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arXiv:1909.02532 · Journal of Physics Communications

Efficient unitary method for simulation of driven quantum dot systems

Spenser Talkington, H. W. Jiang

Published May 1, 2020• 1 citation
arXivHTMLPDF

Abstract

Abstract Density matrices evolved according the von Neumann equation are commonly used to simulate the dynamics of driven quantum systems. However, computational methods using density matrices are often too slow to explore the large parameter spaces of solid state quantum systems. Here we develop a unitary computation method to quickly perform simulations for closed quantum systems, where dissipation to the environment can be ignored. We use three techniques to optimize simulations, apply them to six time-dependent pulses for a semiconductor quantum dot qubit system, and predict the dynamic evolutions. We compare computational times between our unitary method and the density matrix method for a variety of image sizes. As an example, we implement our unitary method for a realistic four-state system (Z. Shi et al, Nat. Commun. 5, 3020 (2014)), and find that it is over two orders of magnitude faster than the corresponding density matrix method implemented in the popular quantum simulation software QuTiP.

DOI
10.1088/2399-6528/ab8ff8

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Abstract Density matrices evolved according the von Neumann equation are commonly used to simulate the dynamics of driven quantum systems. However, computational methods using density matrices are often too slow to explore the large parameter spaces of solid state quantum systems. Here we develop a unitary computation method to quickly perform simulations for closed quantum systems, where dissipation to the environment can be ignored. We use three techniques to optimize simulations, apply them to six time-dependent pulses for a semiconductor quantum dot qubit system, and predict the dynamic evolutions. We compare computational times between our unitary method and the density matrix method for a variety of image sizes. As an example, we implement our unitary method for a realistic four-state system (Z. Shi et al, Nat. Commun. 5, 3020 (2014)), and find that it is over two orders of magnitude faster than the corresponding density matrix method implemented in the popular quantum simulation software QuTiP.
The rapid development of quantum computers has enabled demonstrations of quantum advantages on various tasks. However, real quantum systems are always dissipative due to their inevitable interaction with the environment, and the resulting non-unitary dynamics make quantum simulation challenging with only unitary quantum gates. In this work, we present an innovative and scalable method to simulate open quantum systems using quantum computers. We define an adjoint density matrix as a counterpart of the true density matrix, which reduces to a mixed-unitary quantum channel and thus can be effectively sampled using quantum computers. This method has several benefits, including no need for auxiliary qubits and noteworthy scalability. Moreover, some long-time properties like steady states and the thermal equilibrium can also be investigated as the adjoint density matrix and the true dissipated one converge to the same state. Finally, we present deployments of this theory in the dissipative quantum XY model for the evolution of correlation and entropy with short-time dynamics and the disordered Heisenberg model for many-body localization with long-time dynamics. This work promotes the study of real-world many-body dynamics with quantum computers, highlighting the potential to demonstrate practical quantum advantages.
Electronic structure and transport in realistically-sized systems often require an open quantum system (OQS) treatment, where the system is defined in the context of an environment. As OQS evolution is non-unitary, implementation on quantum computers -- limited to unitary operations -- is challenging. We present a general framework for OQSs where the system's d×dd \times dd×d density matrix is recast as a d2d^{2}d2 wavefunction which can be evolved by unitary transformations. This theory has two significant advantages over conventional approaches: (i) the wavefunction requires only an nnn-qubit, compared to 2n2n2n-qubit, bath for an nnn-qubit system and (ii) the purification includes dynamics of any pure-state universe. We demonstrate this method on a two-level system in a zero temperature amplitude damping channel and a two-site quantum Ising model. Quantum simulation and experimental-device results agree with classical calculations, showing promise in simulating non-unitary operations on NISQ quantum devices.
Basing on our hierarchical equations of motion for time-dependent quantum transport [X. Zheng, G. H. Chen, Y. Mo, S. K. Koo, H. Tian, C. Y. Yam, and Y. J. Yan, J. Chem. Phys. 133, 114101 (2010)], we develop an efficient and accurate numerical algorithm to solve the Liouville-von-Neumann equation. We solve the real-time evolution of the reduced single-electron density matrix at the tight-binding level. Calculations are carried out to simulate the transient current through a linear chain of atoms, with each represented by a single orbital. The self-energy matrix is expanded in terms of multiple Lorentzian functions, and the Fermi distribution function is evaluated via the Padè spectrum decomposition. This Lorentzian-Padè decomposition scheme is employed to simulate the transient current. With sufficient Lorentzian functions used to fit the self-energy matrices, we show that the lead spectral function and the dynamics response can be treated accurately. Compared to the conventional master equation approaches, our method is much more efficient as the computational time scales cubically with the system size and linearly with the simulation time. As a result, the simulations of the transient currents through systems containing up to one hundred of atoms have been carried out. As density functional theory is also an effective one-particle theory, the Lorentzian-Padè decomposition scheme developed here can be generalized for first-principles simulation of realistic systems.
This report is concerned with the efficiency of numerical methods for simulating quantum spin systems, with the aim to implement an improved method for simulation of a time-dependent Hamiltonian that displays chirped pulses at a high frequency. Working in the density matrix formulation of quantum systems, we study evolution under the Liouville-von Neumann equation, presenting analysis of and benchmarking current numerical methods. The accuracy of existing techniques is assessed in the presence of chirped pulses. We also discuss the Magnus expansion and detail how a truncation of it is used to solve differential equations. The results of this work are implemented in the Python package MagPy to provide a better error-to-cost ratio than current approaches allow for time-dependent Hamiltonians.
Abstract We introduce variational hybrid classical-quantum algorithms to simulate the Lindblad master equation and its adjoint for time-evolving Markovian open quantum systems and quantum observables. Our methods are based on a direct representation of density matrices and quantum observables as quantum superstates. We design and optimize low-depth variational quantum circuits that efficiently capture the unitary and non-unitary dynamics of the solutions. We benchmark and test the algorithms on different models and system sizes, showing their potential for utility with near-future hardware.
We present an efficient, nearly optimal quantum algorithm for solving linear matrix differential equations, with applications to the simulation of open quantum systems and beyond. For unitary or dissipative dynamics, the algorithm computes an entry of the solution matrix with query complexity O~(νLt/ε)\widetilde{\mathcal{O}}(ν\mathcal{L} t/ε)O(νLt/ε), where the constant ννν depends on the problem parameters, L\mathcal{L}L involves a time integral of upper bounds on the norms of evolution operators, and εεε is the error. In particular, νLν\mathcal{L}νL is linear in ttt for unitary dynamics and can be a constant for dissipative dynamics. Our result contrasts prior quantum approaches for differential equations that typically require exponential time for this problem due to the encoding in a quantum state, which can lead to exponentially small amplitudes. We demonstrate the utility of the algorithm through an end-to-end application, namely the simulation of dissipative dynamics for non-interacting fermions, which can be extended to other quantum and classical systems. We compare with classical algorithms and give evidence of polynomial quantum speedups for systems in a lattice, which become more pronounced for systems with long-range interactions and can be shown to be exponential in general. We also provide a lower bound of Ω(νLt/ε)Ω(ν\mathcal{L} t/ε)Ω(νLt/ε) for unitary or dissipative dynamics that proves our algorithm is optimal up to logarithmic factors.