Scientific Series

Recently, Roland van der Veen and myself found that there are sequences of solvable Lie algebras "converging" to any given semi-simple Lie algebra (such as sl(2) or sl(3) or E8). Certain computations are much easier in solvable Lie algebras; in particular, using solvable approximations we can compute in polynomial time certain projections (originally discussed by Rozansky) of the knot invariants arising from the Chern-Simons-Witten topological quantum field theory. This provides us with the first strong knot invariants that are computable for truly large knots.  But sl(2) and sl(3) and similar algebras occur in physics (and in mathematics) in many other places, beyond the Chern-Simons-Witten theory. Do solvable approximations have further applications?

This is a repeat of a talk I gave in McGill University in February, 2017. A video recording, a handout, and some further links are at McGill-1702

I'll ask whether the knowledge of a single eigenstate of a local lattice Hamiltonian is sufficient to uniquely determine the Hamiltonian. I’ll present evidence that the answer is yes for generic local Hamiltonians, given either the ground state or an excited state. In fact, knowing only the correlation functions of local observables with respect to the eigenstate appears generically sufficient to exactly recover both the eigenstate and the Hamiltonian, with efficient numerical algorithms. I may also comment on the extension of these ideas to the continuum, pose questions about the behavior of generic local spin systems, or discuss the question of what happens if you don't know a preferred factorization of the Hilbert space.

The strong interaction of quarks and gluons is described theoretically within the framework of Quantum Chromodynamics (QCD). The most promising way to evaluate QCD for all energy ranges is to formulate the theory on a 4 dimensional Euclidean space-time grid, which allows for numerical simulations on state of the art supercomputers. We will review the status of lattice QCD calculations providing examples such as the hadron spectrum and the inner structure of nucleons. We will then point to problems that cannot be solved by conventional Monte Carlo simulation techniques, i.e. chemical potentials and understanding the large amount of charge and parity symmetry violation. It will be demonstrated at the example of the 1+1 dimensional Schwinger model that tensor network techniques are able to overcome these problems opening thus a possible path for a solution also in QCD.

Red supergiants (RSGs) are the helium-fusing evolved descendants of moderately massive (10-25Mo) stars, the result of a near-horizontal evolution across the top of the Hertzsprung-Russell diagram following their time on the main sequence. As the coldest and largest (in physical size) members of the massive star population, these stars represent a significant evolutionary extreme and serve as ideal "magnifying glasses" for scrutinizing our current understanding of massive stars and their role in the universe. RSGs are significant dust producers in young stellar populations, the observationally-confirmed progenitors of core-collapse supernovae, and a crucial step in the formation and population statistics of massive interacting binaries (including those that will ultimately produce compact object binaries and gravitational waves). Observations of RSGs can also be used to test stellar evolution and population models and as metallicity indicators in nearby galaxies. This talk will provide an overview of our field's knowledge of RSGs, identify some of the most pressing current questions about these stars, and consider the importance of RSGs in the coming decade as the next generation of observatories - including JWST, WFIRST, and the ELTs - comes online.

The distillation of quantum resources such as entanglement and coherence forms one of the most fundamental protocols in quantum information and is of outstanding operational significance, but it is often characterized in the idealized asymptotic limit where an unbounded number of independent and identically distributed copies of a quantum system are available. Physical considerations necessarily limit the number of copies available as well as restrict our ability to perform coherent state manipulations over a large numbers of systems, which makes it crucial to be able to characterize how well we can distill resources in realistic, non-asymptotic settings.

We introduce a framework for the non-asymptotic characterization of two related operational tasks, the distillation as well as environment-assisted distillation of quantum coherence. We establish a complete description of the achievable rates of distillation in the one-shot setting under several different classes of free operations, which we show to correspond to the optimization of smoothed entropic quantities as semidefinite programs. We introduce a class of coherence measures which quantify the best achievable fidelity of distillation, and use them to obtain an explicit analytical characterization of coherence distillation for all pure states. Further, we provide insight into the distillation of entanglement, revealing new operational similarities and differences between the resource theories of coherence and entanglement.

This talk is based on joint work with Kun Fang, Xin Wang, and Gerardo Adesso (arxiv:1711.10512) as well as with Ludovico Lami and Alexander Streltsov (in preparation).

The modern conception of phases of matter has undergone tremendous developments since the first observation of topologically ordered states in fractional quantum Hall systems in the 1980s. In this paper, we explore the question: In principle, how much detail of the physics of topological orders can be observed using state of the art technologies? We find that using surprisingly little data, namely the toric code Hamiltonian in the presence of generic disorders and detuning from its exactly solvable point, the modular matrices -- characterizing anyonic statistics that are some of the most fundamental fingerprints of topological orders -- can be reconstructed with very good accuracy solely by experimental means. This is a first experimental realization of these fundamental signatures of a topological order, a test of their robustness against perturbations, and a proof of principle -- that current technologies have attained the precision to identify phases of matter and, as such, probe an extended region of phase space around the soluble point before its breakdown. Given the special role of anyonic statistics in quantum computation, our work promises myriad applications in both probing and realistically harnessing these exotic phases of matter.

Topological defect operators are extended operators in a quantum field theory (QFT) whose correlation functions are independent of continuous changes of the ambient space. They satisfy nontrivial fusion relations and put nontrivial constraints on the QFT itself and its deformations (such as renormalization group (RG) flows). Canonical examples of topological defect operators include generators of (higher-form) global symmetries whose constraints on the QFT have been well-studied. However they constitute just a tip of a much richer framework which we explore in detail in two spacetime dimensions. 

More specifically, we study the crossing relations of general topological defect lines (TDL) in two dimensions, discuss their relation to the usual 't Hooft anomalies associated to global symmetries, and use them to constrain RG flows to either conformal fixed points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.

In 2020 the European Space Agency (ESA) will launch the Euclid satellite mission. Euclid is an ESA medium class astronomy and astrophysics space mission, and will undertake a galaxy redshift survey over the redshift range 0.9 < z < 1.8, while simultaneously performing an imaging survey in both visible and near infrared bands. The complete survey will provide hundreds of thousands images and several tens of Petabytes of data. About 10 billion sources will be observed by Euclid out of which several tens of million galaxy redshifts will be measured and used to make galaxy clustering measurements. These observations will be used to help to understand Dark Energy, the physical mechanism causing the current acceleration in the expansion of the Universe.

I will give a brief survey of the study of decomposable Specht modules for the symmetric group and its Hecke algebra, which includes results of Murphy, Dodge and Fayers, and myself. I will then report on an ongoing project with Louise Sutton, in which we are studying decomposable Specht modules for the Hecke algebra of type $B$ indexed by `bihooks’.

Quantum computers can only offer a computational advantage when they have sufficiently many qubits operating with sufficiently small error rates. In this talk, I will show how both these requirements can be practically characterized by variants of randomized benchmarking protocols. I will first show that a simple modification to protocols based on randomized benchmarking allows multiplicative-precision estimates of error rates.  I will then outline a new protocol for estimating the fidelity of arbitrarily large quantum systems using only single-qubit randomizing gates.

We  examine 1D spin-chains evolving under random local unitary circuits and prove a number of exact results on the behavior of out-of-time-ordered commutators (OTOCs), and entanglement growth. These results follow from the observation that the spreading of operators in random circuits is described by a ``hydrodynamical'' equation of motion. In this hydrodynamic picture quantum information travels in a front with a `butterfly velocity' $v_{\text{B}}$ that is smaller than the light cone velocity of the system, while the front itself broadens diffusively in time. The OTOC increases sharply after the arrival of the light cone, but we do \emph{not} observe a prolonged exponential regime of the form $\sim e^{\lambda_\text{L}(t-x/v)}$ for a fixed Lyapunov exponent $\lambda_\text{L}$, in disagreement with the existing QFT literature on OTOCs. We find that the diffusive broadening of the front has important consequences for entanglement growth, leading to an entanglement velocity that can be significantly smaller than the butterfly velocity. We conjecture that the hydrodynamical description applies to more generic ergodic systems and support this with numerical simulations. When the circuits are constrained so as to conserve a U$(1)$ charge, we show that the OTOC acquires a diffusively decaying component.