Scientific Series

Fault-tolerant quantum computers will compute by applying
a sequence of elementary unitary operations, or gates, to an
error-protected subspace.   While algorithms are typically expressed
over arbitrary local gates, there is unfortunately no known theory
that can correct errors for a continuous set of quantum gates.
However, theory does support the fault-tolerant construction of
various finite gate sets, which in some cases generate circuits that
can approximate arbitrary gates to any desired precision.   In this
talk, I will present a framework for approximating arbitrary qubit
unitaries over a very general but natural class of gate sets derived
from the theory of integral quaternions over number fields, where the
complexity of a unitary is algebraically encoded in the length of a
corresponding quaternion.  Then I will explore the role played by
higher-dimensional generalizations of the Pauli gates in various
physical and mathematical settings, from classifying bulk-boundary
correspondences of abelian fractional quantum Hall states to
generating optimal symmetric quantum measurements with surprising
connections to Hilbert's 12th problem on explicit class field theory
for real quadratic number fields.

Throughout the development of quantum mechanics, the striking refusal of nature to obey classical reasoning and intuition has driven both curiosity and confusion. From the apparent inescapably probabilistic nature of the theory, to more subtle issues such as entanglement, nonlocality, and contextuality, it has always been the `nonclassical’ features that present the most interesting puzzle. More recently, it has become apparent that these features are also the primary resource for quantum information processing.
In this talk, we will introduce several types of nonclassical logical structures contained within the N-qubit Pauli group, corresponding in general to preparation and/or measurement schemes for systems of several qubits that demonstrate violation of some notion of classical reasoning. These structures are geometric in nature, and we identify the primitive elements from which more elaborate types are constructed. We will review the key types of structures that are available and explain their hierarchy. Finally we will use them to give simple and transparent proofs of entanglement correlations, quantum contextuality via the Kochen-Specker theorem, and quantum nonlocality via the Bell-GHZ theorem.
These structures have many applications in quantum information processing, but the real purpose of this talk is simply to introduce unfamiliar researchers to the simplest known logical proofs of these nonclassicality theorems, which can be understood using only the algebra of the Pauli spin matrices and simple counting arguments.

We investigate the quantum trajectories of jointly monitored transmon qubits, tracking measurement-induced entanglement creation as a continuous process.  The quantum trajectories naturally split into low and high entanglement classes corresponding to partial parity collapse.  We theoretically calculate the distribution of concurrence at any given time and show good agreement with the constructed histogram of measured concurrence trajectories.  The distribution exhibits a sharp cut-off in the high concurrence limit, defining a maximal concurrence boundary.  The most probable paths of the two classes, starting in a separable state and ending in either an entangled state or separable state, are found and compared with the experimentally constructed paths of the sub-ensemble.  The most likely time for the transmon qubits to reach their highest concurrence can also be extracted from the most likely path analysis.

In the device-independent paradigm, the labeling of parties/inputs/outputs has no physical meaning and thus the behavior of the system should be studied up to symmetry. We conduct the first formal study of relabelings appearing in Bell scenarios. The talk includes a review of previous works, a definition of Bell relabeling groups illustrated by examples, and applications, including the classification of Bell inequalities, the generalization of binary correlators to d outcomes and the computation of exact bounds using the NPA hierarchy.

We have great certainty on how gravity works around our solar system: General Relativity (GR) has been found to be very accurate at these small scales. On large scales though, we still have a considerable lack of understanding about the evolution of the universe, and its constituents. While the LCDM model is in good agreement with cosmological data, this might change in the future. For this reason, we need to test GR on these scales.

There are a number of proposals on how to characterize deviations from GR on larges scales, by parametrizing different cosmological evolutions of the universe - the PPF, EFT and EA approaches. The objective is to use experimental data to constrain these parameters, and thus identify the most accurate cosmological model. In this talk I will show an alternative, systematic and general, parametrization method, in which we construct the most general quadratic action for linear cosmological perturbations, given some field content and gauge symmetries. I will show the example of linearly diffeomorphism-invariant scalar-tensor theories, in which case the parametrization encompasses well-known theories such as Horndeski and Beyond Horndeski. The method can straightforwardly be applied to any gravity theory with any fields and gauge symmetries.

We discuss how to formulate a quantum field theory of dark energy interacting with dark matter. We show that the proposals based on the assumption that dark matter is made up of heavy particles with masses which are very sensitive to the value of dark energy are strongly constrained. Quintessence-generated long range forces and radiative stability of the quintessence potential require that such dark matter and dark energy are completely decoupled. However, if dark energy and a fraction of dark matter are very light axions, they can have significant mixings which are radiatively stable and perfectly consistent with quantum field theory. Such models can naturally occur in multi-axion realizations of monodromies. The mixings yield interesting signatures which are observable and are within current cosmological limits but could be constrained further by future observations.

Questions of nonlinear stability in global AdS space have recently received a significant amount of attention, both as an interesting problem in mathematical general relativity and nonlinear dynamics, and in relation to thermalization studies within the AdS/CFT paradigm. Working with nonlinear perturbation theory (the main technique available for analytic studies in this area) requires a thorough understanding of the properties of linearized AdS fields'

mode functions, which are the fundamental building blocks in perturbative treatments of nonlinearities. While complicated explicit expressions for these mode functions are available in the literature, they hide a good deal of the elegant underlying structure dictated by the AdS symmetries. Extending the mode functions in the flat embedding space (in which the AdS space can be realized as a hyperboloid) results in families of homogeneous polynomials, on which the AdS isometries act in a straightforward manner. This suggests a simple proof of important selection rules in nonlinear perturbation theory.

Studies of multiplet structures of the mode functions furthermore reveal a relation to the Higgs oscillator, a well-known quantum-mechanical superintegrable system. This AdS connection leads to an explicit construction of the hidden symmetry generators for the Higgs oscillator, a long-standing problem in mathematical quantum mechanics.

I will discuss the stability and breakdown of the topological classification of gapped ground states of non-interacting fermions, the tenfold way, in the presence of quartic fermion-fermion interactions.  In our approach [1], the effects of interactions on the boundary gapless modes are encoded in terms of boundary dynamical masses. Breakdown of the non-interacting topological classification occurs when the quantum nonlinear sigma models for the boundary dynamical masses favor quantum disordered phases.  The non-interacting topological classification $Z$ in odd spatial dimensions is unstable and reduces to $Z_N$ that can be identified explicitly for any dimension and any defining symmetries.
[1] T. Morimoto, A. Furusaki, and C. Mudry, Phys. Rev. B 91, 235111 (2015).

I will discuss how we constrain properties of the universe using two
tracers of large scale structure measured by the BOSS (Baryon
Oscillations Spectroscopic Experiment): galaxies and Lyman-alpha
forest. I will show recent results from baryonic acoustic oscillations
measured in both tracers and discuss cosmological implications. I
will briefly mention other measurements and consider forecasts for
quasar-forest bispectrum to constrain primordial non-Gaussianity.

Topological factorization homology is an invariant of manifolds which enjoys a hybrid of the structures in topological field theory, and in singular homology.  These invariants are especially interesting when we restrict attention to the factorization homology of surfaces, with coefficients in braided tensor categories.  In this talk, I would like to explain a technique, related to Beck monadicity, which allows us to compute these abstractly defined categories, as modules for explicitly computable, and in many cases well-known, algebras.

The algebras produced in this way for the annulus and punctured torus are the so-called ``reflection equation algebra" and "quantum differential equation algebra", respectively.  When we close up punctures, a variation on the our formalism naturally reproduces the framework of quantum Hamiltonian reduction, and leads to simultaneous deformations of categories D(g/G) of character sheaves on g, on the one hand, and categories QC(Ch_G(T^2)), of quasi-coherent sheaves on the character variety of T^2, on the other.  We call these deformations "quantum character varieties", and they form the two-dimensional part of a four dimensional TFT related to Kapustin-Witten's geometric Langlands TFT, or "topologically twisted N=4 SYM".