Ergodic theorems for continuous-time quantum walks on crystal lattices and the torus

We give several quantum dynamical analogs of the classical Kronecker-Weyl theorem, which says that the trajectory of free motion on the torus along almost every direction tends to equidistribute. As a quantum analog, we study the quantum walk $\exp(-i t \Delta) \psi$ starting from a localized initial state $\psi$. Then the flow will be ergodic if this evolved state becomes equidistributed as time goes on. We prove that this is indeed the case for evolutions on the flat torus, provided we start from a point mass, and we prove discrete analogs of this result for crystal lattices. On some periodic graphs, the mass spreads out non-uniformly, on others it stays localized. Finally, we give examples of quantum evolutions on the sphere which do not equidistribute.Comment: 26 pages, 4 figure

Event generation on quantum computers

The synthesis of high quality simulated data from event generators is essential in the search for new physics at collider experiments. Modern event generator algorithms use Monte Carlo processes to simulate the evolution of an event from the collision of high energy particles to the formation of long-lived particles. One of the major building blocks of the event generation process is the QCD parton shower. However, despite being a key aspect of modern event generation, the core algorithms which simulate the showering process have remained unchanged since the 1980s, and will become a limiting factor as we move to an era of higher energy and higher luminosity experiments. With the rapid development of quantum computation, dedicated algorithms are required which exploit the potential that quantum computing provides to address problems in high energy physics. In this thesis, we present three novel quantum algorithms for the simulation of a QCD parton shower. The first algorithm provides a proof-of-principle, classical Monte Carlo inspired approach with the ability to simulate two shower steps of a collinear QCD model. By exploiting the compact circuit architecture of the quantum walk, one can drastically reduce the quantum resources required to simulate a shower step. The second algorithm shows that, in this framework, the quantum parton shower can be extended to simulate realistic shower depths whilst using fewer quantum resources. Finally, the third algorithm utilises a discrete QCD approach to parton showering to include kinematics in the shower, simulating a dipole cascade. In this construction, the algorithm has achieved the first data comparison between synthetic data produced using a Noisy Intermediate-Scale Quantum (NISQ) device, and ``real-life" archival collider data from the Large Electron Positron collider. The three algorithms represent the development of quantum algorithms for the simulation of parton showers, acting as a first step towards a fully quantum simulation of a high energy collision event.Open Acces

Topological classification of symmetric quantum walks. Discrete symmetry types and chiral symmetric protocols

In this thesis, we study the topological classification of symmetric quantum walks. These describe the discrete time evolution of single quantum particles on the lattice with additional locally acting symmetries. The thesis consists of three parts: In the first part, we discuss discrete symmetry types for self-adjoint and unitary operators from an abstract point of view, i.e. without assuming an underlying physical model. We reduce any abstract finite group of involutive symmetries and their projective representations to a smaller set of symmetry types, eliminating elements that are redundant for topological classifications. This reduction process leads to the well-known tenfold way for self-adjoint operators, and for unitary operators, we identify 38 non-redundant symmetry types. For these, we define a symmetry index, which labels equivalence classes of finite-dimensional representations up to trivial direct summands. We show that these equivalence classes naturally carry a group structure and finish the discussion by explicitly computing the corresponding index groups for all non-trivial symmetry types. Second, we develop a topological classification for symmetric quantum walks based on the symmetry index derived in the first part. We begin without a locality condition on the unitary time evolution operator but only assume an underlying discrete spatial structure. Unlike continuous-time systems, quantum walks exhibit non-gentle perturbations, i.e. local or compact perturbations that cannot be undone continuously. Using the symmetry index, we provide a complete topological classification of such perturbations of unitary operators on any lattice or graph. We add a locality condition on the one-dimensional lattice and detail the implications of such assumption on the classification. The spatial structure of the one-dimensional lattice allows us to define the left- and right symmetry index, which characterise a walks topological properties on the two half-chains. The sum of these two indices equals the overall symmetry index, which provides a lower bound on the number of symmetry protected eigenstates of the walk. For the symmetry types of the tenfold way, a subset of three different symmetry indices is complete with respect to norm-continuous deformations and compact perturbations. In the third part, we consider quantum walk protocols instead of single time-step unitaries. We show that any unitary operator with finite jump length on a one-dimensional lattice can be factorised into a sequence of shift and coin operations. We then provide a complete topological classification of such protocols under the influence of chiral symmetry. The classification is in terms of the half-step operator, i.e. the time evolution operator at half of the driving period, which is singled out by the chiral symmetry. We also show that a half-step operator can be constructed for every chiral symmetric single time-step unitary without a pre-defined underlying protocol. This renders the classification via the half-step operator valid for periodically driven continuous-time (Floquet systems), discretely driven protocols, and single time-step quantum walks

On Quantum Speedups for Nonconvex Optimization via Quantum Tunneling Walks

Classical algorithms are often not effective for solving nonconvex optimization problems where local minima are separated by high barriers. In this paper, we explore possible quantum speedups for nonconvex optimization by leveraging the $global$ effect of quantum tunneling. Specifically, we introduce a quantum algorithm termed the quantum tunneling walk (QTW) and apply it to nonconvex problems where local minima are approximately global minima. We show that QTW achieves quantum speedup over classical stochastic gradient descents (SGD) when the barriers between different local minima are high but thin and the minima are flat. Based on this observation, we construct a specific double-well landscape, where classical algorithms cannot efficiently hit one target well knowing the other well but QTW can when given proper initial states near the known well. Finally, we corroborate our findings with numerical experiments

How Fast Do Quantum Walks Mix?

The fundamental problem of sampling from the limiting distribution of quantum walks on networks, known as mixing, finds widespread applications in several areas of quantum information and computation. Of particular interest in most of these applications is the minimum time beyond which the instantaneous probability distribution of the quantum walk remains close to this limiting distribution, known as the "quantum mixing time". However, this quantity is only known for a handful of specific networks. In this Letter, we prove an upper bound on the quantum mixing time for almost all networks, i.e. the fraction of networks for which our bound holds, goes to one in the asymptotic limit. To this end, using several results in random matrix theory, we find the quantum mixing time of Erdös-Renyi random networks: networks of n nodes where each edge exists with probability p independently. For example, for dense random networks, where p is a constant, we show that the quantum mixing time is O(n3/2+o(1)). In addition to opening avenues for the analytical study of quantum dynamics on random networks, our work could find applications beyond quantum information processing. Owing to the universality of Wigner random matrices, our results on the spectral properties of random graphs hold for general classes of random matrices that are ubiquitous in several areas of physics. In particular, our results could lead to novel insights into the equilibration times of isolated quantum systems defined by random Hamiltonians, a foundational problem in quantum statistical mechanics.info:eu-repo/semantics/publishe