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Every unitary map with a factorisation of domain and codomain into subsystems has a well-defined causal structure given by the set of influence relations between its input and output subsystems. A causal decomposition of a unitary map U is, roughly, one that makes all there is to know about U in terms of causal structure evident and understandable in compositional terms. We'll argue that this is more than just about drawing more intuitive pictures for the causal structure of U -- it is about really understanding it at all. Now, it has been known for a while that decompositions in terms of ordinary circuit diagrams do not suffice to this end and that at least so called 'extended circuit diagrams', or 'routed circuit diagrams' are necessary, revealing a close connection between causal structure and algebraic structures that involve a particular interplay of direct sum and tensor product. Though whether or not these sorts of routed circuit diagrams suffice has been an open question since. I will give an introduction and overview of this business of causal decompositions of unitary maps, and share what is an on-going thriller.

Zoom link:  https://pitp.zoom.us/j/95689128162?pwd=RFNqWlVHMFV0RjRaakszSTBsWkZkUT09

The aim of the presentation is to review the beautiful geometry underlying the standard model of particle physics, as captured by the framework of "SO(10) grand unification." Some new observations related to how the Standard Model (SM) gauge group sits inside SO(10) will also be described. 

I will start by reviewing the SM fermion content, organising the description in terms of 2-component spinors, which give the cleanest picture. 

I will then explain a simple and concrete way to understand how spinors work in 2n dimensions, based on the algebra of differential forms in n dimensions.

This will be followed by an explanation of how a single generation of standard model fermions (including the right-handed neutrino) is perfectly described by a spinor in a 10 ("internal") dimensions. 

I will review how the two other most famous "unification" groups -- the SU(5) of Georgi-Glashow and the SO(6)xSO(4) of Pati-Salam -- sit inside SO(10), and how the SM symmetry group arises as the intersection of these two groups, when they are suitably aligned.

I will end by explaining the more recent observation that the choice of this alignment, and thus the choice of the SM symmetry group inside SO(10), is basically the choice of two Georgi-Glashow SU(5) such that the associated complex structures in R^{10} commute. This means that the SM gauge group arises from SO(10) once a "bihermitian" geometry in R^{10} is chosen. I will end with speculations as to what this geometric picture may be pointing to. 

Zoom link:  https://pitp.zoom.us/j/95984379422?pwd=SE1ybktzQzcreWREblhEUkZWWElMUT09

We study the behavior of errors in the quantum simulation of  spin systems with long-range multibody interactions resulting from the  Trotter-Suzuki decomposition of the time evolution operator. We  identify a regime where the Floquet operator underlying the Trotter  decomposition undergoes sharp changes even for small variations in the  simulation step size. This results in a time evolution operator that  is very different from the dynamics generated by the targeted  Hamiltonian, which leads to a proliferation of errors in the quantum  simulation. These regions of sharp change in the Floquet operator,  referred to as structural instability regions, appear typically at  intermediate Trotter step sizes and in the weakly interacting regime,  and are thus complementary to recently revealed quantum chaotic  regimes of the Trotterized evolution [L. M. Sieberer et al. npj  Quantum Inf. 5, 78 (2019); M. Heyl, P. Hauke, and P. Zoller, Sci. Adv.  5, eaau8342 (2019)]. We characterize these structural instability  regimes in p-spin models, transverse-field Ising models with  all-to-all p-body interactions, and analytically predict their  occurrence based on unitary perturbation theory. We further show that  the effective Hamiltonian associated with the Trotter decomposition of  the unitary time-evolution operator, when the Trotter step size is  chosen to be in the structural instability region, is very different  from the target Hamiltonian, which explains the large errors that can  occur in the simulation in the regions of instability. These results  have implications for the reliability of near-term gate based quantum  simulators, and reveal an important interplay between errors and the  physical properties of the system being simulated.

Zoom link:  https://pitp.zoom.us/j/92045582127?pwd=WDUxcnlIeXdnVWM3WGJoSFVMNDE2dz09