Application
Quantum-inspired machine learning algorithms and their applications in privacy
GICC + MATHQI
Group description:
The Complutense University of Madrid (UCM) has two research groups specializing in quantum computing, QICC and MATHQI, and is interested in collaborating with other project partners for the development, expansion, and dissemination of this technology. In order to advance research in quantum computing within the fields of its scientific interest, UCM will undertake the activities described in the following section, within the framework of the Quantum Spain project.
The QICC and MATHQI groups, led respectively by professors Miguel Ángel Martín-Delgado and David Pérez-García, have extensive experience in the mathematical analysis of concepts related to quantum computing, as well as in the simulation of quantum systems with tensor networks. The research tasks of these UCM groups will focus on the study of tensor networks and the use of quantum Markov chains in processes related to artificial intelligence.
Activity description:
Tensor networks are a mathematically inspired quantum technology with broad utility in the study of complex quantum systems. Since 2016, this technology has been applied to the field of artificial intelligence with notable results in image processing and anomaly detection. In this context, the objective of this activity is to delve into this utility by analyzing the privacy guarantees obtained when using machine learning algorithms based on tensor network architectures.
Results
Florido-Llinàs, M.; Alhambra, A. M.; Pérez-García, D.; Cirac, J. I.
Regular language quantum states Journal Article
In: Quantum, 2026.
Abstract | Links | BibTeX | Tags: UCM-4.2
@article{nokey,
title = {Regular language quantum states},
author = {Florido-Llinàs, M. and Alhambra, A.M. and Pérez-García, D. and Cirac, J.I. },
url = {https://quantum-journal.org/papers/q-2026-04-29-2089/#},
doi = {doi.org/10.22331/q-2026-04-29-2089},
year = {2026},
date = {2026-04-29},
urldate = {2024-07-24},
journal = {Quantum},
abstract = {We introduce regular language states, a family of quantum many-body states. They are built from a special class of formal languages, called regular, which has been thoroughly studied in the field of computer science. They can be understood as the superposition of all the words in a regular language and encompass physically relevant states such as the GHZ-, W- or Dicke-states. By leveraging the theory of regular languages, we develop a theoretical framework to describe them. First, we express them in terms of matrix product states, providing efficient criteria to recognize them. We then develop a canonical form which allows us to formulate a fundamental theorem for the equivalence of regular language states, including under local unitary operations. We also exploit the theory of tensor networks to find an efficient criterion to determine when regular languages are shift-invariant.
},
keywords = {UCM-4.2},
pubstate = {published},
tppubtype = {article}
}
We introduce regular language states, a family of quantum many-body states. They are built from a special class of formal languages, called regular, which has been thoroughly studied in the field of computer science. They can be understood as the superposition of all the words in a regular language and encompass physically relevant states such as the GHZ-, W- or Dicke-states. By leveraging the theory of regular languages, we develop a theoretical framework to describe them. First, we express them in terms of matrix product states, providing efficient criteria to recognize them. We then develop a canonical form which allows us to formulate a fundamental theorem for the equivalence of regular language states, including under local unitary operations. We also exploit the theory of tensor networks to find an efficient criterion to determine when regular languages are shift-invariant.
Florido-Llinàs, M.; Alhambra, A. M.; Trivedi, R.; N. Pérez-García Schuch, D.; Cirac, J. I.
The Product Structure of Matrix Product States under Permutations Journal Article
In: 2025.
Abstract | Links | BibTeX | Tags: UCM-4.2
@article{nokey,
title = {The Product Structure of Matrix Product States under Permutations},
author = {Florido-Llinàs, M. and Alhambra, A.M. and Trivedi, R. and Schuch, N. Pérez-García, D. and Cirac, J.I. },
url = {https://journals.aps.org/prxquantum/abstract/10.1103/8sbs-t24w},
doi = {doi.org/10.1103/8sbs-t24w},
year = {2025},
date = {2025-11-11},
urldate = {2024-10-25},
abstract = {Tensor network methods have proved to be highly effective in addressing a wide variety of physical scenarios, including those lacking an intrinsic one-dimensional geometry. In such contexts, it is possible for the problem to exhibit a weak form of permutational symmetry, in the sense that entanglement behaves similarly across any arbitrary bipartition. In this paper, we show that translationally-invariant (TI) matrix product states (MPSs) with this property are trivial, meaning that they are either product states or superpositions of a few of them. The results also apply to non-TI generic MPSs, as well as further relevant examples of MPSs including the 𝑊 state and the Dicke states in an approximate sense. Our findings motivate the usage of Ansätze simpler than tensor networks in systems whose structure is invariant under permutations.},
keywords = {UCM-4.2},
pubstate = {published},
tppubtype = {article}
}
Tensor network methods have proved to be highly effective in addressing a wide variety of physical scenarios, including those lacking an intrinsic one-dimensional geometry. In such contexts, it is possible for the problem to exhibit a weak form of permutational symmetry, in the sense that entanglement behaves similarly across any arbitrary bipartition. In this paper, we show that translationally-invariant (TI) matrix product states (MPSs) with this property are trivial, meaning that they are either product states or superpositions of a few of them. The results also apply to non-TI generic MPSs, as well as further relevant examples of MPSs including the 𝑊 state and the Dicke states in an approximate sense. Our findings motivate the usage of Ansätze simpler than tensor networks in systems whose structure is invariant under permutations.
Styliaris, G.; Trivedi, R.; Pérez-García, D.; Cirac, J. I.
Matrix-product unitaries: Beyond quantum cellular automata Journal Article
In: 2025.
Abstract | Links | BibTeX | Tags: UCM-4.2
@article{nokey,
title = {Matrix-product unitaries: Beyond quantum cellular automata},
author = {Styliaris, G. and Trivedi, R. and Pérez-García, D. and Cirac, J. I.
},
url = {https://quantum-journal.org/papers/q-2025-02-25-1645/},
doi = {doi.org/10.22331/q-2025-02-25-1645},
year = {2025},
date = {2025-02-25},
urldate = {2025-02-20},
abstract = {Matrix-product unitaries (MPU) are 1D tensor networks describing time evolution and unitary symmetries of quantum systems, while their action on states by construction preserves the entanglement area law. MPU which are formed by a single repeated tensor are known to coincide with 1D quantum cellular automata (QCA), i.e., unitaries with an exact light cone. However, this correspondence breaks down for MPU with open boundary conditions, even if the resulting operator is translation-invariant. Such unitaries can turn short- to long-range correlations and thus alter the underlying phase of matter. Here we make the first steps towards a theory of MPU with uniform bulk but arbitrary boundary. In particular, we study the structure of a subclass with a direct-sum form which maximally violates the QCA property. We also consider the general case of MPU formed by site-dependent (nonuniform) tensors and show a correspondence between MPU and locally maximally entanglable states.},
keywords = {UCM-4.2},
pubstate = {published},
tppubtype = {article}
}
Matrix-product unitaries (MPU) are 1D tensor networks describing time evolution and unitary symmetries of quantum systems, while their action on states by construction preserves the entanglement area law. MPU which are formed by a single repeated tensor are known to coincide with 1D quantum cellular automata (QCA), i.e., unitaries with an exact light cone. However, this correspondence breaks down for MPU with open boundary conditions, even if the resulting operator is translation-invariant. Such unitaries can turn short- to long-range correlations and thus alter the underlying phase of matter. Here we make the first steps towards a theory of MPU with uniform bulk but arbitrary boundary. In particular, we study the structure of a subclass with a direct-sum form which maximally violates the QCA property. We also consider the general case of MPU formed by site-dependent (nonuniform) tensors and show a correspondence between MPU and locally maximally entanglable states.
Security of quantum position-verification limits Hamiltonian simulation via holography Journal Article
In: Journal of High Energy Physics, vol. 2024, iss. 8, no. 152, 2024.
Abstract | Links | BibTeX | Tags: UCM-4.2
@article{nokey,
title = {Security of quantum position-verification limits Hamiltonian simulation via holography},
url = {https://link.springer.com/article/10.1007/JHEP08(2024)152},
doi = {doi.org/10.1007/JHEP08(2024)152},
year = {2024},
date = {2024-08-20},
journal = {Journal of High Energy Physics},
volume = {2024},
number = {152},
issue = {8},
abstract = {We investigate the link between quantum position-verification (QPV) and holography established in [1] using holographic quantum error correcting codes as toy models. By inserting the “temporal” scaling of the AdS metric by hand via the bulk Hamiltonian interaction strength, we recover a toy model with consistent causality structure. This leads to an interesting implication between two topics in quantum information: if position-based verification is secure against attacks with small entanglement then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another.},
keywords = {UCM-4.2},
pubstate = {published},
tppubtype = {article}
}
We investigate the link between quantum position-verification (QPV) and holography established in [1] using holographic quantum error correcting codes as toy models. By inserting the “temporal” scaling of the AdS metric by hand via the bulk Hamiltonian interaction strength, we recover a toy model with consistent causality structure. This leads to an interesting implication between two topics in quantum information: if position-based verification is secure against attacks with small entanglement then there are new fundamental lower bounds for resources required for one Hamiltonian to simulate another.
Pozas-Kerstjens, A.; Hernández-Santana, S.; Pareja Monturiol, J. R.; Castrillón López, M.; Scarpa, G.; González-Guillén, C. E.; Pérez-García, D.
Privacy-preserving machine learning with tensor networks Journal Article
In: Quantum, 2024.
Abstract | Links | BibTeX | Tags: UCM-4.2
@article{nokey,
title = {Privacy-preserving machine learning with tensor networks},
author = {Pozas-Kerstjens, A. and Hernández-Santana, S. and Pareja Monturiol, J.R. and Castrillón López, M. and Scarpa, G. and González-Guillén, C.E. and Pérez-García, D.},
url = {https://quantum-journal.org/papers/q-2024-07-25-1425/#},
doi = {doi.org/10.22331/q-2024-07-25-1425},
year = {2024},
date = {2024-07-25},
journal = {Quantum},
abstract = {Tensor networks, widely used for providing efficient representations of low-energy states of local quantum many-body systems, have been recently proposed as machine learning architectures which could present advantages with respect to traditional ones. In this work we show that tensor-network architectures have especially prospective properties for privacy-preserving machine learning, which is important in tasks such as the processing of medical records. First, we describe a new privacy vulnerability that is present in feedforward neural networks, illustrating it in synthetic and real-world datasets. Then, we develop well-defined conditions to guarantee robustness to such vulnerability, which involve the characterization of models equivalent under gauge symmetry. We rigorously prove that such conditions are satisfied by tensor-network architectures. In doing so, we define a novel canonical form for matrix product states, which has a high degree of regularity and fixes the residual gauge that is left in the canonical forms based on singular value decompositions. We supplement the analytical findings with practical examples where matrix product states are trained on datasets of medical records, which show large reductions on the probability of an attacker extracting information about the training dataset from the model's parameters. Given the growing expertise in training tensor-network architectures, these results imply that one may not have to be forced to make a choice between accuracy in prediction and ensuring the privacy of the information processed.
},
keywords = {UCM-4.2},
pubstate = {published},
tppubtype = {article}
}
Tensor networks, widely used for providing efficient representations of low-energy states of local quantum many-body systems, have been recently proposed as machine learning architectures which could present advantages with respect to traditional ones. In this work we show that tensor-network architectures have especially prospective properties for privacy-preserving machine learning, which is important in tasks such as the processing of medical records. First, we describe a new privacy vulnerability that is present in feedforward neural networks, illustrating it in synthetic and real-world datasets. Then, we develop well-defined conditions to guarantee robustness to such vulnerability, which involve the characterization of models equivalent under gauge symmetry. We rigorously prove that such conditions are satisfied by tensor-network architectures. In doing so, we define a novel canonical form for matrix product states, which has a high degree of regularity and fixes the residual gauge that is left in the canonical forms based on singular value decompositions. We supplement the analytical findings with practical examples where matrix product states are trained on datasets of medical records, which show large reductions on the probability of an attacker extracting information about the training dataset from the model's parameters. Given the growing expertise in training tensor-network architectures, these results imply that one may not have to be forced to make a choice between accuracy in prediction and ensuring the privacy of the information processed.
Pareja Monturiol, J. R.; Pérez-García, D.; Pozas-Kerstjens, A.
TensorKrowch: Smooth integration of tensor networks in machine learning Journal Article
In: Quantum, vol. 8, pp. 1364 , 2024.
Abstract | Links | BibTeX | Tags: UCM-4.2
@article{nokey,
title = {TensorKrowch: Smooth integration of tensor networks in machine learning},
author = {Pareja Monturiol, J.R. and Pérez-García, D. and Pozas-Kerstjens, A. },
url = {https://quantum-journal.org/papers/q-2024-06-11-1364/},
doi = {doi.org/10.22331/q-2024-06-11-1364},
year = {2024},
date = {2024-06-11},
journal = {Quantum},
volume = {8},
pages = {1364 },
abstract = {Tensor networks are factorizations of high-dimensional tensors into networks of smaller tensors. They have applications in physics and mathematics, and recently have been proposed as promising machine learning architectures. To ease the integration of tensor networks in machine learning pipelines, we introduce TensorKrowch, an open source Python library built on top of PyTorch. Providing a user-friendly interface, TensorKrowch allows users to construct any tensor network, train it, and integrate it as a layer in more intricate deep learning models. In this paper, we describe the main functionality and basic usage of TensorKrowch, and provide technical details on its building blocks and the optimizations performed to achieve efficient operation.},
keywords = {UCM-4.2},
pubstate = {published},
tppubtype = {article}
}
Tensor networks are factorizations of high-dimensional tensors into networks of smaller tensors. They have applications in physics and mathematics, and recently have been proposed as promising machine learning architectures. To ease the integration of tensor networks in machine learning pipelines, we introduce TensorKrowch, an open source Python library built on top of PyTorch. Providing a user-friendly interface, TensorKrowch allows users to construct any tensor network, train it, and integrate it as a layer in more intricate deep learning models. In this paper, we describe the main functionality and basic usage of TensorKrowch, and provide technical details on its building blocks and the optimizations performed to achieve efficient operation.
