Many dualities in mathematics arise from the inherent duality of ‘syntax’ and ‘semantics’ in logic. Classical Stone duality, for example, is the syntax-semantics duality for theories of classical propositional logic, with Boolean algebras encoding the syntax of a propositional theory. The logical perspective on these ‘syntax-semantics’ dualities gives both an intuitive understanding for why mathematicians (or at least logicians) would expect these dualities to hold in the first place, as well as a framework to generalise to nearby logics.

In recent years, new ‘non-commutative’ generalisations of Stone duality have been discovered, involving inverse monoids and étale groupoids. Interestingly, this branch of duality theory was developed in the absence of a logical description. In this talk, we describe a class of logical theories whose syntax-semantics duality is given by a version of non-commutative Stone duality. Rather than originating in an exotic fragment of logic, these are theories of first-order logic which share many of the same properties as the theory of vector spaces, suggesting that non-commutative Stone duality is not so distant from classical logic as one might expect.

In this talk I will present the status on quantum communications in Portugal. From the Laboratory to the operational network what has been done through the main actors in industry, academia and public institutions in Portugal. Starting with the development of national technology under a European Defence project, DISCRETION, to the deployment of the first EuroQCI (the European Quantum Communication Infrastructure) segment in Portugal, PTQCI. I will show that quantum communications is no longer a science project, but it is on the heart of sovereignty in Europe.

Identical quantum particles exhibit only two types of statistics: bosonic and fermionic. Theoretically, this restriction is commonly established through the symmetrizationpostulate or (anti)commutation constraints imposed on the algebra of creation and annihilation operators. The physical motivation for these axioms remains poorly understood, leading to various generalizations by modifying the mathematical formalism in somewhat arbitrary ways. In this talk, I will take an opposing route and show how to classify quantum particle statistics based on operationally well-motivated assumptions. Specifically, I will consider that a) the standard (complex) unitary dynamics defines the set of single-particle transformations, and b) phase transformations act locally in the space of multi-particle systems. In this way, a complete characterization, which includes bosons and fermions as basic statistics with minimal symmetry is being developed. Interestingly, we have discovered whole families of novel statistics (dubbed transtatistics) accompanied by hidden symmetries, generic degeneracy of ground states, and spontaneous symmetry breaking — effects that are (typically) absent in ordinary statistics.

Reference: N. M. Sánchez and B. Dakić, Quantum 8, 1473 (2024).

This presentation will report on a number of studies conducted in my laboratory, mainly on the ability of large language models to solve analogies between sentences. We show that the task does not necessarily require a large number of parameters for fairly formal analogies. By defining constraints and introducing a measure of analogy quality, we are able to collect a large number of more or less robust analogies between sentences. We show that analogy quality prevails over quantity in fine-tuning and demonstrate this in machine translation tasks with low-resource languages. We also mention experiments on solving arithmetic problems by answer verification (binary classification), multiple-choice questions (selection from candidate answers), and generation of answer. The results clearly show that binary classification is the easiest task, and that there is little correlation between the three tasks.

Quantum Singular Value Transformation (QSVT) is a unified framework that describes most quantum algorithms discovered thus far. This method implements polynomial transformations to any matrix embedded in the top-left block of a unitary, known as a block encoding. Given such a block-encoding U as an input, QSVT leads to optimal complexities in terms of the number of queries to U. However, the reliance on block encoding is a serious bottleneck for this framework. For instance, consider any Hamiltonian H that is a sum of L local operators. Constructing a block encoding of H itself requires O(log L) ancilla qubits and several sophisticated multi-qubit controlled gates which detrimentally affects the applicability of QSVT.

In my talk, I will discuss our recent work where we develop new quantum algorithms for implementing QSVT without relying on block encodings. Surprisingly, our methods make use of only basic Hamiltonian simulation techniques such as Trotterization to achieve near-optimal complexities while needing only a single ancilla qubit. Central to our method is a novel use of Richardson extrapolation, enabling systematic error cancellation in any interleaved sequences of arbitrary unitaries and Hamiltonian evolution operators, establishing a broadly applicable framework beyond QSVT. As applications, we develop new quantum algorithms for solving quantum linear systems and ground state property estimation, both achieving near optimal complexities without oracular access or sophisticated quantum subroutines, and consuming very few ancilla qubits. This represents a major simplification over existing algorithms for solving these problems.

We also develop two randomized quantum algorithms for QSVT in a setting where there is only sampling access to the terms of the Hamitonian. We prove concrete lower bounds for the complexity of QSVT within this access model which extends to other randomized quantum linear algebra-based techniques. Overall, our results provide a new framework for quantum algorithms, reducing hardware overhead while maintaining near-optimal performance, with implications for both near-term and fault-tolerant quantum computing.