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Quantum algorithms have been found which are able to solve important problems exponentially faster than any known classical algorithm. The most well known example is Shor's algorithm which would be able to break all RSA encryption if fault tolerant quantum computers existed for it to be run on. It is currently not believed that quantum computers will be able to efficiently solve NP-Complete problems, but the answer is still unknown. I present a novel quantum algorithm which is able to solve 3-SAT along with numerical simulations to see how it performs on small instances, but new methods of analyzing the complexity of quantum algorithms will need to be developed before we can say exactly how it performs. This will be the goal of my PSI essay.

Zoom Link: https://pitp.zoom.us/j/95795655548?pwd=aU5jNFhFZHEzUWpLK1FvWjd2Q1c2Zz09

We describe a geometric framework to study Newton's equations on infinite-dimensional configuration spaces of diffeomorphisms and smooth probability densities. It turns out that several important PDEs of hydrodynamical origin can be described in this framework in a natural way. In particular, the so-called Madelung transform between the Schrödinger-type equations on wave functions and Newton's equations on densities turns out to be a Kähler map between the corresponding phase spaces, equipped with the Fubini-Study and Fisher Rao information metrics. This is a joint work with G.Misiolek and K.Modin.

Zoom Link: https://pitp.zoom.us/j/92283820673?pwd=dXQzYWppaUdBVnhtVG1TZTFQQmx4UT09

We show that if a massive (or charged) body is put in a quantum superposition of spatially separated states in the vicinity of any (Killing) the mere presence of the horizon will eventually destroy the coherence of the superposition. This occurs because, in effect, the long-range fields sourced by the superposition registers on the horizon which forces the radiation of entangling soft gravitons/photons through the horizon. This allows the horizon to harvest “which path” information about the superposition. The electromagnetic decoherence arises only when the superposed particle carries electric charge. However, since all matter sources gravity, the quantum gravitational decoherence applies to all superpositions. We provide estimates of the decoherence time for such quantum superpositions. Additionally, we show that this decoherence is distinct from--and larger than--the decoherence resulting from the presence of thermal radiation from the horizon (i.e. Hawking/Bunch-Davies/Unruh radiation). We believe that the fact that Killing horizons will eventually decohere any quantum superposition may be of fundamental significance for our understanding of the nature of black holes and horizons in quantum gravity. (Based on arXiv:2205.06279 and arXiv: 2301.00026).

Zoom link:  https://pitp.zoom.us/j/95263116767?pwd=amZ6SkROV1lxckVpdzhNbFhYc1ZiQT09

Tidal interactions in coalescing binary neutron stars modify the dynamics of the inspiral, and hence imprint a signature on their gravitational-wave (GW) signals in the form of an extra phase shift. We need accurate models for the tidal phase shift in order to constrain the supranuclear equation of state from observations. In previous studies, GW waveform models were typically constructed by treating the tide as a linear response to a perturbing tidal field. In this work, we incorporate non-linear corrections due to hydrodynamic three- and four-mode interactions and show how they can improve the accuracy and explanatory power of waveform models. We set up and numerically solve the coupled differential equations for the orbit and the modes, and analytically derive solutions of the system's equilibrium configuration. Our analytical solutions agree well with the numerical ones up to the merger and involve only algebraic relations, allowing for fast phase shift and waveform evaluations for different equations of state over a large parameter space. We find that, at Newtonian order, nonlinear fluid effects can enhance the tidal phase shift by  >~ 1 radian at a GW frequency of 1000 Hz, corresponding to a  10−20%  correction to the linear theory. The scale of the additional phase shift near the merger is consistent with the difference between numerical relativity and theoretical predictions that account only for the linear tide. Nonlinear fluid effects are thus important when interpreting the results of numerical relativity, and in the construction of waveform models for current and future GW detectors.

Zoom link:  https://pitp.zoom.us/j/91730134841?pwd=U0JXNHhFbWdQYno3aUpDWHRxWEtkUT09

We present an efficient machine learning (ML) algorithm for predicting any unknown quantum process over n qubits. For a wide range of distributions D on arbitrary n-qubit states, we show that this ML algorithm can learn to predict any local property of the output from the unknown process, with a small average error over input states drawn from D. The ML algorithm is computationally efficient even when the unknown process is a quantum circuit with exponentially many gates. Our algorithm combines efficient procedures for learning properties of an unknown state and for learning a low-degree approximation to an unknown observable. The analysis hinges on proving new norm inequalities, including a quantum analogue of the classical Bohnenblust-Hille inequality, which we derive by giving an improved algorithm for optimizing local Hamiltonians. Overall, our results highlight the potential for ML models to predict the output of complex quantum dynamics much faster than the time needed to run the process itself.

Zoom link:  https://pitp.zoom.us/j/93857777354?pwd=c044blZuQVhLS200ME4vN25uaGJudz09