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Hopfield neural network dynamics in open quantum systems

Periodic Reporting for period 1 - HopeQNet (Hopfield neural network dynamics in open quantum systems)

Reporting period: 2017-11-03 to 2019-11-02

In the last 10 years Machine Learning has been one of the hottest topic in Science because of its impact on many aspects of our society, with applications ranging from computer vision and time series prediction in Finance, to computer-aided diagnosis in Medicine.

More recently, scientists (and physicists in particular) are wondering whether Machine Learning algorithms can benefit of the counterintuitive laws of Quantum Physics. This expectation is supported by the recent progresses in Quantum computing, which include, for instance, the achievement of quantum supremacy claimed by Google in October 2019.

Unfortunately, Research in Quantum Machine Learning is only at the embryonal stage and many questions still need to be answered: which is the best theoretical framework to incorporate Machine Learning in the quantum domain? Can simple models of neural networks be implemented in quantum many-body systems/simulators? Do Machine Learning algorithms benefit in some way of quantum effects?

During my project I contributed to this rapidly growing research field, working at the interface between Statistical Physics and Quantum many-body systems. The overall objectives of the action have been:

(i) Formulating a solid theoretical framework based on open quantum systems to incorporate the unitary dynamics typical of quantum systems and the intrinsic non-linear evolution of neural networks, using the Hopfield model of associative memory as a benchmark.

(ii) Investigating with the analytical and numerical methods of Statical and quantum many-body physics the dissipative dynamics of the open quantum Hopfield model defined in (i), also addressing possible experimental scenarios for its realisation in a lab.

These concrete steps will contribute in the long term to understand whether Machine Learning can benefit of the laws of Quantum Physics and which platforms for quantum simulation are the most effective for this purpose.
During the project I have addressed the objectives (i) and (ii) reported above, thanks also to a regular interaction with my Supervisor and other members of the group at UoN involved in the research.

-Research summary for Objective (i)

During the first months of my work as a MC fellow, I have been involved in a training-through-research activity where I learnt the basic facts of open quantum systems (that I was not familiar with). On my side I have contributed bringing my previous experience on disordered systems both in classical and quantum systems.

Exploiting this unique background, I have developed a theoretical framework, based on open quantum systems, to deal with a simple model of neural network (the Hopfield model) and generalise it at the quantum level. Later I have applied disordered systems techniques to study the typical properties and to identify the phase diagram of the model. In this way I have been able to identify a novel phase of the quantum generalisation of the model, not present at the classical level.

Additionally, I have used similar techniques to investigate the relevant timescales to approach stationarity in this open quantum system and how the retrieval phase of the associate memory can benefit of quantum effects.

-Research summary for Objective (ii)

Here the goal has been to identify suitable platforms for the quantum simulation of associative memories, in the attempt to go beyond the purely theoretical framework proposed in (i). Among the many possible candidates, that include atoms in multimodal cavities and superconducting qubits placed in transmission line resonators, we have focused on trapped ions, a versatile platform where long-range interactions can be engineered.

As in many other quantum systems, we had to understand the role of the environment and its influence on the possibility of retrieving memory patterns encoded in the trapped ions simulator. We have developed a perturbative framework that gave us the possibility to treat effectively the strong coupling interaction responsible of memory retrieval in the isolated case.

The outcome of this approach has been a non-equilibrium effective dynamics that I have investigated by means of kinetic Monte Carlo simulations. This analysis has allowed to identify the best regime to obtain retrieval in the open quantum system under exam, setting useful thresholds for future implementations in lab.
During the project I investigated the possibility of implementing associative memories in quantum many-body systems, which is by itself a novel and timely research topic.
My main initial contribution to the field has been the formulation of a theoretical framework based on open quantum systems that allowed to combine quantum effects with the non-linear dynamics typical of artificial intelligence. As a key technical advance, we have pioneered analytical disordered systems techniques in the field of open quantum systems, that gave us the possibility to investigate the phase diagram of the open quantum Hopfield network, highlighting the regimes where the retrieval of the memory patterns is possible.

Later on we have explored viable modern quantum platforms for the implementation of this associative memory in a lab, exploiting the versatility of trapped ions, that naturally exhibit the long-range interactions needed for a neural network to work. More importantly we have established solid analytical and numerical framework to study the effect of the environment on this system, that will guide experimentalists in the correct regime of parameters to make the quantum many-body system useful as an associative memory.

This project has been a preliminary concrete step in the research field of quantum machine learning and it will hopefully lead in the long term to the development of quantum technologies that mimic the behaviour of the human brain, placing EU in a leading position in the race to get commercial quantum platforms for computation.
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