# Research

Research efforts in the MathQuantum RTG draw from a wide range of areas in Pure and Applied Mathematics: Algebra/Representation Theory, Nonlinear Analysis, PDEs, Scientific Computing, and Stochastic Analysis, as well as Data Science and Machine Learning. These areas play a vital role in Quantum Information Science (QIS), and our group focuses on three specific quantum areas: Algorithms, Cryptography, and Dynamics/Systems.

Our faculty come from the Departments of Mathematics, Chemistry, Computer Science, and Physics, as well as QuICS (Joint Center for Quantum Information and Computer Science), JQI (Joint Quantum Institute), IPST (Institute for Physical Science & Technology), and QTC (Quantum Technology Center).

## Quantum Algorithms

The field of quantum algorithms studies the possibility of solving computational problems dramatically faster with computers that store and process information quantum-mechanically. Such exponential quantum speedup often relies on the “structure” of a problem arising from concepts such as algebra and graph theory. This shows the essential connection between mathematics (e.g., algebraic problems) and quantum information. Within this area, we aim to address central questions such as:

- From a theoretical point of view, can we offer exponential quantum speedups for general differential equations?
- Can we investigate end-to-end quantum applications for differential equations and related real-world problems? In particular, how can we encode the equations and understand what useful properties can be efficiently extracted from the output state?
- Can one construct Quantum Fourier Transforms (QFT) for compact Lie groups? The QFT is probably the most important ingredient for dramatic quantum speedups.

## Quantum Cryptography

Cryptography studies interactive information processing in scenarios where not all participants are trusted. Of all the fields of theoretical computer science, it is one of the most impacted by the development of quantum technology. On the one hand, if Shor’s algorithm were ever implemented on a large-scale quantum computer, it would destroy the security of the most commonly used public-key encryption algorithms. This has motivated, in the last several years, a new stage of development in cryptography: “post-quantum cryptography.” On the other hand, quantum information science offers an alternative approach to cryptosystem design. Quantum cryptography exploits the unique properties of quantum devices to secure information. In the present day, the original scenario of concern in quantum cryptography — interaction between two parties in a minimal physical model — is well understood at a theoretical level. A natural next direction for the field is to improve cryptographic results in multi-party settings (e.g., voting, secret sharing, etc.) to see if they can reach the same level of security as in the two-party setting. But perhaps an even more interesting direction (from the mathematical perspective) is: What new cryptographic capabilities can we achieve if additional assumptions are introduced? Position-based cryptography is based on the additional physical assumption that communication between distant parties cannot travel faster than the speed of light. This subfield of quantum cryptography has been under development for several years, but basic problems remain open. This project will aim to tackle central questions such as:

- Is it possible to verify the location of an untrusted party in space?
- How could the basic construction of trapdoor claw-free functions developed by Prof. Urmila Mahadev be reconfigured to achieve more powerful results? This question is part of a rich subfield originated by Mahadev et al. on quantum cryptography in a classical communication model with post-quantum cryptographic assumptions.

## Quantum Dynamics/Systems

The study of quantum phenomena has acquired significant momentum in recent years. Two particular areas of quantum research bearing exciting projects for students and postdocs are rare events in quantum chemical systems and quantum oscillators. Rare events in quantum systems are principally different than in classical ones. They happen not due to a streak of random forcing driving the system out of a metastable state but due to the wave nature of quantum particles enabling quantum tunneling. The development of a mathematical framework for the description of rare events in quantum chemical systems, relies on a quantum counterpart of Transition Path Theory (TPT), as well as numerical methods for sampling and quantification. Within this area, we aim to address questions such as the concept of a quantum analog of the committor function, the key function of the TPT. We also investigate applications to real-life molecular systems. The interest in quantum oscillators is promoted by envisioning the development of quantum technologies. While the dynamics of the linear quantum oscillator are well-understood, the dynamics of nonlinear quantum oscillators are the subject of active research. Phenomena such as synchronization, oscillation collapse, and critical response to external forcing have been studied. Single quantum van der Pol and Rayleigh oscillators served as the prototypical models. The study of quantum oscillator arrays will be the subject of our research. Some fundamental questions that our team is poised to address are:

- What are the quantum analogs of a bistable oscillator and oscillator arrays?
- What are the transition mechanisms between the attractors?
- How can they be found numerically? These questions are motivated by recent successes in the study of circular arrays of classical nonlinear oscillators with external periodic forcing and small white noise. The team will also investigate many-body theories (including Fock-space techniques) as well as quantum Boson dynamics of atomic gases beyond mean-field approaches; the electronic transport in novel 2D materials and topological insulators as well as models of quantum fluids; and the connections between the physics/engineering of devices and mathematical topics such as non-Euclidean geometry and spectral graphs.