a real IBMQ device using up to 12 qubits. Q-TEDOPA shows promise in enhancing quantum simulation capabilities, providing a more resource-efficient approach compared to classical TEDOPA.

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[1] José D. Guimarães, James Lim, Mikhail I. Vasilevskiy, Susana F. Huelga, and Martin B. Plenio, "Noise-Assisted Digital Quantum Simulation of Open Systems Using Partial Probabilistic Error Cancellation", PRX Quantum 4 4, 040329 (2023).

[2] Anthony W. Schlimgen, Kade Head-Marsden, LeeAnn M. Sager-Smith, Prineha Narang, and David A. Mazziotti, "Quantum state preparation and nonunitary evolution with diagonal operators", Physical Review A 106 2, 022414 (2022).

[3] Jonathon P. Misiewicz and Francesco A. Evangelista, "Implementation of the Projective Quantum Eigensolver on a Quantum Computer", Journal of Physical Chemistry A 128 11, 2220 (2024).

[4] Takanori Nishi and Kaoru Yamanouchi, "Simulation of a spin-boson model by iterative optimization of a parametrized quantum circuit", AVS Quantum Science 6 2, 023801 (2024).

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