High-accuracy model reduction for open quantum systems – HAMROQS

Classical systems theory has developed several model reduction methods. We will apply them to quantum systems with their specificities: large dimension, linear parts, positivity requirement, extreme timescales. Rather standard applied mathematics is thus to be used for scientific objectives 1 and 2.
Objective 3 combines stochastic differential equations and geometry.

We have demonstrated how quantum adiabatic elimination can be carried out at higher orders. This has enabled us to understand new phenomena in several quantum experiments, among others counterintuitive effects like stronger decoupling from perturbations as they get stronger, and quantum measurement rates (thanks to analyzing their average back-action).
The identification of nonlinear invariant manifolds has enabled us to write closed-form solutions for several quantum systems; their key applications are still underway.

Strong driving with time-dependent signals has taken even more importance in the physics community during the project. Boundaries of chaos, optimization of trajectories, and many other things in this context can benefit from more specific approaches than model reduction, or could consider adaptive model reduction.
Numerical simulation also seems to need our results now, as the complexity of «standard« experiments is hitting the limits of what a complete modeling can do.

see Google Scholar page of Alain Sarlette

The success of quantum engineering during the last decades has heavily relied on approximate models, like Rotating Wave Approximation (RWA), Markov assumption or simplified noise models, to tractably quantify dominant properties in an experiment design context. Providing a true quantum advantage in applications will require to engineer larger systems with still more precision. From the modeling viewpoint this will require, at the design stage already, to capture more efficiently the effects of strong drives and pumps, higher-order interactions, and refined noise models. Experiments have indeed reached a stage where the limitations of current modeling and design guidelines are becoming visible in the most advanced platforms, like superconducting qubits.?The models currently used in quantum system design mainly correspond to first-order approximations in terms of advanced control and systems theory tools like averaging, algebraic (singular or regular) perturbations, Markovian assumptions or center manifolds, which in principle could be carried out to arbitrary order. The purpose of the HAMROQS project is to make these higher orders explicitly available to quantum experimentalists. More precisely, we will adapt the general and abstract systems theory tools to the particular structure of quantum systems: positivity of the density operator and related dynamics, tensor product and entanglement for interacting systems, dual Heisenberg/Schrödinger pictures, measurement via operator observables; and we will distill guidelines and quantitative formulas that can be directly interpreted and exploited in a quantum context. The results shall thus be framed in terms of associated frequency shifts, coherence losses, dissipation operators, noise correlations, and other features directly accessible in experiments. By quantifying in this way the performance of complex quantum systems, we aim at providing a more accurate basis towards understanding the most advanced quantum hardware, and tractable tools towards improving and optimizing their design.
The HAMROQS project will construct high-accuracy reduced models with three angles of attack. A first one is the modeling of “non-Markovian” effects induced by environments with memory. This will build on a rigorous progressive elimination of the environment, without completely discarding its dynamics, and on systems theory tools for modeling input-output relations in an efficient and robust way. A second point is the development of more accurate rotating wave type approximations, typically for studying oscillating Hamiltonians. Besides a gain in pure precision, this will enable stronger drives for noise protection or fast operation, and to scale up quantum systems with high numbers of drives and of addressable frequencies. We will confront the various quantum-specific series developments for integrating time-dependent Hamiltonian dynamics, to the system-theoretic averaging technique, in order to extract and combine the best parts of each approach in a systematic way. A major aim is also to rigorously include the decoherence and measurement terms into this RWA-type model simplification, which is less standard in existing quantum formulas. The third point concerns a more tractable characterization of the nonlinear behavior due to measurement back-action, with a particular eye on improving estimation filters and feedback controller design. Its starting point is a novel nonlinear deterministic structure that we have recently observed in quantum stochastic differential equations.
The project will be carried out in close collaboration with physicists who are encountering these needs in their current experiments. It will thereby have direct implications on overcoming coherence time limitations due to TLS baths and on error protection based on powerful reservoir engineering. Other examples will be actively pursued during the project lifetime to make these tools available to the quantum physics community.