Mathematics and Quantum Technology: Talk titles and abstracts
Weibin Li (University of Nottingham)
Title: Hilbert space fragmentation in a driven-dephasing Rydberg atom array
Abstract: We investigate the onset and mechanism of Hilbert space fragmentation (HSF) in a chain of strongly interacting Rydberg atoms subjected to local dephasing. It is found that the emergence of multiple long-lived metastable states is fundamentally tied to the HSF of the driven-dephasing Rydberg atom system. We demonstrate that the manifesting HSF is captured by a dephasing PXP model that supports multiple degenerate zero modes. These modes form disconnected block-diagonal subspaces of maximally mixed states, which consist of many-body spin states sharing the same symmetry. A key result is the identification of the underlying symmetry in the HSF, where the conserved quantities in each subspace are defined by the consecutive double excitation addressing operator. Moreover, we show explicitly that the number of fragmented Hilbert spaces grows exponentially with the chain length, following a modified Fibonacci sequence. Our work provides insights into many-body dynamics under dynamical constraints and opens avenues for controlling and manipulating HSF in Rydberg atom systems.
Matthias Caro (University of Warwick)
Title: Privacy and Verification for Quantum Learning
Abstract: Quantum data access and quantum processing can make certain classically intractable learning tasks feasible. However, advanced quantum capabilities will only be available to a select few for the foreseeable future. Thus, schemes that allow a client to privately and verifiably delegate learning to an untrusted quantum server are required to facilitate widespread access to quantum learning advantages. I will discuss how one can mathematically formalise the challenges of privacy and verification, drawing on frameworks from cryptography and interactive proofs, and I will present some recent progress along these lines.
Vedran Sohinger (University of Warwick)
Title: Gibbs measures of 1D quintic nonlinear Schrödinger equations as limits of many-body quantum Gibbs states
Abstract: Gibbs measures of nonlinear Schrödinger equations (NLS) are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We study the problem of the derivation of Gibbs measures as mean-field limits of Gibbs states in many-body quantum mechanics.
In earlier joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein, we studied this problem for variants of the cubic NLS with defocusing (positive) interactions. The latter models physically correspond to pair interactions of bosons. In these works, the problem was studied in dimensions d=1,2,3.
In this talk, I will explain how one can obtain an analogous result for the 1D quintic NLS, which corresponds to three-body interactions of bosons. In this setting, we consider focusing interactions, due to which we need to add a truncation in the mass and rescaled particle number. Our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the 1D quintic NLS. This is joint work with Andrew Rout.
Randa Herzallah (University of Warwick)
Title: Fully Probabilistic Control for Quantum Systems
Abstract: This talk presents a generalised fully probabilistic control (FPC) solution that addresses functional uncertainties and develops a decentralised control framework for large-scale dynamical systems. The focus will be on two key areas: (1) the generalisation of FPC, (2) the application of FPC to quantum systems. The first part of the talk will introduce the concept of fully probabilistic control and demonstrate how it differs from deterministic control approaches, emphasising its explorative nature. We will discuss the advantages of a cautious and dual design strategy over the certainty equivalence design and highlight the importance of accounting for functional uncertainties in control systems. In the second part, we will delve into applying the FPC to quantum systems, showcasing the potential benefits and challenges of implementing FPC in such complex environments.
Animesh Datta (University of Warwick)
Title: Credibility of real-world quantum computers and simulators
Abstract: The value of quantum computers and simulators lie in solving efficiently and correctly problems that are hard classically. This is particularly relevant for simulation, sampling, and optimisation problems whose solutions cannot be verified efficiently classically, unlike integer factoring. I will present methods that can provide, for a given problem, an upper bound on the variation distance between an experimentally obtained output from a noisy quantum computer and the ideal output from a noiseless device. I will show how this can be achieved without the inherently infeasible method of simulating the quantum computation classically. I will highlight the vital interplay between empirical experimental observations and mathematical assumptions.
Grega Saksida (University of Warwick)
Title: Correlation functions of the quantum XY model
Abstract: The quantum XY model describes the behaviour of spin particles on a lattice. We study the spin-1/2 system, where at each site the spin measured along any axis can be in two states: "up" or "down". The model is quantum, meaning the spin at each site is a complex linear combination of the two possible "measured" states. The spins at adjacent sites interact in a way that favours alignment. A natural question is whether this alignment persists across the entire system. This motivates the study of so-called correlation functions.
In this talk, I will define the quantum XY model, and present some new results on the correlation functions. I will explain how we derived these results by representing the quantum XY model as an Ising model with some additional interactions. This is an ongoing joint work with Vedran Sohinger and Daniel Ueltschi.