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Simulation methods for open quantum many-body systems

Hendrik Weimer, Augustine Kshetrimayum, Román Orús

arXiv:1907.07079v2quant-phcond-mat.str-elphysics.comp-ph

TL;DR

Open quantum many-body systems require specialized simulation tools because equilibrium methods cannot generally capture their steady states and dynamics. This review surveys Markovian approaches spanning stochastic, tensor-network, variational, phase-space, and linked-cluster methods. No method is universally optimal, but recent advances have enabled progress across different regimes while leaving important problems inaccessible.

  • Problem

    Open-system steady states and relaxation dynamics cannot generally be simulated by directly adapting closed-system ground-state methods, because steady states may be nonthermal and distinct from Hamiltonian ground states.

  • Method

    The review surveys Markovian master-equation methods including stochastic trajectories, tensor networks, variational methods, quantum Monte Carlo, truncated Wigner, BBGKY equations, and linked-cluster expansions.

  • Results

    The review finds promising developments across different regimes, but no method that is universally optimal for all cases.

  • Takeaways & Limitations

    Tensor networks have successfully addressed hard open-system problems, while open two-dimensional systems remain largely unexplored.

  • Takeaways & Limitations

    Many problems remain inaccessible to current state-of-the-art numerical techniques.

Abstract

from arXiv · show

Coupling a quantum many-body system to an external environment dramatically changes its dynamics and offers novel possibilities not found in closed systems. Of special interest are the properties of the steady state of such open quantum many-body systems, as well as the relaxation dynamics towards the steady state. However, new computational tools are required to simulate open quantum many-body systems, as methods developed for closed systems cannot be readily applied. We review several approaches to simulate open many-body systems and point out the advances made in recent years towards the simulation of large system sizes.

I. INTRODUCTION

Open quantum many-body systems support engineered steady states and nonequilibrium dynamics, but their statistical states and environment-induced memory constraints make them harder to simulate than closed systems. The review frames steady-state properties and relaxation dynamics as complementary targets and surveys methods developed for these settings.

  • Controlled dissipation can engineer interesting quantum many-body states as stationary states, while open-system dynamics exhibits features absent from equilibrium systems.
  • Open systems are computationally harder than closed systems because their states are statistical, even though stationary states can be easier to prepare experimentally.
  • The review restricts its discussion to Markovian dynamics, whose generator depends only on the present state, under weak-coupling and timescale-separation assumptions.
  • Driven quantum-optical systems can realize nonequilibrium steady states, while their platforms include polaritonic, circuit-QED, and artificial-atom systems.
  • The review studies both steady states and the dynamical evolution toward them, including cases where relaxation displays complex glassy quantum dynamics despite a trivial steady state.
  • For many-body systems, the long-time and thermodynamic limits need not commute, so computing full time evolution can sometimes be more efficient than directly finding the steady state.

D. Differences to equilibrium problems

Methods for open systems cannot generally be obtained by directly adapting equilibrium ground-state techniques, because steady states need not be thermal or Hamiltonian ground states. The review therefore examines distinct models and simulation strategies, including stochastic trajectories and tensor-network approaches, while highlighting finite-size and approximation limits.

  • D. Differences to equilibrium problems: Adapting ground-state methods can fail because open-system steady states are often nonthermal and can differ from Hamiltonian ground states even for weak dissipation.
  • D. Differences to equilibrium problems: Open-system simulations are computationally harder than closed-system simulations because density operators encode statistical states.
  • E. Paradigmatic models: The dissipative Ising model uses longitudinal spin flips that break the Hamiltonian's Z2 symmetry, and mean-field theory predicts a bistable region.
  • E. Paradigmatic models: The driven-dissipative Bose-Hubbard model combines hopping, onsite interactions, coherent driving, and particle loss, producing mean-field multistability islands reminiscent of Mott lobes.
  • II. STOCHASTIC METHODS: Wave-function Monte Carlo replaces density-matrix propagation with independent pure-state trajectories, reducing the cost to O(Md) but requiring many repetitions for low statistical error.
  • II. STOCHASTIC METHODS: Quantum-jump trajectories evolve under an effective non-Hermitian Hamiltonian until a random threshold triggers a jump, enabling highly parallel computations.
  • II. STOCHASTIC METHODS: Finite-size stochastic simulations found no bistable phase in the one-dimensional dissipative Ising model and evidence for a first-order transition in a related two-dimensional model.

A. One spatial dimension

In one dimension, tensor-network methods extend MPS representations from pure states to mixed states and simulate open-system dynamics while preserving physical constraints such as positivity. The review covers purification, Choi-vectorization, locally purified evolution, and variational steady-state methods based on the Liouvillian.

  • MPS and MPDO representations: MPS provide an efficient one-dimensional tensor-network ansatz whose tensor contractions can be performed efficiently and exactly.The ansatz uses a bounded bond dimension D and physical dimension d.
  • MPS and MPDO representations: MPDOs extend MPS from pure to mixed states and automatically ensure positivity of the represented density matrix.They can also be constructed by purifying an MPS on an enlarged system with ancillas and tracing out the ancillas.
  • Purification methods: Purification evolves an MPS with ancillas and reconstructs the density operator and observables after tracing out the ancillas.The approach applies to mixed-state evolution under dissipation and to thermal equilibrium, with Trotter and truncation errors as main error sources.
  • Choi-vectorization methods: Choi isomorphism vectorizes a density matrix so that nearest-neighbor Liouvillian dynamics can be treated with standard TEBD on an MPO.The method reshapes a matrix index into a vector index and was applied to driven-dissipative Bose-Hubbard dynamics.
  • Locally purified methods: Locally purified tensor networks keep the density matrix as ρ = XX†, avoiding contraction of the two layers and preserving positivity during evolution.The approach also provides control of approximation error with respect to the trace norm.
  • Steady-state methods: Variational steady-state methods target the null eigenvector of the Liouvillian by finding the ground state of the positive semidefinite operator L†L.For infinite one-dimensional systems, hybrid imaginary- and real-time evolution improves convergence and then refines the stationary state.

B. Extensions to higher dimensions

Higher-dimensional tensor-network methods extend mixed-state simulations through PEPO/PEPS representations, corner-space truncation, and real-time Liouvillian evolution. These methods improve access to larger systems but face contraction, entanglement-growth, positivity, and nonlocality limitations.

  • PEPS-based methods: PEPS generalize MPS to higher dimensions, but their algorithms require substantial programming effort and exact contraction is mathematically hard.Approximate contraction schemes are therefore necessary, despite continuing advances in PEPS algorithms.
  • Corner Space Renormalization: Corner Space Renormalization merges small systems while retaining the χ most probable product states, increasing χ until observables converge.The method can determine the corner-space steady state by direct integration for small χ or stochastic wave-function Monte Carlo for large χ.
  • Corner Space Renormalization: The proposed Corner Space Renormalization method reached 16 × 16 sites for the two-dimensional driven-dissipative Bose-Hubbard model.It was also applied to the critical Heisenberg model up to 6 × 6 sites and the critical Bose-Hubbard regime up to 8 × 8 sites.
  • PEPO-based methods: PEPOs represent two-dimensional mixed states, and vectorization converts a PEPO with physical dimension d into a PEPS for |ρ⟩♯ with physical dimension d2.The vectorized master equation uses a local Liouvillian representation, enabling tensor-network real-time evolution in principle.
  • PEPO-based methods: Real-time evolution can cause rapid operator-entanglement growth, especially in two dimensions, limiting tensor-network simulations at finite bond dimension.Strong enough dissipation can drive the system to the steady state before entanglement becomes too large, and stronger dissipation lowers entanglement growth and speeds convergence.
  • Limitations and extensions: Higher-dimensional PEPO evolution does not automatically preserve positivity, while positivity-preserving PEPDO constructions may require very high bond dimension.An alternative based on the ground state of L♯†L♯ is hindered by nonlocal crossed products that make standard time-evolution algorithms difficult to implement without further approximations.

IV. VARIATIONAL METHODS

Variational methods provide a powerful framework for analyzing quantum many-body systems and can be applied successfully to open systems.

  • Variational techniques have been powerful tools for analyzing quantum many-body systems.The paper highlights successes including density functional theory and matrix product state approaches for ground-state problems.
  • Variational methods can be successfully applied to open quantum many-body systems.
  • The section motivates variational approaches as an extension of established methods for closed-system problems.

A. The variational principle for open quantum systems

For open systems, variational methods parameterize the density matrix and optimize a functional related to the quantum master equation. The approach can target steady states, time evolution, and effective critical theories, while practical evaluation may require approximations.

  • Variational methods parameterize the density matrix with variational parameters and optimize a suitable functional.The steady-state objective is based on the quantum master equation.
  • Because the exact steady state generally lies outside the variational manifold, optimization minimizes ||Lρ|| under a suitable norm.
  • The trace norm ||Lρ|| = Tr{| ˙ρ|} is identified as the natural norm for variational optimization.It is motivated by the role of trace distance for density matrices.
  • Evaluating the trace norm remains exponentially hard, motivating upper bounds that preserve the variational character and introduce small quantitative deviations near phase transitions.The upper bound depends on the variational manifold and its tangent space.
  • The variational principle has been applied to steady states of dissipative Ising, Bose-Hubbard, Heisenberg, Rydberg, and other open-system models.
  • The framework also extends to full time evolution through variational integration of the quantum master equation in small time steps τ.The implicit midpoint method is exact up to second order in τ while requiring one Liouvillian application.

B. Comparison with mean-field methods

Mean-field methods can produce multiple steady-state solutions and bistability, but beyond-mean-field and variational approaches often replace these predictions with first-order transitions. Several systematic extensions improve treatment of correlations and dynamics.

  • Mean-field methods: Open-system mean-field theory uses self-consistent effective single-site master equations obtained by tracing out the rest of the system.
  • Mean-field methods: Nonlinear mean-field equations can yield two or more independent steady-state solutions, producing apparent bistability.
  • Mean-field methods: The variational solution predicts a first-order transition where mean-field theory displays bistability for the dissipative Ising model.The variational and mean-field solutions become identical only in infinite dimensions.
  • Beyond mean field: Tensor-network simulations and field-theoretic calculations provide further confirmation that mean-field bistability can be replaced by a first-order transition.
  • Beyond mean field: Cluster mean-field theory enlarges the self-consistent cluster to treat short-range physics more accurately than bare mean-field theory.
  • Beyond mean field: Open-system DMFT maps a many-body lattice model onto a self-consistent single-impurity problem.The approach uses a dynamical Green’s function and self-energy to impose self-consistency.
  • Beyond mean field: Projection-operator methods treat the rest of the system as a non-Markovian environment and expand corrections beyond mean field.They have been applied to dissipative XY and Heisenberg models.

C. Variational tensor network methods

Combining variational methods with tensor networks is promising but is constrained by the difficulty of evaluating the natural trace norm. Representing density matrices as ensembles of pure states offers an efficient alternative.

  • The natural trace norm cannot be calculated efficiently in tensor-network representations, motivating alternative norms.Using a non-natural norm can introduce errors that are not controlled by the variational algorithm.
  • The choice of norm becomes especially consequential when computational limits prevent reaching an arbitrarily low variational norm.This difficulty is prominent in higher-dimensional systems and can also occur with long relaxation times in one dimension.
  • A density matrix can instead be represented as an ensemble of pure variational tensor-network states with an associated probability distribution.
  • The resulting variational norm can be computed efficiently with tensor-network methods and equals the natural trace norm over the full ensemble.Quantum-jump terms can be treated similarly.

D. Variational quantum Monte-Carlo methods

Variational and quantum Monte-Carlo methods extend sampling-based approaches to open quantum systems, where sign problems and density-matrix representations create distinctive challenges. Recent methods use full-configuration-interaction Monte Carlo and restricted Boltzmann machines, with norms chosen for time evolution or steady states.

  • Quantum Monte-Carlo: Quantum Monte-Carlo rewrites quantum many-body problems as sampling over classical probability distributions, but destructive interference can produce negative probabilities.Sampling absolute probabilities avoids negative weights at the cost of exponential computational complexity with system size.
  • Quantum Monte-Carlo: Open systems are especially prone to sign problems because Liouvillian eigenvalues can be complex.
  • Quantum Monte-Carlo: The first open-system quantum Monte-Carlo simulation used a non-variational full-configuration-interaction algorithm designed to mitigate, rather than eliminate, the sign problem.For magnetization in dissipative XYZ models on small lattices, it agreed closely with wave-function Monte-Carlo results.
  • Variational Monte-Carlo: Variational Monte-Carlo approaches use restricted Boltzmann-machine wave functions whose hidden layer introduces parameters for quantum many-body correlations.RBMs are closely connected to matrix product states and may describe long-range entangled states.
  • Variational Monte-Carlo: Different variational norms target time evolution or steady states, including Hilbert-Schmidt norms, purity-normalized norms, L†L minimization, and vectorized-density-matrix expectation values.One steady-state norm is not biased toward the maximally mixed state; RBM approaches behave similarly to tensor-network simulations with respect to the trace norm.

V. PHASE SPACE AND RELATED METHODS

Phase-space and hierarchy methods provide general alternatives for simulating open quantum many-body systems. Truncated Wigner methods map quantum dynamics to stochastic equations under a controlled truncation, while BBGKY hierarchies organize reduced-density-matrix correlations and permit dimensionality-independent approximations.

  • Truncated Wigner approximation: The truncated Wigner approximation has been applied to driven-dissipative microcavity polaritons in the optical parametric oscillator regime and to two-dimensional non-equilibrium phase transitions.
  • Truncated Wigner approximation: Truncated Wigner methods represent quantum fields by quasiprobability distributions whose Fokker-Planck dynamics are mapped to stochastic differential equations.For driven-dissipative polaritons, the Wigner representation is solved on a finite grid after truncating to second-order derivatives when (gX/κX,Ca^2) ≪ 1.
  • BBGKY hierarchy: The Bogoliubov-Born-Kirkwood-Yvon hierarchy describes interacting-particle systems through equations for different reduced density matrices and can be applied to open dissipative systems.
  • BBGKY hierarchy: BBGKY equations separate correlated parts of reduced density matrices and can be expanded in powers of 1/Z to approximate one- and two-particle behavior.Here Z is the coordination number of the Hamiltonian.
  • BBGKY hierarchy: The hierarchy approach applies to spins, bosons, and fermions and is independent of spatial dimensionality.It supports analytical expansions and efficient numerical simulations, including applications to lattice Bose-Hubbard models.

VI. LINKED CLUSTER EXPANSION METHODS

Linked-cluster expansions target steady-state observables by decomposing them into contributions from finite clusters. Truncation yields an approximation that can capture dissipative phase-transition properties in some models but may fail to converge in others.

  • Method and scope: Linked-cluster methods numerically target steady-state expectation values of observables in two-dimensional spin systems with incoherent spin relaxation.
  • Cluster construction: The method expands observables in local coupling strengths and reorganizes the series into cluster weights containing contributions associated with specific clusters.
  • Cluster construction: The thermodynamic-limit expectation value per site sums contributions from topologically distinct clusters and can be truncated at cluster size R.
  • Results and limitations: The expansion works well for the dissipative Heisenberg model, where an exact product-state solution enables calculation of phase boundaries and critical exponents.
  • Results and limitations: For the dissipative Ising model, the linked-cluster series failed to converge even at 10th order.

VII. SUMMARY AND OUTLOOK

The review surveys a broad set of simulation methods for Markovian open quantum master equations and finds substantial progress without a universally optimal technique. Complementary methods remain important because difficult regimes—including long-range interactions and dissipative criticality—remain challenging.

  • Scope and progress: The review covers mean-field, stochastic, tensor-network, variational, quantum Monte-Carlo, truncated-Wigner, BBGKY, and linked-cluster methods for Markovian master equations in the weak-coupling limit.
  • Scope and progress: No method is universally optimal, and some problems remain inaccessible to current numerical techniques.The review identifies long-ranged Rydberg interactions and dissipative phase transitions as concrete difficult cases.
  • Method comparison: Confidence in simulation results is greatest when they are reproducible with a complementary simulation approach.
  • Method comparison: Mean-field methods are less reliable for open systems than for closed systems, whereas tensor networks have successfully addressed many difficult open-system problems.Open two-dimensional systems are identified as a particularly promising but largely unexplored case.
  • Method comparison: Variational methods face a tradeoff between the formal suitability of their norm and the efficiency with which it can be computed.
  • Outlook: Recent progress has produced a broad toolkit for systematic comparison with experiments, while the experimentally accessible steady state supports continued study of strongly correlated open systems.
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