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 phase maps, as we explicitly show by linking 2D networks to 1D fermions, to a Z2-nontrivial 2D insulator. However, beyond a rotation angle ϕth, instead of a Z2-trivial insulator as for incoherent errors, coherent errors map to a Majorana metal. This ϕth is the theoretically achievable storage threshold. We numerically find ϕth≈0.14π. The corresponding bit-flip rate sin2(ϕth)≈0.18 exceeds the known incoherent threshold pth≈0.11.

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 the final ramp of the fully mixed chaotic Hamiltonian. The transitional behaviour is governed by a universal process that we call Hilbert space diffusion, with a diffusion constant related to the relaxation rate of the associated nearly conserved charge.

I shall discuss nonreciprocal particle and energy transport in linear systems with balanced gain and loss of particles or energy in the bulk  [1]. 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 [2]. Next, I shall present our studies of these systems within an open-quantum system and a classical master equation description. These descriptions surprisingly lead to nonreciprocal transport in linear systems, even without magnetic fields. Previous studies have found nonreciprocity in such models only due to nonlinearity or magnetic fields. Our results suggest that these systems with broken parity and time-reversal symmetry show an arrow of space manifested through nonreciprocal transport.

References:

1. Nonreciprocal electrical transport in linear systems with balanced gain and loss in the bulk, Rupak Bag and Dibyendu Roy, arXiv: 2409.12510 (2024)

2. Quantum noise induced nonreciprocity for single photon transport in parity-time symmetric systems, Dibyendu Roy and G. S. Agarwal, arXiv: 2407.00758 (2024)

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 impact of PT-symmetric phase transitions on wave function dynamics using Spread Complexity. Finally, I will propose a potential connection between classical integrability, quantum chaos, and the topology of phase space flow through the lens of Spread Complexity.

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 the SINDy (Sparse Identification of Nonlinear Dynamics) framework, a weak formulation of the dynamics and a novel model selection measure. Using this method, we were able to extract partial differential equations governing the dynamics of a simple fluid from simulations based on a particle model — dissipative particle dynamics. These equations were the mass continuity equation and a form of the Navier-Stokes equation, the latter containing the correct pressure equation of state. The talk is based on our recently published article (https://doi.org/10.1016/j.cma.2024.117379).

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 one-site Markov transfer operator, which is generally of infinite dimensionality. The theory is tested for a specific family of dual-symplectic circuits, describing the dynamics of a classical Floquet spin chain. Remarkably, expressing these models in the form of a composition of rotations leads to a transfer operator with a block diagonal form in the basis of spherical harmonics. This allows for obtaining, analytical predictions for simple local observables. 

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 an infinite complement. For initial times, shorter than the subsystem’s size, spatial locality and (dual) unitarity implies constant partial spectral form factor, clearly deviating from the linear ramp of the partial spectral form factor in random matrix theory. In contrast, for larger times we prove, that the partial spectral form factor follows the random matrix result up to exponentially suppressed corrections. We supplement those exact results by semi-analytic computations performed directly in the thermodynamic limit.

Many physical systems admit a simplified description of their dynamics when examined at macroscopic scales. This simplified description—generally referred to as hydrodynamics—is governed by a restricted set of macroscopic observables that includes conserved quantities, Goldstone modes, and order parameters. An outstanding challenge in quantum many-body physics is finding this hydrodynamic description in terms of the microscopic variables. I will present a method inspired by holography for constructing the effective hydrodynamic description in the form of a transfer matrix and a set of hydrodynamically-relevant variables. The method proceeds by constructing an alternative representation of the operator dynamics in the form of a local (1+1)d “bulk” theory. I will show how the properties of the auxiliary bulk encode the existence of an effective local equation of motion of a given model, allowing for the extraction of hydrodynamic parameters like diffusion constants and characteristic thermalization scales. I will show results for various qubit and fermionic systems, and compare to the known literature.