André Melo, Gaspard Beugnot, and Fabrizio Minganti of Alice & Bob have detailed a new approach to simulating quantum systems in volume 10, page 2246 of Quantum. Their Variational Perturbation Theory (VPT) overcomes longstanding limitations of traditional methods by eliminating the need for costly pseudo-inverse calculations and expanding the radius of convergence for steady-state computations.
This advancement allows for more efficient exploration of quantum phenomena, even those involving problematic dissipative phase transitions, previously inaccessible with standard techniques. The researchers developed two new numerical strategies, one using a single LU decomposition and another reformulating VPT as a Krylov space recycling problem with preconditioned iterative methods.
Variational Perturbation Theory Extends Steady State Calculations
This new approach bypasses the need for computationally expensive operations by introducing two distinct numerical strategies, each designed to accelerate steady-state computation. 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. These methods allow for efficient reuse of information from previous calculations, significantly reducing the computational burden when exploring parameter spaces.
This is a departure from traditional methods, which often require recalculating the pseudo-inverse for each new set of parameters, a process that scales poorly with system size. Beyond computational efficiency, VPT demonstrably extends the range of parameters over which perturbation theory remains valid. Standard perturbation theory struggles with systems undergoing dissipative phase transitions, where non-analytic effects render the expansion unreliable.
The new theory achieves a wider radius of convergence, enabling the simulation of phenomena previously inaccessible to standard methods. The practical implications of this extended convergence radius are significant, particularly for systems exhibiting critical behavior. Researchers can now explore parameter regimes where the system’s properties change dramatically, such as the onset of a new phase or a transition between different behaviors.
This is particularly relevant to studies of driven-dissipative systems, where the interplay between energy input and dissipation can lead to complex and often non-intuitive phenomena, as demonstrated by the work of Vicentini, Biella, Regnault and Ciuti on neural network ansatze for steady states. The team’s work builds on earlier investigations into dissipative phase transitions by Kessler, Giedke, Imamoglu, Yelin, Lukin and Cirac, and by Carmichael, who explored the breakdown of photon blockade. These methods, combined with the variational approach, provide a robust and efficient framework for simulating open quantum systems.
LU Decomposition and Krylov Methods Eliminate Pseudo-Inverse Computation
A single LU decomposition now streamlines steady-state computation in quantum simulations, circumventing a traditionally expensive step. This advancement allows for faster and more accurate exploration of quantum phenomena, particularly in systems with many interacting components. This reformulation addresses the inherent inefficiencies in sequential simulations, where repeated calculations on similar systems become computationally burdensome. By efficiently reusing information from previous iterations, the Krylov method significantly reduces the overall computational cost, enabling simulations of larger and more complex quantum systems than previously feasible.
The work builds on earlier investigations by Parks, de Sturler, Mackey, Johnson and Maiti into recycling Krylov subspaces for sequences of linear systems. The practical implications extend to the study of Schrödinger cat states in driven-dissipative systems, as demonstrated by Minganti, Bartolo, Lolli, Casteels, and Ciuti in their 2016 report.
The ability to accurately simulate these fragile quantum states is essential for developing robust quantum technologies. The approach is applicable to a wide range of quantum systems, from optical cavities to many-body systems, as evidenced by the benchmarking across various models detailed in the publication.
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