Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2601.17718 · Physical review. B./Physical review. B

Tree tensor network impurity solver based on Cayley-tree mapping

Bo Zhan, Chen Jia-lin, Zhen Fan, Tao Xiang

Published Apr 23, 2026• 0 citations
arXivHTMLPDF

Abstract

We introduce a tree tensor network (TTN) impurity solver that enables highly efficient and accurate real-time simulations of quantum impurity models. By decomposing a noninteracting bath Hamiltonian into a Cayley tree, the method provides a tensor network representation that naturally captures the multiscale entanglement structure intrinsic to impurity-bath systems. This geometry differs from conventional chain-based mappings and yields a substantial reduction of entanglement, allowing accurate ground-state properties and long-time dynamics to be captured at significantly lower bond dimensions. Benchmark calculations for the single-impurity Anderson model demonstrate that the TTN solver achieves markedly enhanced resolution of real-frequency spectral functions, without invoking analytic continuation. This impurity solver provides a balanced, scale-uniform description of impurity physics and offers a versatile approach for real-time dynamical mean-field theory and related applications involving quantum impurity models.

DOI
10.1103/ycty-d5f9

Related papers

Tree tensor-network real-time multiorbital impurity solver: Spin-orbit coupling and correlation functions in Sr2RuO4

Xiaodong Cao, Yi Lu, P. Hansmann +1 more

We present a tree tensor-network impurity solver suited for general multiorbital systems. The network is constructed to efficiently capture the entanglement structure and symmetry of an impurity problem. The solver works directly on the real-time/frequency axis and generates spectral functions with energy-independent resolution of the order of one percent of the correlated bandwidth. Combined with an optimized representation of the impurity bath, it efficiently solves self-consistent dynamical mean-field equations and calculates various dynamical correlation functions for systems with off-diagonal Green's functions. For the archetypal correlated Hund's metal ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$, we show that both the low-energy quasiparticle spectra related to the van Hove singularity and the high-energy atomic multiplet excitations can be faithfully resolved. In particular, we show that while the spin-orbit coupling has only minor effects on the orbital-diagonal one-particle spectral functions, it has a more profound impact on the low-energy spin and orbital response functions.

arXiv:2103.05545•Physical review. B./Physical review. B•Sep 9, 2021•22 citations

pyTTN: An open-source toolbox for open and closed system quantum dynamics simulations using tree tensor networks

Lachlan P. Lindoy, Daniel Rodrigo-Albert, Yannic Rath +1 more

We present the Python Tree Tensor Network (pyTTN) package for the evaluation of dynamical properties of closed and open quantum systems that makes use of Tree Tensor Network (TTN) based representations of wave functions. This package includes several features allowing for easy setup of zero- and finite-temperature calculations for general Hamiltonians using single and multi-set TTN ansätze with an adaptive bond dimension using subspace expansion techniques. In addition to these core features, pyTTN provides several tools for setting up efficient simulations of open quantum system dynamics, including the use of the TTN ansatz to represent the auxiliary density operator space for the simulation of the hierarchical equation of motion method and generalized quasi-Lindblad pseudomode methods. We present a set of applications for the package, starting with the widely used benchmark case of the photo-excitation dynamics of 24 mode pyrazine, after which we consider a more challenging model describing the exciton dynamics at the interface of an n-oligothiophene donor-C60 fullerene acceptor system. Finally, we consider applications to open quantum systems, including the spin-boson model, a set of extended dissipative spin models, and an Anderson impurity model. By combining ease of use and an efficient implementation, along with an extendable design that allows for the addition of future extensions, pyTTN can be integrated into a wide range of computational modeling software.

The Journal of Chemical Physics•Nov 24, 2025•6 citations

pyTTN: An Open Source Toolbox for Open and Closed System Quantum Dynamics Simulations Using Tree Tensor Networks

Lachlan P. Lindoy, Daniel Rodrigo-Albert, Yannic Rath +1 more

We present the Python Tree Tensor Network package (pyTTN) for the evaluation of dynamical properties of closed and open quantum systems that makes use of Tree Tensor Network (TTN), or equivalently the multi-layer multiconfiguration time-dependent Hartree (ML-MCTDH), based representations of wavefunctions. This package includes several features allowing for easy setup of zero- and finite-temperature calculations for general Hamiltonians using single and multi-set TTN ansätze with an adaptive bond dimension through the use of subspace expansion techniques. All core features are implemented in C++ with Python bindings provided to simplify the use of this package. In addition to these core features, pyTTN provides several tools for setting up efficient simulation of open quantum system dynamics, including the use of the TTN ansatz to represent the auxiliary density operator space for the simulation of the Hierarchical Equation of Motion (HEOM) method and generalised pseudomode methods; furthermore we demonstrate that the two approaches are equivalent up to a non-unitary normal mode transformation acting on the pseudomode degrees of freedom. We present a set of applications of the package, starting with the widely used benchmark case of the photo-excitation dynamics of 24 mode pyrazine, following which we consider a more challenging model describing the exciton dynamics at the interface of a $n$-oligothiophene donor-C$_{60}$ fullerene acceptor system. Finally, we consider applications to open quantum systems, including the spin-boson model, a set of extended dissipative spin models, and an Anderson impurity model. By combining ease of use, an efficient implementation, as well as an extendable design allowing for the addition of future extensions, pyTTN can be integrated in a wide range of computational modelling software.

arXiv:2503.15460•Mar 19, 2025•0 citations

From tree tensor network to multiscale entanglement renormalization ansatz

Xiangjian Qian, Mingpu Qin

Tensor network states (TNSs) offer an efficient representation for the ground state of quantum many-body systems and play an important role in the simulations of them. Numerous TNSs have been proposed in the past few decades. However, due to the high cost of TNS for two-dimensional systems, a balance between the encoded entanglement and computational complexity of TNS is yet to be reached. In this work we introduce a new tree tensor network-based (TTN-based) TNS dubbed a ``fully augmented tree tensor network (FATTN) by releasing the constraint in augmented tree tensor network (ATTN). When disentanglers are augmented in the physical layer of TTN, FATTN can provide more entanglement than TTN and ATTN. At the same time, FATTN maintains the scaling of computational cost with bond dimension in TTN and ATTN. Benchmark results on the ground-state energy for the transverse Ising model are provided to demonstrate the improvement of accuracy of FATTN over TTN and ATTN. Moreover, FATTN is quite flexible, which can be constructed as an interpolation between TTN and the multiscale entanglement renormalization ansatz to reach a balance between the encoded entanglement and the computational cost.

arXiv:2110.08794•Physical review. B./Physical review. B•May 4, 2022•11 citations

Anderson impurity solver integrating tensor network methods with quantum computing

François Jamet, Lachlan P. Lindoy, Yannic Rath +6 more

Solving the Anderson impurity model typically involves a two-step process, where one first calculates the ground state of the Hamiltonian and then computes its dynamical properties to obtain Green’s function. Here, we propose a hybrid classical/quantum algorithm where the first step is performed using a classical computer to obtain the tensor network ground state as well as its quantum circuit representation and the second step is executed on the quantum computer to obtain Green’s function. Our algorithm exploits the efficiency of tensor networks for preparing ground states on classical computers and takes advantage of quantum processors for the evaluation of the time evolution, which can become intractable on classical computers. We demonstrate the algorithm using 24 qubits on a quantum computing emulator for SrVO3 with a multi-orbital Anderson impurity model within the dynamical mean field theory. The tensor network based ground state quantum circuit preparation algorithm can also be performed for up to 60 qubits with our available computing resources, while the state vector emulation of the quantum algorithm for time evolution is beyond what is accessible with such resources. We show that, provided that the tensor network calculation is able to accurately obtain the ground state energy, this scheme does not require a perfect reproduction of the ground state wave function on the quantum circuit to give an accurate Green’s function. This hybrid approach may lead to quantum advantage in materials simulations where the ground state can be computed classically, but where the dynamical properties cannot.

arXiv:2304.06587•APL Quantum•Feb 24, 2025•8 citations

Real-time evolution of Anderson impurity models via tensor network influence functionals

Nathan Ng, Gunhee Park, Andrew J. Millis +2 more

In this work, we present and analyze two tensor network-based influence functional approaches for simulating the real-time dynamics of quantum impurity models such as the Anderson model. Via comparison with recent numerically exact simulations, we show that such methods accurately capture the long-time nonequilibrium quench dynamics. The two parameters that must be controlled in these tensor network influence functional approaches are a time discretization (Trotter) error and a bond dimension (tensor network truncation) error. We show that the actual numerical uncertainties are controlled by an intricate interplay of these two approximations, which we demonstrate in different regimes. Our work opens the door to using these tensor network influence functional methods as general impurity solvers.

arXiv:2211.10430•Physical review. B./Physical review. B•Mar 1, 2023•52 citations
We introduce a tree tensor network (TTN) impurity solver that enables highly efficient and accurate real-time simulations of quantum impurity models. By decomposing a noninteracting bath Hamiltonian into a Cayley tree, the method provides a tensor network representation that naturally captures the multiscale entanglement structure intrinsic to impurity-bath systems. This geometry differs from conventional chain-based mappings and yields a substantial reduction of entanglement, allowing accurate ground-state properties and long-time dynamics to be captured at significantly lower bond dimensions. Benchmark calculations for the single-impurity Anderson model demonstrate that the TTN solver achieves markedly enhanced resolution of real-frequency spectral functions, without invoking analytic continuation. This impurity solver provides a balanced, scale-uniform description of impurity physics and offers a versatile approach for real-time dynamical mean-field theory and related applications involving quantum impurity models.
We present a tree tensor-network impurity solver suited for general multiorbital systems. The network is constructed to efficiently capture the entanglement structure and symmetry of an impurity problem. The solver works directly on the real-time/frequency axis and generates spectral functions with energy-independent resolution of the order of one percent of the correlated bandwidth. Combined with an optimized representation of the impurity bath, it efficiently solves self-consistent dynamical mean-field equations and calculates various dynamical correlation functions for systems with off-diagonal Green's functions. For the archetypal correlated Hund's metal Sr2RuO4{\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}Sr2​RuO4​, we show that both the low-energy quasiparticle spectra related to the van Hove singularity and the high-energy atomic multiplet excitations can be faithfully resolved. In particular, we show that while the spin-orbit coupling has only minor effects on the orbital-diagonal one-particle spectral functions, it has a more profound impact on the low-energy spin and orbital response functions.
We present the Python Tree Tensor Network (pyTTN) package for the evaluation of dynamical properties of closed and open quantum systems that makes use of Tree Tensor Network (TTN) based representations of wave functions. This package includes several features allowing for easy setup of zero- and finite-temperature calculations for general Hamiltonians using single and multi-set TTN ansätze with an adaptive bond dimension using subspace expansion techniques. In addition to these core features, pyTTN provides several tools for setting up efficient simulations of open quantum system dynamics, including the use of the TTN ansatz to represent the auxiliary density operator space for the simulation of the hierarchical equation of motion method and generalized quasi-Lindblad pseudomode methods. We present a set of applications for the package, starting with the widely used benchmark case of the photo-excitation dynamics of 24 mode pyrazine, after which we consider a more challenging model describing the exciton dynamics at the interface of an n-oligothiophene donor-C60 fullerene acceptor system. Finally, we consider applications to open quantum systems, including the spin-boson model, a set of extended dissipative spin models, and an Anderson impurity model. By combining ease of use and an efficient implementation, along with an extendable design that allows for the addition of future extensions, pyTTN can be integrated into a wide range of computational modeling software.
We present the Python Tree Tensor Network package (pyTTN) for the evaluation of dynamical properties of closed and open quantum systems that makes use of Tree Tensor Network (TTN), or equivalently the multi-layer multiconfiguration time-dependent Hartree (ML-MCTDH), based representations of wavefunctions. This package includes several features allowing for easy setup of zero- and finite-temperature calculations for general Hamiltonians using single and multi-set TTN ansätze with an adaptive bond dimension through the use of subspace expansion techniques. All core features are implemented in C++ with Python bindings provided to simplify the use of this package. In addition to these core features, pyTTN provides several tools for setting up efficient simulation of open quantum system dynamics, including the use of the TTN ansatz to represent the auxiliary density operator space for the simulation of the Hierarchical Equation of Motion (HEOM) method and generalised pseudomode methods; furthermore we demonstrate that the two approaches are equivalent up to a non-unitary normal mode transformation acting on the pseudomode degrees of freedom. We present a set of applications of the package, starting with the widely used benchmark case of the photo-excitation dynamics of 24 mode pyrazine, following which we consider a more challenging model describing the exciton dynamics at the interface of a nnn-oligothiophene donor-C60_{60}60​ fullerene acceptor system. Finally, we consider applications to open quantum systems, including the spin-boson model, a set of extended dissipative spin models, and an Anderson impurity model. By combining ease of use, an efficient implementation, as well as an extendable design allowing for the addition of future extensions, pyTTN can be integrated in a wide range of computational modelling software.
Tensor network states (TNSs) offer an efficient representation for the ground state of quantum many-body systems and play an important role in the simulations of them. Numerous TNSs have been proposed in the past few decades. However, due to the high cost of TNS for two-dimensional systems, a balance between the encoded entanglement and computational complexity of TNS is yet to be reached. In this work we introduce a new tree tensor network-based (TTN-based) TNS dubbed a ``fully augmented tree tensor network (FATTN) by releasing the constraint in augmented tree tensor network (ATTN). When disentanglers are augmented in the physical layer of TTN, FATTN can provide more entanglement than TTN and ATTN. At the same time, FATTN maintains the scaling of computational cost with bond dimension in TTN and ATTN. Benchmark results on the ground-state energy for the transverse Ising model are provided to demonstrate the improvement of accuracy of FATTN over TTN and ATTN. Moreover, FATTN is quite flexible, which can be constructed as an interpolation between TTN and the multiscale entanglement renormalization ansatz to reach a balance between the encoded entanglement and the computational cost.
Solving the Anderson impurity model typically involves a two-step process, where one first calculates the ground state of the Hamiltonian and then computes its dynamical properties to obtain Green’s function. Here, we propose a hybrid classical/quantum algorithm where the first step is performed using a classical computer to obtain the tensor network ground state as well as its quantum circuit representation and the second step is executed on the quantum computer to obtain Green’s function. Our algorithm exploits the efficiency of tensor networks for preparing ground states on classical computers and takes advantage of quantum processors for the evaluation of the time evolution, which can become intractable on classical computers. We demonstrate the algorithm using 24 qubits on a quantum computing emulator for SrVO3 with a multi-orbital Anderson impurity model within the dynamical mean field theory. The tensor network based ground state quantum circuit preparation algorithm can also be performed for up to 60 qubits with our available computing resources, while the state vector emulation of the quantum algorithm for time evolution is beyond what is accessible with such resources. We show that, provided that the tensor network calculation is able to accurately obtain the ground state energy, this scheme does not require a perfect reproduction of the ground state wave function on the quantum circuit to give an accurate Green’s function. This hybrid approach may lead to quantum advantage in materials simulations where the ground state can be computed classically, but where the dynamical properties cannot.
In this work, we present and analyze two tensor network-based influence functional approaches for simulating the real-time dynamics of quantum impurity models such as the Anderson model. Via comparison with recent numerically exact simulations, we show that such methods accurately capture the long-time nonequilibrium quench dynamics. The two parameters that must be controlled in these tensor network influence functional approaches are a time discretization (Trotter) error and a bond dimension (tensor network truncation) error. We show that the actual numerical uncertainties are controlled by an intricate interplay of these two approximations, which we demonstrate in different regimes. Our work opens the door to using these tensor network influence functional methods as general impurity solvers.