Scientific Series

A central question in quantum computation is to identify which problems can be solved faster on a quantum computer. A Holy Grail of the field would be to have a theory of quantum speed-ups that delineates the physical mechanisms sustaining quantum speed-ups and helps in the design of new quantum algorithms. In this talk, we present such a toy theory for the study of a class of quantum algorithms for algebraic problems, including Shor’s celebrated factoring algorithm. Our theory is an extension of Gottesman’s stabilizer formalism based on elements of group and hypergroup theory. Using our methods, we develop classical simulation algorithms for Clifford-like circuits containing quantum Fourier transforms as well as new quantum algorithms for hidden symmetries and hyper-symmertry problems. During the talk, we will discuss the role of resources such as entanglement, interference and contextuality within our formalism and connect quantum speed-ups therein to the presence of precise algebraic structures.

Based on the following works:

[1] https://arxiv.org/abs/1210.3637
[2] https://arxiv.org/abs/1409.3208
[3] https://arxiv.org/abs/1409.4800
[4] https://arxiv.org/abs/1509.05806

Tensor networks offer an efficient representation of many-body wave-functions in an exponentially large Hilbert space by exploiting the area law of ground state quantum entanglement. I will start with a gentle introduction to the tensor network formalism. Then I will describe its application to realizing Wilson's renormalization group directly on quantum lattice models (e.g. quantum spin chains), with emphasis on the RG fixed points corresponding to conformal field theories. We will see how to define both global and local scale transformations on the lattice in such a way that, intriguingly, conformal invariance can be tested (and conformal data extracted) right at the UV cut-off scale.

Can we decompose the information of a composite system into terms arising from its parts and their interactions?
For a bipartite system (X,Y), the joint entropy can be written as an algebraic sum of three terms: the entropy of X alone, the entropy of Y alone, and the mutual information of X and Y, which comes with an opposite sign. This suggests a set-theoretical analogy: mutual information is a sort of "intersection", and joint entropy is a sort of "union".
The same picture cannot be generalized to three or more parts in a straightforward way, and the problem is still considered open. Is there a deep reason for why the set-theoretical analogy fails?
Category theory can give an alternative, conceptual point of view on the problem. As Shannon already noted, information appears to be related to symmetry. This suggests a natural lattice structure for information, which is compatible with a set-theoretical picture only for bipartite systems.
The categorical approach favors objects with a structure in place of just numbers to describe information quantities. We hope that this can clarify the mathematical structure underlying information theory, and leave it open to wider generalizations.

Abstract: We present the foundation for a holographic dictionary with depth perception. The dictionary consists of natural operators associated with CFT bilocals whose duals are simple, diffeomorphism-invariant bulk operators.  These objects admit a description as fields in kinematic space, a phase space for such probes.  The framework of kinematic space allows for conceptually simple derivations of many results known in the literature, including linearized Einstein’s equations, the relationship between conformal blocks and geodesic Witten diagrams, and the CFT representation of bulk local operators.   Reference: https://arxiv.org/abs/1604.03110

Soft theorems for the scattering of low energy photons and gravitons and cosmological consistency conditions on the squeezed-limit correlation functions are both understood to be consequences of invariance under large gauge transformations. I apply the same method used in cosmology -- based on the identification of an infinite set of "adiabatic modes" and the corresponding conserved currents -- to derive flat space soft theorems for electrodynamics and gravity. I discuss how the recent derivations based on the asymptotic symmetry groups (BMS) can be continued to a finite size sphere surrounding the scattering event. Finally, I comment on Hawking-Perry-Strominger proposal of "soft hair on black holes".

I discuss the phenomenology of models of inflation with periodic particle production. Particle production occurs as the mass of heavy particles goes through non-adiabatic modulations as the inflaton field rolls. I show that this process can lead to significant emission of scalar and gravitational waves during inflation, with distinct observational signatures.

We consider d=2 fermions at finite density coupled to a critical boson. In the quenched or Bloch-Nordsieck approximation, where one takes the limit of fermion flavors N_f→0, the fermion spectral function can be determined {exactly}. We show that one can obtain this non-perturbative answer thanks to a specific identity of fermionic two-point functions in the planar local patch approximation. The resulting spectrum is that of a non-Fermi liquid: quasiparticles are not part of the exact fermionic excitation spectrum of the theory. Instead one finds continuous spectral weight with power law scaling excitations. Moreover, at low energies there are three such excitations at three different Fermi surfaces, two with a low energy Green's function G∼(ω−v k)^−1/2 and one with G∼|ω+k|^−1/3. We proceed to study this model with a finite N_f but still neglecting fermionic loops with 3 or more vertices, motivated by multiloop cancellations at large k_F. We still find a non-Fermi liquid but with a different IR spectrum.

I describe how, within the group field theory (GFT) formalism for quantum gravity, we can: 

1) provide a candidate description of the quantum building blocks of spacetime, bringing together ideas and mathematical structures from other quantum gravity formalisms;

2) apply powerful tools from quantum field theory, like the (perturbative and non-perturbative) renormalization group, to establish the quantum consistency of given GFT models and to study their continuum limit and phase structure;

3) extract, from the full theory, an effective cosmological dynamics for the universe described as a quantum condensate of GFT building blocks; in the simplest approximation, this dynamics reduces to the Friedmann equations at large scales but replaces the classical big bang singularity with a quantum bounce.

Galaxy mergers are a standard aspect of galaxy formation and evolution, and most (likely all) large galaxies contain supermassive black holes.  As part of the merging process, the supermassive black holes should in-spiral together and eventually merge, generating a background of gravitational radiation in the nanohertz to microhertz regime.  Processes in the early Universe such as relic gravitational waves and cosmic strings may also generate gravitational radiation in the same frequency band.  An array of precisely timed pulsars spread across the sky can form a galactic-scale gravitational wave detector in the nanohertz band.  I describe the current efforts to develop and extend the pulsar timing array concept, together with recent limits which have emerged from North American and international efforts to constrain astrophysical phenomena at the heart of supermassive black hole mergers.

Non-abelian anyons have drawn much interest due to their suspected existence in two-dimensional condensed matter systems and for their potential applications in quantum computation. In particular, a quantum computation can in principle be realized by braiding and fusing certain non-abelian anyons. These operations are expected to be intrinsically robust due to their topological nature. Provided the system is kept at a
temperature T lower than the spectral gap, the density of thermal excitations is suppressed by an exponential Boltzman factor. In contrast to the topological protection however, this thermal protection is not scalable: thermal excitations do appear at constant density for any non-zero temperature and so their presence is unavoidable as the size of the computation increases. Thermally activated anyons can corrupt the encoded
data by braiding or fusing with the computational anyons.

In the present work, we generalize a fault-tolerant scheme introduced by Harrington for the toric-code to the setting of non-cyclic modular anyons. We prove that the quantum information encoded in the fusion space of a non-abelian anyon system can be preserved for arbitrarily long times with poly-log overhead. In particular, our model accounts for noise processes which lead to the creation of anyon pairs from the vacuum, anyon diffusion, anyon fusion as well as errors in topological charge measurements.
Related Arxiv #: arXiv:1607.02159

Scale invariant transfer matrices and Hamiltonians Abstract We investigate the possibility of strictly scale invariant transfer matrices in quantum spin chains based on a certain planar algebra, both as operators and as quadratic forms.