arXiv subjects

Zeno Toffano

Publications and source records attributed to Zeno Toffano.

13 recordsLinked to original sources

arXiv2026

Learning PDEs for Portfolio Optimization with Quantum Physics-Informed Neural Networks

Partial differential equations (PDEs) play a crucial role in financial mathematics, particularly in portfolio optimization, and solving them using classical numerical or neural network methods has always posed significant challenges. Here, we investigate the potential role of quantum circuits for solving PDEs. We design a parameterized quantum circuit (PQC) for implementing a polynomial based on tensor rank decomposition, reducing the quantum resource complexity from exponential to polynomial when the corresponding tensor rank is moderate. Building on this circuit, we develop a Quantum Physics-Informed Neural Network (QPINN) and a Quantum-inspired PINN, both of which guarantee the existence of an approximation of the PDE solution, and this approximation can be represented as a polynomial that incorporates tensor rank decomposition. Numerical experiments are conducted on the Hamilton--Jacobi--Bellman (HJB) PDE arising from the Merton portfolio optimization problem, which determines the optimal investment fraction between a risky and a risk-free asset. The results show that our quantum models achieve lower losses and approximation errors than a classical fully connected PINN while using substantially fewer trainable parameters. Our quantum models further outperform a classical PINN constructed to share a similar inductive bias, providing experimental evidence of quantum-induced improvement in the tested settings and highlighting a resource-efficient pathway toward classical and near-term quantum solvers for PDEs with exploitable solution structure.

quant-ph↗

arXiv2026

Geometric Algebra Quantum Gate Decomposition

Quantum gates are usually described through matrix and tensor-product formalisms that often obscure their geometric structure. In this work, we formulate the Pauli and Clifford groups within the complex Geometric Algebra (GA) framework. We show that the Pauli group is naturally identified with the group of blades up to a global phase, thereby providing a geometric interpretation of Pauli operators and their commutation relations in terms of oriented subspaces. We further prove that Clifford operators are generated by products of {\pi}/4-Pauli rotors and introduce a greedy Pauli rotor decomposition algorithm whose empirical behavior suggests unexpectedly compact decompositions for Clifford operators. Finally, we show that Clifford+T universality admits a natural geometric interpretation through {\pi}/8-rotors within this framework.

quant-ph↗

arXiv2026

Distributionally Robust Complex Chance-Constrained Optimization

This paper introduces a framework for Chance-Constrained Optimization with Complex Variables, addressing complex linear programming for both individual and joint probabilistic constraints in the complex domain. We first analyze the 3CP model in the density-based setting under the assumption that the random parameters follow a Complex Elliptically Symmetric distribution. The framework is then extended to distributionally robust settings, which include a moment-based model where the moments are known or bounded; a support-based model, where the ambiguity set contains distributions supported on norm-bounded uncertainty sets; and a data-driven model where moments are estimated empirically. The individual constraints are transformed into a convex deterministic second-order cone problem. We employ copula theory to the joint probability constraints and derive both upper and lower approximations. Finally, we demonstrate the proposed framework on the minimum variance distortionless response beamforming problem in signal processing. We further evaluate empirical out-of-sample rates and show that the observed behavior closely matches the prescribed probabilistic guarantees.

math.OC↗

arXiv2026

A Complex-Valued Continuous-Variable Quantum Approximation Optimization Algorithm (CCV-QAOA)

Continuous-variable (CV) quantum systems offer a natural framework for continuous optimization through their infinite-dimensional Hilbert spaces. In this paper, we propose the Complex Continuous-Variable Quantum Approximate Optimization Algorithm (CCV-QAOA), a variational framework operating in the complex domain that optimizes over complex decision variables. The method efficiently solves real and complex multivariate optimization problems. To demonstrate its versatility, we apply CCV-QAOA across a broad suite of optimization use cases, including convex quadratic minimization, scaling studies with circuit depth and cutoff dimension, constrained quadratic programs using penalty constructions, and non-convex benchmarks such as the Styblinski-Tang function and complex quartic landscapes.

quant-ph↗

arXiv2026

Robust Chance Constrained Complex Zero-Sum Games

This paper develops a unified framework for zero-sum games in which both the pure strategies and the payoff matrices contain complex-valued entries. By leveraging a linear isomorphism between complex and real vector spaces, we extend key results from real-valued convex analysis to the complex domain, establishing the validity of the minimax theorem and the preservation of saddle-point structure. Building on this foundation, we formulate a complex zero-sum game model that enables mixed strategies to interact with the real and imaginary components of the payoff matrix, and we characterize its saddle-point equilibrium through associated primal and dual problems. To incorporate uncertainty, we introduce a complex chance-constrained zero-sum game model (3CP) that handles individual probabilistic constraints defined by complex linear functionals. We first study the 3CP formulation under known exact distributions, focusing on Complex Elliptically Symmetric random variables, which generalize the complex Gaussian family. The framework is then extended to moments-based ambiguity sets, including: (i) distributions with known first two moments, (ii) distributions with unknown second-order moments, and (iii) fully distributed with unknown moments. In all cases, the probabilistic constraints admit deterministic second-order cone representations, ensuring convex feasible strategy sets and enabling explicit characterization of the complex game value. Numerical experiments, including a transmitter--jammer waveform interaction model, show how the proposed framework captures the behavior of complex mixed strategies. Additionally, we evaluate out-of-sample rates and confirm that practical behavior closely aligns with the theoretical guarantees.

math.GM↗

arXiv2025

Eigenlogic and Probabilistic Inference; when Bayes meets Born

This paper shows how inference is treated within the context of Eigenlogic projection operators in linear algebra. In Eigenlogic operators represent logical connectives, their eigenvalues the truth-values and the associated eigenvectors the logical models. By extension, a probabilistic interpretation is proposed using vectors outside the eigensystem of the Eigenlogic operators. The probability is calculated by the quantum mean value (Born rule) of the logical projection operators. We look here for possible connections between the Born rule in quantum mechanics and Bayes' theorem from probability theory and show that Eigenlogic offers an innovative approach to address the probabilistic version of logical inference (material implication) in a quantum context.

quant-ph↗

arXiv2018

Quantum Semantic Correlations in Hate and Non-Hate Speeches

This paper aims to apply the notions of quantum geometry and correlation to the typification of semantic relations between couples of keywords in different documents. In particular we analysed texts classified as hate / non hate speeches, containing the keywords "women", "white", and "black". The paper compares this approach to cosine similarity, a classical methodology, to cast light on the notion of "similar meaning".

cs.IR↗

arXiv2018

Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis

A method for synthesizing quantum gates is presented based on interpolation methods applied to operators in Hilbert space. Starting from the diagonal forms of specific generating seed operators with non-degenerate eigenvalue spectrum one obtains for arity-one a complete family of logical operators corresponding to all the one-argument logical connectives. Scaling-up to n-arity gates is obtained by using the Kronecker product and unitary transformations. The quantum version of the Fourier transform of Boolean functions is presented and a Reed-Muller decomposition for quantum logical gates is derived. The common control gates can be easily obtained by considering the logical correspondence between the control logic operator and the binary propositional logic operator. A new polynomial and exponential formulation of the Toffoli gate is presented. The method has parallels to quantum gate-T optimization methods using powers of multilinear operator polynomials. The method is then applied naturally to alphabets greater than two for multi-valued logical gates used for quantum Fourier transform, min-max decision circuits and multivalued adders.

quant-ph↗

arXiv2018

Eigenlogic in the spirit of George Boole

This work presents an operational and geometric approach to logic. It starts from the multilinear elective decomposition of binary logical functions in the original form introduced by George Boole. A justification on historical grounds is presented bridging Boole's theory and the use of his arithmetical logical functions with the axioms of Boolean algebra using sets and quantum logic. It is shown that this algebraic polynomial formulation can be naturally extended to operators in finite vector spaces. Logical operators will appear as commuting projection operators and the truth values, which take the binary values {0,1}, are the respective eigenvalues. In this view the solution of a logical proposition resulting from the operation on a combination of arguments will appear as a selection where the outcome can only be one of the eigenvalues. In this way propositional logic can be formalized in linear algebra by using elective developments which correspond here to combinations of tensored elementary projection operators. The original and principal motivation of this work is for applications in the new field of quantum information, differences are outlined with more traditional quantum logic approaches.

cs.LO↗

arXiv2017

Eigenlogic: Interpretable Quantum Observables with applications to Fuzzy Behavior of Vehicular Robots

This work proposes a formulation of propositional logic, named Eigenlogic, using quantum observables as propositions. The eigenvalues of these operators are the truth-values and the associated eigenvectors the interpretations of the propositional system. Fuzzy logic arises naturally when considering vectors outside the eigensystem, the fuzzy membership function is obtained by the Born rule of the logical observable.This approach is then applied in the context of quantum robots using simple behavioral agents represented by Braitenberg vehicles. Processing with non-classical logic such as multivalued logic, fuzzy logic and the quantum Eigenlogic permits to enlarge the behavior possibilities and the associated decisions of these simple agents.

cs.AI↗

arXiv2016

Eigenlogic: a Quantum View for Multiple-Valued and Fuzzy Systems

We propose a matrix model for two- and many-valued logic using families of observables in Hilbert space, the eigenvalues give the truth values of logical propositions where the atomic input proposition cases are represented by the respective eigenvectors. For binary logic using the truth values {0,1} logical observables are pairwise commuting projectors. For the truth values {+1,-1} the operator system is formally equivalent to that of a composite spin 1/2 system, the logical observables being isometries belonging to the Pauli group. Also in this approach fuzzy logic arises naturally when considering non-eigenvectors. The fuzzy membership function is obtained by the quantum mean value of the logical projector observable and turns out to be a probability measure in agreement with recent quantum cognition models. The analogy of many-valued logic with quantum angular momentum is then established. Logical observables for three-value logic are formulated as functions of the Lz observable of the orbital angular momentum l=1. The representative 3-valued 2-argument logical observables for the Min and Max connectives are explicitly obtained.

quant-ph↗

arXiv2013

Contextual Query Using Bell Tests

Tests are essential in Information Retrieval and Data Mining in order to evaluate the effectiveness of a query. An automatic measure tool intended to exhibit the meaning of words in context has been developed and linked with Quantum Theory, particularly entanglement. "Quantum like" experiments were undertaken on semantic space based on the Hyperspace Analogue Language (HAL) method. A quantum HAL model was implemented using state vectors issued from the HAL matrix and query observables, testing a wide range of windows sizes. The Bell parameter S, associating measures on two words in a document, was derived showing peaks for specific window sizes. The peaks show maximum quantum violation of the Bell inequalities and are document dependent. This new correlation measure inspired by Quantum Theory could be promising for measuring query relevance.

cs.IR↗

arXiv2012

Quantum-like Tests for Contextual Querying

Tests are essential in Information Retrieval (IR), in order to evaluate the effectiveness of a query. Tests intended to exhibit the sense of words in con-text were undertaken and linked with Quantum Mechanics (QM). Poll tests were undertaken on heterogeneous media such as music and polysemy in foreign languages. Interference effects are shown in the results. Bell inequality was used leading to a significant spread in the results of the poll tests but without violating the classical limit. Then an automatic pertinence measure tool on texts has been developed using the HAL algorithm using an orthonormal vector decomposition model. In this case the spread in the values can lead to the violation of the Bell inequality even beyond Cirel'son bound.

cs.IR↗