Quantum systems engineering

Abstract:

We introduce a widely applicable tensor network-based framework for developing reduced order models describing wall-bounded fluid flows. As a paradigmatic example, we consider the incompressible Navier-Stokes equations and the lid-driven cavity in two spatial dimensions. We benchmark our solution against published reference data for low Reynolds numbers and find excellent agreement. In addition, we investigate the short-time dynamics of the flow at high Reynolds numbers for the liddriven and doubly-driven cavities. We represent the velocity components by matrix product states and find that the bond dimension grows logarithmically with simulation time. The tensor network algorithm requires at most a few percent of the number of variables parameterizing the solution obtained by direct numerical simulation, and approximately improves the runtime by an order of magnitude compared to direct numerical simulation on similar hardware. Our approach is readily transferable to other flows, and paves the way towards quantum computational fluid dynamics in complex geometries.

Abstract:

Explaining quantum many-body dynamics is a long-held goal of physics. A rigorous operator algebraic theory of dynamics in locally interacting systems in any dimension is provided here in terms of time-dependent equilibrium (Gibbs) ensembles. The theory explains dynamics in closed, open, and time-dependent systems, provided that relevant pseudolocal quantities can be identified, and time-dependent Gibbs ensembles unify wide classes of quantum nonergodic and ergodic systems. The theory is applied to quantum many-body scars, continuous, discrete, and dissipative time crystals, Hilbert space fragmentation, lattice gauge theories, and disorder-free localization, among other cases. Novel pseudolocal classes of operators are introduced in the process: projected-local, which are local only for some states, cryptolocal, whose locality is not manifest in terms of any finite number of local densities, and transient ones, that dictate finite-time relaxation dynamics. An immediate corollary is proving saturation of the Mazur bound for the Drude weight. This proven theory is intuitively the rigorous algebraic counterpart of the weak eigenstate thermalization hypothesis and has deep implications for thermodynamics: Quantum many-body systems “out of equilibrium” are actually always in a time-dependent equilibrium state for any natural initial state. The work opens the possibility of designing novel out-of-equilibrium phases, with the newly identified scarring and fragmentation phase transitions being examples.