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Interacting quantum particles can form non-trivial states of matter characterized by topological order, which features several unconventional properties such as topological degeneracy and fractionalized quasiparticles. In addition, it also provides a promising platform for realizing quantum computing in a robust manner. In this series of lectures, I will introduce the basics of topological order and its connection to quantum computing from various aspects involving lattice models, symmetry, and entanglement structure. Several frontier topics such as fracton topological phases, self-correcting quantum memory, state preparation, and quantum LDPC codes will be briefly discussed.

Kronecker coefficients appear in representation of the symmetric group in the decomposition of tensor products of irreducible representations. They are notoriously difficult to compute and it is a long standing problem to find a combinatorial expression for them. 

We study the problem of computing Kronecker coefficients from quantum computational perspective. First, we show that the coefficients can be expressed as a dimension of a subspace given by intersection of two commuting, efficiently implementable projectors and relate their computation to the recently introduced quantum approximate counting class (QAPC). Using similar construction, we show that deciding positivity of Kronecker coefficients is contained in QMA. We give similar results for a related problem of approximating row sums in a character table of the symmetric group and show that its decision variant is in QMA. We then discuss two quantum algorithms - one that samples a distribution over squared characters and another one that approximates normalized Kronecker coefficients to inverse-polynomial additive error. We show that under a conjecture about average-case hardness of computing Kronecker coefficients, the resulting distribution is hard to sample from classically. 

Our work explores new structures for quantum algorithms and improved characterization of the quantum approximate counting.

Joint work with David Gossett, Sergey Bravyi, Anirban Chowdhury and Guanyu Zhu 

Zoom link:  https://pitp.zoom.us/j/95976938016?pwd=eDV3TXZReHo5UHdvZ0hIbkhXOFcxQT09

The question of whether the holomorphic collinear singularities of graviton amplitudes define a consistent chiral algebra has garnered much recent attention. I will discuss a version of this question for infinitesimal perturbations around the self-dual sector of 4d Einstein gravity. The singularities of tree amplitudes in such perturbations do form a consistent chiral algebra. However, at loop level new poles are generated, the simplest of which are described the 1-loop effective graviton vertex. These quantum corrections violate associativity of the operator product. I will argue that this failure can be traced to an anomaly in the twistor uplift of self-dual gravity. Associativity can be restored by coupling to an unusual 4th-order gravitational axion, which cancels the anomaly by a Green-Schwarz mechanism. Alternatively, the anomaly vanishes in certain theories of self-dual gravity coupled to matter, including in self-dual supergravity.

Zoom link:  https://pitp.zoom.us/j/91979544537?pwd=Ykw4Q3lRK2ZicVhCMDhlNUxFaHhlUT09

In this talk I’ll highlight the existence of gaps in the spectrum of GWs poorly explored by current observations and that may contain information about BSM physics or primordial cosmology. I’ll focus on the muHz gap, and explain how to use the resonance absorption of GWs by binary systems (as the Earth-Moon system or binary pulsars) to access this band. I’ll also highlight the potential of high frequency (\omega > kHz) GWs. The focus of this second part will be the use of cavities to detect these signals, together with a brief discussion of sources and what can be learned from them.

Zoom Link:  https://pitp.zoom.us/j/94501758748?pwd=amFsTjV5Z1c5SHJrVVBCanVCRHRMUT09