Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences最新文献

In recent work, the author has developed a general category-theoretic framework for decision theory. This article applies this to the category of orthomodular lattices (OMLs). Every Boolean algebra is an OML, so this yields a new (syntactic) model of decision-making with classical uncertainty. The lattice of closed subspaces of a Hilbert space is also an OML, so this also yields a new model of decision-making with quantum uncertainty. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

Quantum randomness evidently transcends the classical framework of random variables defined on a single comprehensive Kolmogorov probability space. One prominent example is the quantum double-slit experiment owing to Feynman (Feynman 1951 In Second Berkeley Symposium on Mathematical Statistics and Probability (ed. J. Neyman), pp. 533-541 (doi:10.1525/9780520411586-039)). A related non-quantum example, inspired by Boole (Boole 1862 Phil. Trans. R. Soc. Lond. 152, 225-252 (doi:10.1098/rstl.1862.0015)) and Vorob'ev (Vorob'ev 1962 Theory Probab. Appl.7, 147-163 (doi:10.1137/1107014)), has three two-valued random variables X, Y and Z, where the pairs X, Y and X, Z are perfectly correlated, yet Y, Z are perfectly anti-correlated. Such examples can be accommodated using a 'multi-measurable' space with several different sigma-algebras of measurable events. This concept, owing to Vorob'ev (Vorob'ev 1962 Theory Probab. Appl.7, 147-163 (doi:10.1137/1107014)), allows construction of (i) a measurable metaspace whose elements combine a point in the original sample space with a variable 'contextual' Boolean algebra; (ii) a parametric family of 'probability metaspaces', each of which is a Kolmogorov probability space that represents a two-stage stochastic process where a random choice from the original sample space is preceded by the random choice of a contextual Boolean algebra in the multi-measurable space. Subsequent work will explore how quantum experimental results can be described using a quantum measurement tree with one or more preparation nodes, where an experimental configuration is determined that governs the probability distribution of relevant quantum observables. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

Combinatorial problems pose significant challenges, especially for large-scale instances. A promising approach to addressing these challenges involves quadratic unconstrained binary optimization (QUBO) models, a key framework in quantum computing. QUBO formulations require tailored approaches owing to their unique characteristics and the hardware constraints of quantum systems. This paper focuses on the QUBO formulations of the Hamiltonian cycle problem (HCP). We analyse three QUBO formulations for the HCP, including two inspired by Lagrangian relaxation techniques, where penalties may result in negative costs for certain solutions but are balanced within the model. A preprocessing methodology is also introduced. To evaluate these models, we employ D-Wave quantum annealers, which feature a fixed qubit topology and impose strict resource constraints. The mapping of QUBOs on to this architecture highlights the importance of efficient model design. Through experimental studies, we analyse the performance of the proposed QUBOs with and without preprocessing. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

Several quantum algorithms have been proposed for solving discrete optimization problems. In this paper, we focus on the quantum counting algorithm (QCB) of Brassard G, Høyer P, Tapp A. 1998 Quantum counting. In Automata, languages and programming (eds KG Larsen, S Skyum, G Winksel), pp. 820-831. Berlin Heidelberg, Berlin, Heidelberg: Springer. (doi:10.1007/BFb0055105), the nested quantum search algorithm of Cerf NJ, Grover LK, Williams CP. 2000 Nested quantum search and structured problems. Phys. Rev. 61. (doi:10.1103/physreva.61.032303), and amplitude amplification. We discuss potential limitations of these quantum algorithms and investigate how they can be used to solve discrete optimization problems. Our goal is to provide a set of practical guidelines that can help researchers to effectively implement Grover-based quantum algorithms. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

Given a quantum state in the finite-dimensional Hilbert space Cn, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such observable is identified with (i) an 'ortho-measurable' function defined on the Boolean 'ortho-algebra' generated by the eigenspaces that form an orthogonal decomposition of Cn, and (ii) a 'numerically identified' orthogonal decomposition of Cn. The latter means that each subspace of the orthogonal decomposition can be uniquely identified by its own attached real number, just as each eigenspace of a Hermitian matrix can be uniquely identified by the corresponding eigenvalue. Furthermore, any density matrix on Cn is identified with a Bayesian prior 'ortho-probability' measure defined on the linear subspaces that make up the Boolean ortho-algebra induced by its eigenspaces. Then any pure quantum state is identified with a degenerate density matrix, and any mixed state with a probability measure on a set of orthogonal pure states. Finally, given any quantum observable, the relevant Bayesian posterior probabilities of measured outcomes can be found by the usual trace formula that extends Born's rule. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

In a series of recent scientific contributions, the role of bosonic and fermionic ladder operators in a macroscopic realm has been investigated. Creation, annihilation and number operators have been used in very different contexts, all sharing the same common main feature, i.e. the relevance of discrete changes in the description of the system. The main problem when using this approach is that computations are easy for Hamiltonians which are quadratic in the ladder operators but become very complicated, both at the analytical and at the numerical level, when the Hamiltonian is not quadratic. In this paper, we propose a possible alternative approach, again based on some sort of ladder operators, but for which an analytic solution can often be deduced without particular difficulties. We describe our proposal with a few applications, mostly related to different versions of a predator-prey model and to love affairs (from a decision-making point of view). This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

This study explores the application of a hybrid quantum-classical algorithm for solving the parallel machine scheduling problem (PMSP) with sequence-dependent set-up times, a pivotal scheduling problem that has applications in multiple industries. Using a column generation-based approach, we propose a heuristic that combines a classical linear relaxation for the master problem with quantum annealing (QA) for solving the pricing subproblem (PSP). Whereas the PSP generates columns (i.e. a sequence of jobs that are assigned to a machine), the master problem selects which columns to use in order to minimize the makespan of the schedule. To generate columns, the PSP solves a travelling salesman problem (TSP) that is formulated as a quadratic unconstrained binary optimization (QUBO) problem. The big advantage thereof is that subtours can be eliminated by use of quadratic terms in the objective function. In addition, our approach also leverages the quantum annealer's capability to generate many high-quality solutions (i.e. columns) in a very short time. To assess the performance of our hybrid column generation-based heuristic, we perform a computational experiment. The results of this experiment demonstrate the synergy of hybrid methods for tackling complex decision-making problems, achieving competitive high-quality solutions and computational advantages when compared with classical solution methods. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

This article 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. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

After briefly recalling some of the main theses of quantum probability concerning the foundational aspects of quantum mechanics and of probability theory, we describe the various categories of axioms in classical probability theory and focus our attention on the classical conditioning axioms. Four classical conditioning axioms are proposed, their probabilistic and model-independent meaning is discussed and it is proven that: (i) the only mathematical realization of these axioms is the classical Bayes rule (or equivalently the classical theorem of composite probabilities) and (ii) von Neumann's projection postulate, which can be considered one of the first examples of quantum conditioning axioms, violates two of these axioms. Finally, we construct an example showing that, even in classical probability, an uncritical application of Bayes' rule can lead to non-optimal predictions. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.