While the notion of quantum chaos is tied to random-matrix spectral correlations, also eigenstate properties in chaotic systems are often assumed to be described by random matrix theory. Analytic insights into such eigenstate properties can be obtained by the recently introduced partial spectral form factor, which captures correlations between eigenstates. Here, we study the partial spectral form factor in chaotic dual-unitary quantum circuits. We compute the latter for a finite connected subsystem in a brickwork circuit in the thermodynamic limit, i.e., for…

I am going to talk about a general exact method of calculating dynamical correlation functions in dual symplectic brick-wall circuits in one dimension. These are deterministic classical many-body dynamical systems which can be interpreted in terms of symplectic dynamics in two orthogonal (time and space) directions. In close analogy with quantum dual-unitary circuits, one can prove that two-point dynamical correlation functions are nonvanishing only along the edges of the light cones. The dynamical correlations are exactly computable in terms of a…

We study the phase space of a particle moving in the gravitational field of a rotating black hole described by the Kerr metric from a geometrical perspective. In particular, we show the construction of a multidimensional generalization of the unstable periodic orbits, known as Normally Hyperbolic Invariant Manifolds, and their stable and unstable invariant manifolds that direct the dynamics in the phase space. Those stable and unstable invariant manifolds divide the phase space and are robust under perturbations. To visualize…

Equations describing the macroscopic dynamics of complex materials are traditionally derived by a systematic symmetry-based approach. A model derived in this way usually contains a number of unknown parameters that have to be estimated from data; either from experiments or simulations. With a suitable regression method, not only the parameters, but also the dynamic equations themselves can be extracted directly from data, bypassing the need for a traditional derivation. In this talk, I will present such a method, based on…

In this talk, I will review recent progress in defining a universal measure of quantum complexity based on operator growth and state evolution in Krylov space for both unitary and non-unitary dynamics. After introducing Krylov space techniques, I will focus on the complexity measure associated with quantum state evolution, specifically ‘Spread Complexity,’ along with explicit examples and general properties. I will use Spread Complexity to investigate measurement-induced effects on wave function spreading in tight-binding models. Additionally, I will discuss the…

I shall discuss nonreciprocal particle and energy transport in linear systems with balanced gain and loss of particles or energy in the bulk  . The role of balanced gain and loss of particles or energies has been extensively investigated in recent years in the context of an effective parity-time symmetry in classical and quantum systems. First, I shall point out severe issues with existing theoretical modeling for the time evolution of such systems . Next, I shall present our studies…

Random matrix theory (RMT) universality is the defining property of quantum mechanical chaotic systems, and can be probed by observables like the spectral form factor (SFF). In this talk, I’ll describe systematic deviations from RMT behaviour at intermediate time scales in systems with approximate symmetries. In such systems, the approximate symmetries allow us to organize the Hilbert space into approximately decoupled sectors. At early times, each of which contributes independently to the SFF. At late times, the SFF transitions into…

Statistical mechanics mappings provide key insights on quantum error correction. However, existing mappings assume incoherent noise, thus ignoring coherent errors due to, e.g., spurious gate rotations. We map the surface code with coherent errors, taken as X or Z rotations (replacing bit or phase flips), to a two-dimensional (2D) Ising model with complex couplings, and further to a 2D Majorana scattering network. Our mappings reveal both commonalities and qualitative differences in correcting coherent and incoherent errors. For both, the error-correcting…