Giuseppe Di Molfetta

• Quantum Algorithms and fault tolerance 

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The aim of this internship is to study fault resistant quantum codes, in particular for searching or optimization, on quantum walk/quantum cellular automata based-architectures. There is a huge interest in designing fault-tolerant quantum protocols, especially with limited quantum resources. It is well known that topological protection can be used to encode and process quantum information in a robust way. Using the fact that certain three-dimensional quantum walks possess two-dimensional topologically protected states, we will investigate using numerical simulations and analytical techniques whether, e.g., quantum search can be done optimally in a fault tolerant way. 

 

The successful candidate, during the internship, will profit of an existing and consolidated scientific collaboration with the CPT (centre de physique theorique) à Marseille and the established cooperation between our group and the ULB (Université Libre de Bruxelles, Centre for Quantum Information and Communication). Mobility will be encouraged. 

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• Computational tasks on quantum distributed architectures

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Mid-terms quantum architectures will be likely distributed. Similarly to modern high-performance computing infrastructures, many quantum processors, memories and storage units can be inter-connected via quantum communication networks, and the computational tasks be solved by adopting a distributed approach. However, distributed quantum computing remains challenging from different points of view. Some examples are quantum teleportation as a mean to transfer quantum information between interconnected devices, make secure quantum networks or abstracting and optimizing the execution of the quantum algorithms, based on the characteristics of the underlying distributed system. Our proposal will focus on this last issue. Although the literature is quite scarce, few problems have already been solved in a distributed fashion, as for instance finding the triangle finding problem. Most of them are based on multiparty entanglement and one pivotal question is how to generate efficiently robust non-local states to solve a specific problem. In this respect, the quantum LOCAL model recently introduced by Le Gall has been proven advantageous to build graph states. Quite interestingly such models are nothing but synchronous network of QCA (SQN), from a mathematical perspective. Indeed quantum cellular automata (QCA), usually defined on spatial grids, can be extended to graphs, in the same fashion cellular automata have been extended to Boolean networks. Each node of the graph hosts a finite number of states and the interaction is local and synchronous between connected nodes.

 

Theory of QCA is well-developed and their realizability as well and we believe this may help to identify new fundamental strategies to construct multiparty entanglement and to elaborate quantum-enhanced distributed algorithm. Moreover, one major limitation is that all known (few) results are fault-free and nowadays we do not know how errors propagate in the network, how they depends on the topology and affect the computational power. This will be one of the main open problems to which proposal is aiming to give an answer. 

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The successful candidate, during the internship, will profit of an existing and consolidated scientific collaboration with the Distributed Algorithms groups at LIS (DALGO) and the Department of Mathematics in Marseille (I2M) on the quantum error codings side. 

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• Quantum simulation of Physical theories

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Quantum simulate physical fundamental theories is the first milestone in order to model strongly correlated materials or complex behavior. We collected a long series of results, rigorously proving that quantum walks ought to be used to simulate quantum particles coupled to an arbitrary field (electromagnetic,  weak, strong interaction), where the field is externally encoded by hand in the local rules coupling next neighboring cells. In all these cases, the corresponding field interaction is external, not dynamical. Yet, this result paved the way to quantum simulate more realistic quantum systems. 

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Using interacting Quantum Walks/QCA we are aiming to simulate strongly correlated systems and develop quantum codes to describe such systems at several energy/length scales.  

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The successful candidate, during the internship, will profit of an existing and consolidated scientific collaboration with the CPT (centre de physique théorique) à Marseille and the established cooperation between our group and Saclay Université (Pablo Arrighi), Sorbonne Université (Fabrice Debbasch), LPS ENS (Marc Brachet) and the University of Valencia (Armando C. Perez)

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• Quantum numerical methods and differential geometry 

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Quantum Walks have taken on great importance in many areas of fundamental computer science, particularly in quantum simulation. We proved, in full rigour, using numerical analysis methods, that they efficiently compute geodesics on arbitrary manifolds and converge to arbitrary linear PDE.  These PDE may describe a very broad spectrum of physical phenomena, particularly wave propagation in curved spacetime, in spite of QW fixed background grid.

 

We are aiming to finally construct a universal and minimal set of local quantum gates from which to build arbitrary PDE, aka a minimal quantum circuit for spacetime dynamics. Moreover we would like to study the robustness of such circuits to local noise.

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The successful candidate, during the internship, will profit of an existing and consolidated scientific collaboration with the CPT (centre de physique théorique) à Marseille and the established cooperation between our group and Saclay Université (Pablo Arrighi), Sorbonne Université (Fabrice Debbasch), LPS ENS (Marc Brachet) and the University of Valencia (Armando C. Perez)

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• Quantum Machine Learning  

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We have designed and studied the first amplitude amplification based multi-armed bandit algorithm. The basic idea of a classical bandit algorithm is that an agent, disposing only partial information about the system, has to efficiently identify the arm with the largest expected reward. In the quantum version of such a problem, that we call quantum bandit, the arms are accessed in superposition by applying a quantum oracle and we show that we can find the best arm quadratically faster than in the classical case.  We also introduced a hybrid quantum-classical perceptron algorithm with lower complexity and better generalization ability than the classical perceptron. We showed a quadratic improvement over the classical perceptron in both the number of samples and the margin of the data. In general the big picture is to find machine learning scenario where quantum properties may guarantee a clear advantage respect to the classical case. 

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The successful candidate, during the internship, will profit of an existing and consolidated scientific collaboration with the Machine Learning groups at LIS (QARMA). 

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