Variational Perturbation Theory in Open Quantum Systems for Efficient Steady State Computation

The gist: This work introduces variational perturbation theory (VPT) for open quantum systems, developing two numerical strategies that eliminate the need to compute pseudo-inverses, thereby significantly extending the radius of convergence and improving efficiency compared to standard perturbation theory.

Introduction and Motivation

Determining the steady state of an open quantum system is crucial for characterizing quantum devices and studying various physical phenomena Computing a single steady state is often insufficient, necessitating methods that can explore its dependence on multiple external parameters Perturbation theory (PT) has limitations because it relies on the pseudo-inverse—a numerically costly operation—and has a limited radius of convergence. This paper addresses these roadblocks by introducing VPT and two numerical strategies that avoid computing the pseudo-inverse. The Lindblad master equation is a powerful framework for capturing the effects of an environment [3, 4].

Efficient Steady State Computation

The steady state, denoted by the stationary density matrix ˆρss, satisfies Lρˆss = 0. A common approach to compute ˆρss is to solve Eq. (3) directly. This equation can be solved through standard matrix factorization techniques, specifically LU decomposition, which requires only O(N 2) operations for a matrix of size N × N. The cost to compute the steady state across a phase space of a Liouvillian of size N × N scales as O P RPT N cubed.

Variational Perturbation Theory (VPT)

VPT is introduced to relax the constraints of standard PT by recasting the perturbative approximation as a low-dimensional variational problem, allowing for more efficient computation across parameter regions and a wider convergence radius. The method generalizes Eq. (8) to multiparameter Liouvillians L(θ1, θ2,...), leading to a two-dimensional perturbative corrections grid. The variational approach approximates the coefficients cn(ε) by solving a least-squares problem min cn L˜(ε)X M n=0 cn(ε)ρˆ (n)ss E − b⟩ squared. This variational nature ensures that VPT provides a better approximation than standard PT for any finite order M and hence has a larger convergence radius.

Numerical Strategies for Efficiency

The paper develops two model-independent methods based on VPT that avoid computing the pseudo-inverse of the Liouvillian. The first strategy exploits the properties of the Liouvillian to compute arbitrarily high orders of the perturbative expansion through a single LU decomposition, at a marginal numerical cost of O(N 2). The second strategy reformulates VPT as a Krylov space recycling problem and uses preconditioned iterative methods. This approach allows for an efficient computation of steady states across parameter regions, reducing the computational cost by up to a factor of a hundred compared to direct calculations.

Application and Benchmarking

VPT is demonstrated by applying it to the driven-dissipative Kerr resonator and the dissipative XY Z (Heisenberg) model. In a 1D example, VPT recovers the exact solution in a broader region than standard PT, from the vacuum at large detuning to the coherent-like state at ∆ ≃ 0. In a 2D example involving simultaneous variation of detuning and drive amplitude, VPT significantly outperforms PT near critical points where non-analytical behavior occurs. The performance of VPT is shown to yield significantly larger regions of convergence than PT, requiring seven times fewer points to map the whole phase space.

Conclusion

In summary, this work introduces variational perturbation theory (VPT), a generalization of standard perturbation theory for open quantum systems. The methods allow rapid exploration of phase diagrams and rapid estimation of steady state gradients, making them ideal tools for parameter fitting routines. The method is agnostic to the underlying details of the model under consideration, as it can be combined with other techniques such as cluster expansions or renormalization methods. This work demonstrates that efficient generalizations of standard perturbation theory are both possible and practical. The paper is structured to present these findings through analysis of the driven-dissipative Kerr resonator and the XYZ model. The work concludes by showing that VPT can be used for parameter estimation through efficient gradient computation. The final section details how to use VPT and gradient estimates to fit parameters in a superconducting device, achieving high accuracy within 15 iterations. The paper is structured as follows, beginning with a review of matrix factorization techniques, followed by an overview of perturbation theory for open quantum systems and the introduction of VPT. The paper concludes in Sec. VII with conclusions regarding the efficacy of VPT and its numerical viability. The appendices provide detailed derivations for the modified Liouvillian, PT derivation, tolerance criterion, optimal low-rank basis construction, and efficient gradient computation. The references cited include works on quantum dynamics [3], perturbation theory [33–35], and various numerical simulation techniques [19–27]. The paper is structured as follows, beginning in Sec. II with a brief review of how to determine steady states with matrix factorization techniques, followed by an overview of perturbation theory for open quantum quantum systems. In Sec. III, we present an efficient method for computing the perturbative recurrence relation without relying on a pseudo-inverse, then introduce VPT in Sec. IV and use it to analyze the driven-dissipative Kerr resonator model in both one- and two-dimensional parameter spaces. In Sec. V, we demonstrate how to use VPT for efficient parameter estimation and apply it to a dissipative cat qubit model. Finally, in Sec. VI, we extend VPT to an iterative Krylov-based approach and apply it to an XYZ model on a 3 × 3 lattice. The paper is structured as follows, beginning in Sec. II with a brief review of how to determine steady states with matrix factorization techniques, followed by an overview of perturbation theory for open quantum systems. In Sec. III, we present an efficient method for computing the perturbative recurrence relation without relying on a pseudo-inverse, then introduce VPT in Sec. IV and use it to analyze the driven-dissipative Kerr resonator model in both one- and two-dimensional parameter spaces. In Sec. V, we demonstrate how to use VPT for efficient parameter estimation and apply it to a dissipative cat qubit model. Finally, in Sec. VI, we extend VPT to an iterative Krylov-based approach and apply it to an XYZ model on a 3 × 3 lattice. The paper is structured as follows, beginning in Sec. II with a brief review of how to determine steady states with matrix factorization techniques, followed by an overview of perturbation theory for open quantum systems. In Sec. III, we present an efficient method for computing the perturbative recurrence relation without relying on a pseudo-inverse, then introduce VPT in Sec. IV and use it to analyze the driven-dissipative Kerr resonator model in both one- and two-dimensional parameter spaces. In Sec. V, we demonstrate how to use VPT for efficient parameter estimation and apply it to a dissipative cat qubit model. Finally, in Sec.

1. Bold Header: Variational Perturbation Theory (VPT) for Steady State Computation

This method significantly extends the radius of convergence even in the presence of non-analytic effects such as dissipative phase transitions by reformulating PT as a lowdimensional variational problem, allowing for a much wider convergence radius than standard perturbation theory.

2. Bold Header: Elimination of Pseudo-Inverse Computations

The paper introduces two numerical strategies that eliminate the need to compute pseudo-inverses, specifically by using a single LU decomposition to efficiently construct the steady state within the convergence region and reformulating VPT as a Krylov space recycling problem.

3. Bold Header: Efficient Parameter Space Mapping

The improved system can rapidly explore parameter spaces by scaling computation costs as O P RPT N cubed, which is significantly better than standard PT, enabling the mapping of phase diagrams for a Liouvillian of size N × N.

4. Bold Header: Robust Phase Transition Analysis

The VPT method allows for the accurate capture of non-analytical behavior near critical points, as it significantly outperforms PT in regions where the Liouvillian gap closes, enabling precise characterization of dissipative phase transitions.

5. Bold Header: High-Fidelity Parameter Estimation

The system can perform efficient parameter fitting by using VPT to solve the objective function minimization problem, which involves finding the gradient via solving the following reduced equation derived from implicit differentiation (Eq. 39).

Determining the steady state of an open quantum system is crucial for characterizing quantum devices and studying various physical phenomena. Often, computing a single steady state is insufficient, and it is necessary to explore its dependence on multiple external parameters. In such cases, calculating the steady state independently for each combination of parameters quickly becomes intractable. Perturbation theory (PT) can mitigate this challenge by expanding steady states around reference parameters, minimizing redundant computations across neighboring parameter values. However, PT has two significant limitations: it relies on the pseudo-inverse -- a numerically costly operation -- and has a limited radius of convergence. In this work, we remove both of these roadblocks. First, we introduce a variational perturbation theory (VPT) and its multipoint generalization that significantly extends the radius of convergence even in the presence of non-analytic effects such as dissipative phase transitions. Then, we develop two numerical strategies that eliminate the need to compute pseudo-inverses. The first relies on a single LU decomposition to efficiently construct the steady state within the convergence region, while the second reformulates VPT as a Krylov space recycling problem and uses preconditioned iterative methods. We benchmark these approaches across various models, demonstrating their broad applicability and significant improvements over standard PT.