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:1501.06786 · Physical Review Letters

Variational Matrix Product Operators for the Steady State of Dissipative Quantum Systems

Jian Cui, J. I. Cirac, Mari Carmen Bañuls

Published Jun 4, 2015• 197 citations
arXivHTMLPDF

Abstract

We present a new variational method based on the matrix product operator (MPO) ansatz, for finding the steady state of dissipative quantum chains governed by master equations of the Lindblad form. Instead of requiring an accurate representation of the system evolution until the stationary state is attained, the algorithm directly targets the final state, thus, allowing for a faster convergence when the steady state is a MPO with small bond dimension. Our numerical simulations for several dissipative spin models over a wide range of parameters illustrate the performance of the method and show that, indeed, the stationary state is often well described by a MPO of very moderate dimensions.

DOI
10.1103/physrevlett.114.220601

Related papers

Matrix-product-operator approach to the nonequilibrium steady state of driven-dissipative quantum arrays

Eduardo Mascarenhas, Hugo Flayac, Vincenzo Savona

We develop a numerical procedure to efficiently model the nonequilibrium steady state of one-dimensional arrays of open quantum systems based on a matrix-product operator ansatz for the density matrix. The procedure searches for the null eigenvalue of the Liouvillian superoperator by sweeping along the system while carrying out a partial diagonalization of the single-site stationary problem. It bears full analogy to the density-matrix renormalization-group approach to the ground state of isolated systems, and its numerical complexity scales as a power law with the bond dimension. The method brings considerable advantage when compared to the integration of the time-dependent problem via Trotter decomposition, as it can address arbitrarily long-ranged couplings. Additionally, it ensures numerical stability in the case of weakly dissipative systems thanks to a slow tuning of the dissipation rates along the sweeps. We have tested the method on a driven-dissipative spin chain, under various assumptions for the Hamiltonian, drive, and dissipation parameters, and compared the results to those obtained both by Trotter dynamics and Monte Carlo wave function methods. Accurate and numerically stable convergence was always achieved when applying the method to systems with a gapped Liouvillian and a nondegenerate steady state.

arXiv:1504.06127•Physical Review A•Aug 19, 2015•129 citations

Tensor-network-based variational Monte Carlo approach to the non-equilibrium steady state of open quantum systems

Dawid A. Hryniuk, Marzena Hanna Szymańska

We introduce a novel method of efficiently simulating the non-equilibrium steady state of large many-body open quantum systems with highly non-local interactions, based on a variational Monte Carlo optimization of a matrix product operator ansatz. Our approach outperforms and offers several advantages over comparable algorithms, such as an improved scaling of the computational cost with respect to the bond dimension for periodic systems. We showcase the versatility of our approach by studying the phase diagrams and correlation functions of the dissipative quantum Ising model with collective dephasing and long-ranged power law interactions for spin chains of up to N=100 spins.

arXiv:2405.12044•Quantum•Sep 17, 2024•6 citations

Exact matrix product solutions in the Heisenberg picture of an open quantum spin chain

Clark, S. R., Javier Prior, Michael J. Hartmann +2 more

n recent work, Hartmann et al (2009 Phys. Rev. Lett. 102 057202) demonstrated that the classical simulation of the dynamics of open 1D quantum systems with matrix product algorithms can often be dramatically improved by performing time evolution in the Heisenberg picture. For a closed system this was exemplified by an exact matrix product operator (MPO) solution of the time-evolved creation operator of a quadratic fermi chain with a matrix dimension of just two. In this work, we show that this exact solution can be significantly generalized to include the case of an open quadratic fermi chain subjected to master equation evolution with Lindblad operators that are linear in the fermionic operators. Remarkably even in this open system the time evolution of operators continues to be described by MPOs with the same fixed dimension as that required by the solution of a coherent quadratic fermi chain for all times. Through the use of matrix product algorithms the dynamical behaviour of operators in this non-equilibrium open quantum system can be computed with a cost that is linear in the system size. We present some simple numerical examples that highlight how useful this might be for the more detailed study of open system dynamics. Given that Heisenberg picture simulations have been demonstrated to offer significant accuracy improvements for other open systems that are not exactly solvable, our work also provides further insight into how and why this advantage arises.

•Feb 26, 2010•33 citations

Exact matrix product solutions in the Heisenberg picture of an open quantum spin chain

S R Clark, J Prior, M J Hartmann +2 more

In recent work, Hartmann et al (2009 Phys. Rev. Lett. 102 057202) demonstrated that the classical simulation of the dynamics of open 1D quantum systems with matrix product algorithms can often be dramatically improved by performing time evolution in the Heisenberg picture. For a closed system this was exemplified by an exact matrix product operator (MPO) solution of the time-evolved creation operator of a quadratic fermi chain with a matrix dimension of just two. In this work, we show that this exact solution can be significantly generalized to include the case of an open quadratic fermi chain subjected to master equation evolution with Lindblad operators that are linear in the fermionic operators. Remarkably even in this open system the time evolution of operators continues to be described by MPOs with the same fixed dimension as that required by the solution of a coherent quadratic fermi chain for all times. Through the use of matrix product algorithms the dynamical behaviour of operators in this non-equilibrium open quantum system can be computed with a cost that is linear in the system size. We present some simple numerical examples that highlight how useful this might be for the more detailed study of open system dynamics. Given that Heisenberg picture simulations have been demonstrated to offer significant accuracy improvements for other open systems that are not exactly solvable, our work also provides further insight into how and why this advantage arises.

arXiv:0907.5582•New Journal of Physics•Feb 26, 2010•53 citations

Hybrid algorithm simulating non-equilibrium steady states of an open quantum system

Hongyi Zhou, Mao, Rui, Xiaoming Sun

Non-equilibrium steady states are a focal point of research in the study of open quantum systems. Previous variational algorithms for searching these steady states have suffered from resource-intensive implementations due to vectorization or purification of the system density matrix, requiring large qubit resources and long-range coupling. In this work, we present a novel variational quantum algorithm that efficiently searches for non-equilibrium steady states by simulating the operator-sum form of the Lindblad equation. By introducing the technique of random measurement, we are able to estimate the nonlinear cost function while reducing the required qubit resources by half compared to previous methods. Additionally, we prove the existence of the parameter shift rule in our variational algorithm, enabling efficient updates of circuit parameters using gradient-based classical algorithms. To demonstrate the performance of our algorithm, we conduct simulations for dissipative quantum transverse Ising and Heisenberg models, achieving highly accurate results. Our approach offers a promising solution for effectively addressing non-equilibrium steady state problems while overcoming computational limitations and implementation challenges.

arXiv:2309.06665•Sep 13, 2023•2 citations

Matrix product operator approach to nonequilibrium Floquet steady states

Zihan Cheng, Andrew C. Potter

We present a numerical method to simulate nonequilibrium Floquet steady states of one-dimensional periodically driven many-body systems coupled to a dissipative bath, based on a matrix product operator ansatz for the Floquet density matrix in frequency space. This method enables access to large systems beyond the reach of exact simulations, while retaining the periodic micromotion information. An excited-state extension of this technique allows computation of the dynamical approach to the steady state. We benchmark our method with a driven-dissipative Ising model and apply it to study the possibility of stabilizing prethermal discrete time-crystalline order by coupling to a cold bath.

arXiv:2206.07740•Physical review. B./Physical review. B•Dec 23, 2022•3 citations
We present a new variational method based on the matrix product operator (MPO) ansatz, for finding the steady state of dissipative quantum chains governed by master equations of the Lindblad form. Instead of requiring an accurate representation of the system evolution until the stationary state is attained, the algorithm directly targets the final state, thus, allowing for a faster convergence when the steady state is a MPO with small bond dimension. Our numerical simulations for several dissipative spin models over a wide range of parameters illustrate the performance of the method and show that, indeed, the stationary state is often well described by a MPO of very moderate dimensions.
We develop a numerical procedure to efficiently model the nonequilibrium steady state of one-dimensional arrays of open quantum systems based on a matrix-product operator ansatz for the density matrix. The procedure searches for the null eigenvalue of the Liouvillian superoperator by sweeping along the system while carrying out a partial diagonalization of the single-site stationary problem. It bears full analogy to the density-matrix renormalization-group approach to the ground state of isolated systems, and its numerical complexity scales as a power law with the bond dimension. The method brings considerable advantage when compared to the integration of the time-dependent problem via Trotter decomposition, as it can address arbitrarily long-ranged couplings. Additionally, it ensures numerical stability in the case of weakly dissipative systems thanks to a slow tuning of the dissipation rates along the sweeps. We have tested the method on a driven-dissipative spin chain, under various assumptions for the Hamiltonian, drive, and dissipation parameters, and compared the results to those obtained both by Trotter dynamics and Monte Carlo wave function methods. Accurate and numerically stable convergence was always achieved when applying the method to systems with a gapped Liouvillian and a nondegenerate steady state.
We introduce a novel method of efficiently simulating the non-equilibrium steady state of large many-body open quantum systems with highly non-local interactions, based on a variational Monte Carlo optimization of a matrix product operator ansatz. Our approach outperforms and offers several advantages over comparable algorithms, such as an improved scaling of the computational cost with respect to the bond dimension for periodic systems. We showcase the versatility of our approach by studying the phase diagrams and correlation functions of the dissipative quantum Ising model with collective dephasing and long-ranged power law interactions for spin chains of up to N=100 spins.
n recent work, Hartmann et al (2009 Phys. Rev. Lett. 102 057202) demonstrated that the classical simulation of the dynamics of open 1D quantum systems with matrix product algorithms can often be dramatically improved by performing time evolution in the Heisenberg picture. For a closed system this was exemplified by an exact matrix product operator (MPO) solution of the time-evolved creation operator of a quadratic fermi chain with a matrix dimension of just two. In this work, we show that this exact solution can be significantly generalized to include the case of an open quadratic fermi chain subjected to master equation evolution with Lindblad operators that are linear in the fermionic operators. Remarkably even in this open system the time evolution of operators continues to be described by MPOs with the same fixed dimension as that required by the solution of a coherent quadratic fermi chain for all times. Through the use of matrix product algorithms the dynamical behaviour of operators in this non-equilibrium open quantum system can be computed with a cost that is linear in the system size. We present some simple numerical examples that highlight how useful this might be for the more detailed study of open system dynamics. Given that Heisenberg picture simulations have been demonstrated to offer significant accuracy improvements for other open systems that are not exactly solvable, our work also provides further insight into how and why this advantage arises.
In recent work, Hartmann et al (2009 Phys. Rev. Lett. 102 057202) demonstrated that the classical simulation of the dynamics of open 1D quantum systems with matrix product algorithms can often be dramatically improved by performing time evolution in the Heisenberg picture. For a closed system this was exemplified by an exact matrix product operator (MPO) solution of the time-evolved creation operator of a quadratic fermi chain with a matrix dimension of just two. In this work, we show that this exact solution can be significantly generalized to include the case of an open quadratic fermi chain subjected to master equation evolution with Lindblad operators that are linear in the fermionic operators. Remarkably even in this open system the time evolution of operators continues to be described by MPOs with the same fixed dimension as that required by the solution of a coherent quadratic fermi chain for all times. Through the use of matrix product algorithms the dynamical behaviour of operators in this non-equilibrium open quantum system can be computed with a cost that is linear in the system size. We present some simple numerical examples that highlight how useful this might be for the more detailed study of open system dynamics. Given that Heisenberg picture simulations have been demonstrated to offer significant accuracy improvements for other open systems that are not exactly solvable, our work also provides further insight into how and why this advantage arises.
Non-equilibrium steady states are a focal point of research in the study of open quantum systems. Previous variational algorithms for searching these steady states have suffered from resource-intensive implementations due to vectorization or purification of the system density matrix, requiring large qubit resources and long-range coupling. In this work, we present a novel variational quantum algorithm that efficiently searches for non-equilibrium steady states by simulating the operator-sum form of the Lindblad equation. By introducing the technique of random measurement, we are able to estimate the nonlinear cost function while reducing the required qubit resources by half compared to previous methods. Additionally, we prove the existence of the parameter shift rule in our variational algorithm, enabling efficient updates of circuit parameters using gradient-based classical algorithms. To demonstrate the performance of our algorithm, we conduct simulations for dissipative quantum transverse Ising and Heisenberg models, achieving highly accurate results. Our approach offers a promising solution for effectively addressing non-equilibrium steady state problems while overcoming computational limitations and implementation challenges.
We present a numerical method to simulate nonequilibrium Floquet steady states of one-dimensional periodically driven many-body systems coupled to a dissipative bath, based on a matrix product operator ansatz for the Floquet density matrix in frequency space. This method enables access to large systems beyond the reach of exact simulations, while retaining the periodic micromotion information. An excited-state extension of this technique allows computation of the dynamical approach to the steady state. We benchmark our method with a driven-dissipative Ising model and apply it to study the possibility of stabilizing prethermal discrete time-crystalline order by coupling to a cold bath.