Periodic Reporting for period 1 - CoQHoNet (Connecting Quantum Hopfield Networks)
Berichtszeitraum: 2023-06-01 bis 2025-05-31
Nowadays Classical Neural Networks (NNs) are widely used in Machine Learning (ML) for tasks like pattern recognition, big data analysis, and digitalization. At the same time, the development of quantum technologies is seen as a promising advancement over classical ML, leading to growing interest in quantum NNs as the foundation of quantum ML. However, a comprehensive framework for quantum NNs is still lacking.
In this context, this project aims to contribute to define the core features of quantum NNs and to explore their practical applications. Specifically, the project focuses on quantum versions of associative memory-type NNs, with the Hopfield NN serving as a key example. Associative memories can perform simple tasks like pattern retrieval, but they also play crucial roles in more complex architectures, such the Boltzmann Machines.
Current quantum associative memory research leverages quantum spin NNs and bosonic systems, modelling them as Markovian (or memory-less) open quantum systems. These models are especially relevant to condensed matter and photonic implementations. The goal of this project consists in integrating these proposals into a broader theoretical framework, particularly bridging the gap between Markovian quantum Hopfield-type NNs and the more general formalism of quantum maps. This framing is expected to provide valuable insights to address important questions, such as storing non-classical states, understanding non-Markovian effects on retrieval tasks, and characterizing the storage capacity.
In this context, this project aims to contribute to define the core features of quantum NNs and to explore their practical applications. Specifically, the project focuses on quantum versions of associative memory-type NNs, with the Hopfield NN serving as a key example. Associative memories can perform simple tasks like pattern retrieval, but they also play crucial roles in more complex architectures, such the Boltzmann Machines.
Current quantum associative memory research leverages quantum spin NNs and bosonic systems, modelling them as Markovian (or memory-less) open quantum systems. These models are especially relevant to condensed matter and photonic implementations. The goal of this project consists in integrating these proposals into a broader theoretical framework, particularly bridging the gap between Markovian quantum Hopfield-type NNs and the more general formalism of quantum maps. This framing is expected to provide valuable insights to address important questions, such as storing non-classical states, understanding non-Markovian effects on retrieval tasks, and characterizing the storage capacity.
The first part of the action focused on quantum generalizations of Hopfied Neural Networks that are made of spins ½ particles, and which undergo an evolution ruled by the GKS-Lindblad equation. The model, already introduced in previous research works, features a dissipative part which embeds the Hopfield dynamics, and a coherent driving competing with it. It was already shown that the quantum model, under a mean field hypothesis, undergoes a memory retrieval phase in a certain parameter regime, as well as a novel phase that, due to the competition between dissipation and coherent evolution, turns out to host limit cycles. During this project, the validity of such a result was analysed and rigorously proven, by adopting techniques from operator algebras. As a first step in this direction, the model has been generalised to describe the dynamics of particles in d-dimensional Hilbert spaces, satisfying generic algebras. Here, the mean field theory is proven to hold true in the thermodynamic limit, this confirming the validity of the phase diagram of the generalised open quantum Hopfield model. Importantly, there were derived also bounds to the validity of the mean field approximation in the more realistic scenario of finite size systems. Eventually, by adopting the same model and similar techniques, the impact of quantum fluctuations on the model has been analysed.
In the second part of the action, the research activity focused on the more general formulation of quantum associative memories in terms of completely positive and trace preserving (CPTP) maps, which are key tools to describe generic open quantum systems. Here, by focusing on general properties of CPTP maps, and on the features characterizing classical associative memories, we developed a comprehensive framework for a quantum associative memory based on open quantum system dynamics. On the one hand, this allowed us to compare existing models, as well as to identify the theoretical prerequisites for performing associative memory tasks. On the other hand, it also permitted us to explore general bounds to the storage capacity of the models, which is one of the key features characterizing associative memories.
In the final part of the project we build on some the concepts developed within the theoretical framework of quantum associative memories. Specifically, we explored whether quantum metastable manifolds can be used as code spaces for passive quantum error correction, by exploiting the autonomous recovery realized through the metastable dynamics itself. Concretely, this part of the action focused on two given models, a two qubit system subject to collective dissipation and a driven-dissipative Kerr resonator. We characterized the metastable transient in terms of the model parameters, and we proposed a protocol where the error recovery performance can be analyzed.
In the second part of the action, the research activity focused on the more general formulation of quantum associative memories in terms of completely positive and trace preserving (CPTP) maps, which are key tools to describe generic open quantum systems. Here, by focusing on general properties of CPTP maps, and on the features characterizing classical associative memories, we developed a comprehensive framework for a quantum associative memory based on open quantum system dynamics. On the one hand, this allowed us to compare existing models, as well as to identify the theoretical prerequisites for performing associative memory tasks. On the other hand, it also permitted us to explore general bounds to the storage capacity of the models, which is one of the key features characterizing associative memories.
In the final part of the project we build on some the concepts developed within the theoretical framework of quantum associative memories. Specifically, we explored whether quantum metastable manifolds can be used as code spaces for passive quantum error correction, by exploiting the autonomous recovery realized through the metastable dynamics itself. Concretely, this part of the action focused on two given models, a two qubit system subject to collective dissipation and a driven-dissipative Kerr resonator. We characterized the metastable transient in terms of the model parameters, and we proposed a protocol where the error recovery performance can be analyzed.
The main results of this actions are the following:
• The quantum generalization of the Hopfield networks has been embedded in a more general model, which features all-to-all Hamiltonian interaction as well as jump operators depending on collective rates.
• Based on the above modelling, we derived rigorous results on the validity of mean field results and on the impact of quantum fluctuations, showing that the latter are negligible in the thermodynamic limit.
The above results clearly frame the open quantum generalization of Hopfield neural networks in rigorous mathematical terms, further allowing to analyse the impact on finite-size systems.
Moreover, the presentation of a general framework for quantum associative memories allowed us to obtain the following results:
• define a generic quantum CPTP map that embeds the existing models in a more general theory;
• establish the necessary conditions that a generic open quantum system must satisfy to function as an associative memory;
• highlight the key-role played by symmetries and conserved quantities in relation to pattern storage and retrieval;
• find general bounds to the storage capacity of quantum associative memories.
Future research work could build on such results to identify concrete models that can reach the upper bound of the storage capacity, as well as to develop practical applications in quantum computing and machine learning.
The investigation performed on quantum associative memories found a more applicative research line in the exploration of passive error recovery protocols by leveraging metastable decoherence-free subspaces. Here, the main achievements are summarized by the following results:
• definition of a protocol for investigating error correction within quantum metastable manifolds -- which play the role of code spaces;
• in the first model analyzed, a two-qubit system subject to collective dissipation, only bit-flip error can be recovered. Instead, the correction of phase-flip errors may need the use of hybrid error correction schemes;
• in the second model considered, a two-photon driven-dissipative Kerr resonator, the error induced by a dephasing channel can be recovered. However, with a trade-off between fidelity and recovery time.
• The quantum generalization of the Hopfield networks has been embedded in a more general model, which features all-to-all Hamiltonian interaction as well as jump operators depending on collective rates.
• Based on the above modelling, we derived rigorous results on the validity of mean field results and on the impact of quantum fluctuations, showing that the latter are negligible in the thermodynamic limit.
The above results clearly frame the open quantum generalization of Hopfield neural networks in rigorous mathematical terms, further allowing to analyse the impact on finite-size systems.
Moreover, the presentation of a general framework for quantum associative memories allowed us to obtain the following results:
• define a generic quantum CPTP map that embeds the existing models in a more general theory;
• establish the necessary conditions that a generic open quantum system must satisfy to function as an associative memory;
• highlight the key-role played by symmetries and conserved quantities in relation to pattern storage and retrieval;
• find general bounds to the storage capacity of quantum associative memories.
Future research work could build on such results to identify concrete models that can reach the upper bound of the storage capacity, as well as to develop practical applications in quantum computing and machine learning.
The investigation performed on quantum associative memories found a more applicative research line in the exploration of passive error recovery protocols by leveraging metastable decoherence-free subspaces. Here, the main achievements are summarized by the following results:
• definition of a protocol for investigating error correction within quantum metastable manifolds -- which play the role of code spaces;
• in the first model analyzed, a two-qubit system subject to collective dissipation, only bit-flip error can be recovered. Instead, the correction of phase-flip errors may need the use of hybrid error correction schemes;
• in the second model considered, a two-photon driven-dissipative Kerr resonator, the error induced by a dephasing channel can be recovered. However, with a trade-off between fidelity and recovery time.