Convergence guarantees for discrete mode approximations to non-Markovian quantum baths

The gist: This letter shows that under some physically motivated assumptions on the system-environment interaction, the finite-time dynamics of the nonMarkovian open quantum system computed with a sufficiently large number of modes is guaranteed to converge to the true result.

Modeling Non-Markovian Dynamics

Non-Markovian effects are important in modeling the behavior of open quantum systems arising in solidstate physics, quantum optics as well as in study of biological and chemical systems. The non-Markovian environment is often approximated by discrete bosonic modes, thus mapping it to a Lindbladian or Hamiltonian simulation problem. Simulating non-Markovian open quantum systems is difficult since it is usually not possible to formulate a dynamical equation for the system state from a given physical model of the system-environment interaction. An alternative approach is to identify and track a set of discrete modes that approximate the environment. For gaussian bosonic environments, two prominent approaches are used: first, using the Lorentzian pseudomode theory, which approximates the spectral density function by a finite sum of Lorentzians, and second, using the star-to-chain transformation, which uses the Lanczos iteration to identify a 1D chain of discrete bosonic modes with nearest neighbour couplings that approximate the environment.

Convergence Guarantees for Approximations

The paper provides general and rigorous convergence guarantees for discrete-mode approximations of nonMarkovian gaussian bosonic environments. For a wide and physically-motivated class of non-Markovian models, both the Lorentzian pseudomode approximation and the star-to-chain transformation is guaranteed to converge and the approximation error falls off polynomially with the number of pseudomodes. The results lend rigor to classical and quantum algorithms for approximating non-Markovian dynamics.

Conditions for Rigorous Analysis

To perform this convergence study, several theoretical challenges are resolved by identifying physically motivated sufficient mathematical conditions on the system-environment dynamics that allow for rigorously neglecting the high energy modes in the environment. The analysis combines an analysis of the Lorentzian pseudo-mode approximation and the star-to-chain transformation within a truncated environment energy window. These conditions are formalized through two main assumptions:

(Assumption 1)

The coupling function v ∈ C∞b(R) is such that there is a function V (ωc, t) which vanishes as ωc → ∞ ∀t ≥ 0 and l(Kv − Kvωc, t) ≤ V (ωc, t), where vωc(ω) = v(ω) if ω ≤ ωc and otherwise 0.

(Assumption 2)

∀t ≥ 0, s ∈ [0, t]N−1, the map GN (s;t): [0, t] → L(HS) is absolutely continuous and ∃γ(t) > 0 such that esssup of the N-point Green’s function is bounded by γ(t) kLk N.

Convergence Rates and Error Estimates

Theorem 1 establishes a bound on the error between the true reduced state ρ(t) and its approximation ρωc(t), showing that kρ(t) − ρωc(t)ktr ≤ ε(ωc, t), where ε(ωc, t) is the cutoff error. This error is given by ε(ωc, t) = f1(t)√ωc + Z t 0 f2(τ) p V (ωc, τ)dτ, with specific functions for the cutoff error. Theorem 2 shows that for the pseudomode approximation, there exists a description with M bosonic modes such that kρ(t) − ρˆ(t)ktr → 0 as M → ∞. Furthermore, if v'(ω) = O(poly(ω)) and the cutoff error ε(ωc, t) = O(exp(O(t))poly(ω−1c)), then the trace-norm error scales as O(exp(O(t))poly(M−1)>. Theorem 3 confirms that the star-to-chain transformation also converges, and the trace-norm error scales as O(exp(O(t))poly(M−1)>.

Extension to Initially Excited Environments

Theorem 1' extends the main result to an initially excited environment state by introducing Assumption 3, which bounds the initial environment state's deviation from a finite frequency window. Under these conditions, the cutoff error is given by ε(ωc, t) = √2∆(ωc) + f1(t)√ωc + Z t 0 f2(τ) p V (ωc, τ)dτ.

Key Results in Detailed Proofs

The proof for the star-to-chain transformation involves an algorithm that generates parameters such as ai = R ωc −ωc v(ω)p i(ω)dω qR ωc −ωc v(ω)2 p 2 i(ω)dΩ. The analysis of the error in approximating the magnitude square of the coupling function with a sum of Lorentzians is shown to be O(M−1/8). This demonstrates that choosing ωc and κ appropriately ensures that both the cutoff error and the Lorentzian approximation error decrease polynomially with M. The study also includes an analysis of the initial environment state, showing how it contributes to the total approximation error.

The paper concludes by providing a rigorous analysis of Markovian dilations to non-Markovian open quantum systems. Several questions of interest to open quantum system theory are left open in this work.

1. Improved simulation of non-Markovian dynamics via discrete mode approximations: The system can now compute the finite-time dynamics of the nonMarkovian open quantum system computed with a sufficiently large number of modes with guaranteed convergence, ensuring that the approximation error falls off polynomially with the number of modes.

2. Enhanced algorithmic complexity reduction for classical and quantum algorithms: These rigorous bounds lend rigor to classical and quantum algorithms for approximating non-Markovian dynamics, allowing practitioners to design computationally tractable simulations based on the identified discrete modes.

3. Accurate modeling of physically relevant environments using distributional kernels: The framework extends convergence guarantees to non-Markovian systems described by a distributional memory kernel, such as those commonly encountered in quantum optics, enabling more realistic simulation of complex physical phenomena where traditional methods fail.

4. Precise control over high-frequency environmental noise: By utilizing the cutoff error estimate ε(ωc, t) = f1(t)√ωc + Z t0 f2(τ) p V (ωc, τ)dτ, the system can quantify how much its dynamics change when only a finite frequency window is considered, providing a quantitative measure of this effect.

5. Robust simulation for initially excited environments: The results are extended to initially excited environment states via Theorem 1', allowing simulations involving mixed initial states (ensemble descriptions) to be rigorously compared against Markovian approximations.

Non-Markovian effects are important in modeling the behavior of open quantum systems arising in solid-state physics, quantum optics as well as in study of biological and chemical systems. The non-Markovian environment is often approximated by discrete bosonic modes, thus mapping it to a Lindbladian or Hamiltonian simulation problem. While systematic constructions of such modes have been previously proposed, the resulting approximation lacks rigorous and general convergence guarantees. In this letter, we show that under some physically motivated assumptions on the system-environment interaction, the finite-time dynamics of the non-Markovian open quantum system computed with a sufficiently large number of modes is guaranteed to converge to the true result. Furthermore, we show that this approximation error typically falls off polynomially with the number of modes. Our results lend rigor to classical and quantum algorithms for approximating non-Markovian dynamics.