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Efficient non-Markovian quantum dynamics using time-evolving matrix product operators
Aidan Strathearn, Peter Kirton, Dainius Kilda, Jonathan Keeling, Brendon W. Lovett
TL;DR
Realistic quantum devices require efficient simulation beyond weak-coupling or Markovian assumptions. TEMPO represents environmental history as a compressed tensor network and identifies the spin-boson localisation transition while treating spins in environments with separated timescales.
Problem
Non-Markovian quantum dynamics with memory effects remains difficult to model efficiently for strongly coupled quantum systems in structured environments.
Method
TEMPO encodes finite environmental history in an augmented density tensor represented as a matrix product state and propagated with matrix product operators.
Results
TEMPO accurately identifies spin-boson localisation transitions, including estimates consistent with other techniques, and models spin dynamics with widely separated environmental timescales.
Takeaways & Limitations
The method supports efficient, numerically exact modelling of a wide variety of non-Markovian open-system dynamics.
Takeaways & Limitations
The formulation assumes an initially factorised system–environment state with the environment in thermal equilibrium.
Abstract
from arXiv · showhide
In order to model realistic quantum devices it is necessary to simulate quantum systems strongly coupled to their environment. To date, most understanding of open quantum systems is restricted either to weak system-bath couplings, or to special cases where specific numerical techniques become effective. Here we present a novel general numerical approach to efficiently describe the time evolution of a quantum system coupled to a non-Markovian environment. We demonstrate the power and flexibility of our method by numerically identifying the localisation transition of the Ohmic spin-boson model, and considering a model with widely separated environmental timescales arising for a pair of spins embedded in a common environment.
INTRODUCTION
Structured environments and memory effects make Markovian descriptions inadequate for many quantum systems and technologies. The article introduces TEMPO, a computationally efficient, general, numerically exact approach for non-Markovian dynamics coupled to a harmonic bath.
- Motivation: Memory effects are important in quantum systems including micromechanical resonators, quantum dots, and superconducting qubits, and support emerging quantum technologies.Examples include single-photon sources needed for quantum communication.
- Related work: Existing non-Markovian methods are often exact only for particular problems or limited to certain parameter regimes by perturbative assumptions.Related approaches include unitary transformations, beyond-Born-Markov expansions, and real-time diagrammatic Monte Carlo.
- Contribution: TEMPO is a computationally efficient, general, and numerically exact approach to modelling non-Markovian dynamics for an open quantum system coupled to a harmonic bath.The method is based on the augmented density tensor and represents system history over a finite bath memory time τc.
- Contribution: TEMPO exploits the augmented density tensor to represent system history over a finite bath memory time τc.When the bath is well behaved, singular value decomposition compresses the augmented density tensor on the fly.
RESULTS · Time-Evolving Matrix Product Operators
TEMPO represents non-Markovian system histories as finite-memory augmented density tensors and compresses them as matrix product states for efficient propagation. This enables simulations near localization transitions, including spin-1/2 and spin-1 spin-boson models.
- Time-Evolving Matrix Product Operators: TEMPO extends the system state from a vector to an augmented density tensor that records non-Markovian history through influence functions.The influence functions connect amplitudes separated by an integer number of timesteps.
- Time-Evolving Matrix Product Operators: The finite-memory approximation retains only the previous K = τc/∆ steps because older influence functions no longer affect the dynamics.The resulting augmented density tensor has rank K, with each index spanning d2 values.
- Time-Evolving Matrix Product Operators: TEMPO efficiently represents and propagates the augmented density tensor by decomposing it into a matrix product state and truncating small singular values.The precision is controlled by the singular-value cutoff λc.
- Time-Evolving Matrix Product Operators: The tensor-network propagator has internal legs of dimension d2, allowing direct matrix-product propagation followed by standard singular-value decompositions.At each step, the augmented density tensor is contracted with the next propagator row and truncated.
- Time-Evolving Matrix Product Operators: K = 200 enables spin-1/2 simulations near the localization transition, a memory length an order of magnitude larger than standard augmented-density-tensor implementations.Convergence is obtained by varying the timestep ∆ and SVD cutoff λc.
- Time-Evolving Matrix Product Operators: For α > 0.5, the spin-1/2 polarization decays exponentially to zero, whereas localization requires a nonzero steady-state polarization at sufficiently large α.The crossover near α ≃ 0.5 separates coherent decaying oscillations from incoherent decay.
- Time-Evolving Matrix Product Operators: Extrapolating the decay rate γ to 1/K → 0 identifies the smallest α consistent with γ → 0, requiring simulations up to K = 200.Tensor compression makes these large-memory simulations feasible.
- Time-Evolving Matrix Product Operators: The spin-1 spin-boson model shows complex oscillations followed by exponential decay, with the extracted decay rate vanishing at αc ≃ 0.28.Increasing the local leg dimension from d2 = 4 to d2 = 9 reduces reachable K, while convergence occurs for larger timesteps.
Two Spins in a Common Environment
TEMPO models two coupled spins in a common environment with distinct local-dissipative and environment-mediated timescales, including large separations that challenge standard methods. The dynamics differ by bath dimension: one-dimensional baths produce revivals at t = R, while three-dimensional baths show weaker effects.
- Model: TEMPO studies two identical spin-1/2 particles directly coupled by an isotropic Heisenberg interaction and jointly coupled to a common environment.The spins are located at ra and rb, with position-dependent system-bath coupling phases.
- Timescales: The model contains fast local dissipative dynamics and slower environment-mediated spin-spin interactions, whose timescale ratio is controlled by separation R.Bath dimension also affects the strength of environmental excitations propagating between the spins.
- Method: For large separations R > ωc^-1, standard approaches require a small timestep and a cutoff time exceeding R, whereas TEMPO avoids finite-memory-cutoff error.The ADT requires τc = K∆ > R and therefore a very large K to capture both timescales.
- Method: The dynamics are reduced to the Sz,a+Sz,b = 0 sector, whose effective Hamiltonian maps onto a spin-1/2 SBM with an R-dependent spectral density.This sector contains the two anti-aligned spin states and is the only one with non-trivial dynamics.
- Results: In D = 1, initial oscillations decay on a timescale ∼ωc^-1, followed by revivals at t = R caused by a large bath-correlation peak.For R ≫ωc^-1, the profile of the secondary oscillations is independent of R.
- Results: In D = 3, weaker oscillatory spectral-density components produce smaller correlation peaks and only modest dynamical effects near t ≈R.Small-amplitude oscillations occur for R = 8, while the passage reports differing behavior for R = 16.
DISCUSSION
The paper presents an efficient MPS-based method for modelling general non-Markovian open-system dynamics while reducing the memory demands of high-rank tensor representations. It also demonstrates the method’s ability to locate a phase transition requiring memory times up to K = 200 and suggests extensions to larger systems and longer propagation.
- Method: The method represents the high-rank tensor encoding system history as an MPS, overcoming restrictive memory requirements in established ADT methods.Open-system dynamics are calculated by propagating this MPS via iterative application.
- Phase-transition test: K = 200 memory times were required to precisely locate the Ohmic spin-boson phase transition, providing a rigorous test of numerical methods.Other improved methods have shown enhanced efficiency away from critical coupling, but had not yet been used to precisely locate the transition.
- Tensor-network basis: Tensor networks efficiently represent high-dimensional tensors encoding restricted correlations, linking this approach to methods in quantum systems, statistical physics, and computer science.Related approaches include tensor trains developed in computer science.
- Future extensions: The methods can be extended to larger quantum systems using tensor-network techniques such as the optimal boson basis, which the authors identify as future work.They may also be combined with the tensor transfer method for efficient long-time propagation, subject to an unspecified condition in the supplied passage.
METHODS · TEMPO Algorithm
The TEMPO algorithm represents non-Markovian open-system evolution as tensor-network contractions, using MPS compression and MPO propagation to control computational cost. A finite memory cutoff maintains a fixed history length during propagation, while operator-spectrum degeneracies reduce tensor dimensions.
- TEMPO Algorithm: TEMPO vectorizes the density operator and propagates it with a Liouvillian split into coherent system and environment contributions.Additional Markovian dynamics can be included in the reduced-system Liouvillian.
- TEMPO Algorithm: The discretized path integral forms an augmented density tensor whose components encode non-Markovian correlations across time steps.The coefficients η_k quantify correlations across k timesteps, and the tensor can be expressed as a network of N(N + 1)/2 tensors with at most four legs each.
- TEMPO Algorithm: Iterative contraction grows the augmented density tensor by one history index per timestep, after which bidirectional SVD sweeps truncate singular values below λ_c.This produces an MPS representation while retaining the most efficient compressed form.
- TEMPO Algorithm: The high-rank propagation tensor is represented as an MPO, making iterative augmented-density contractions compatible with standard MPS compression algorithms.The MPO decomposition uses rank-4 tensors in its interior and lower-rank tensors at its ends.
- TEMPO Algorithm: For evenly spaced non-degenerate spin operators, spectral-difference degeneracies reduce the internal-index dimension from d2 to d2 − d + 1 and the b_k tensor size from O(d8) to O(d6).There are only 2d−1 unique values of O−.
- TEMPO Algorithm: A finite memory cutoff τ_c = K∆ discards history older than K timesteps, so propagation retains an MPS of rank K after the initial growth phase.For time-independent problems, the propagation phase repeatedly contracts with the same timestep-independent MPO.
- TEMPO Algorithm: At K = 200, the most demanding Spin-Boson regime occurs near α = 0.5, where obtaining 500 data points required ≈20.5 hours on the HPC Cirrus cluster.This coupling marks the crossover from under- to overdamped oscillations of ⟨S_z⟩, and CPU time is linear in total memory requirement.
single spin model
The pair-of-spins Hamiltonian can be mapped onto a single-spin spin-boson model by exploiting conservation of total z-spin and separating the resulting subspaces. The effective environmental coupling depends on spin separation and bath dimensionality, vanishing at coincident positions and diminishing at large separation for D > 1.
- Mapping to a single spin model: Conservation of total z-spin separates the two-spin problem into one anti-aligned subspace and two one-dimensional aligned-spin subspaces.The conserved quantity is [S_z,a + S_z,b, H] = 0.
- Mapping to a single spin model: The pair Hamiltonian maps onto a single spin-1/2 spin-boson model within the relevant subspace.The mapping connects Eq. (6) for two spins in a common environment to Eq. (4), the single-spin spin-boson model.
- Effective environmental coupling: The effective mode coupling is |g̃_i| = |g_i,a − g_i,b| = 2g_i sin[k_i·(r_a−r_b)/2].The coupling arises from the difference between the two spins’ environmental couplings.
- Effective spectral density: Angular averaging modifies the effective spectral density through F_D(ωR), whose form depends on environmental dimensionality: cos(x), J_0(x), and sinc(x) for D = 1, 2, and 3.J_0(x) is a Bessel function, and J_p(ω) denotes the bath’s actual density of states.
- Separation limits: At R → 0, F_D(ωR) → 1 and J(ω) → 0 for all D, while F_D(ωR) → 0 as R → ∞ for D > 1.The small-separation limit reflects loss of relative phase shift; the large-separation limit reflects diminishing environment-induced coupling in higher dimensions.
- Bare bath model: For a quantum dot in a phonon environment, g_i ∼ √ω_i, with α setting interaction strength and ω_c defining the high-frequency cutoff.These assumptions determine the continuum spectral density for a D-dimensional environment.