The main motivation for this thesis is given by the open problems regarding the axiomatisation of quantum computational logics. This thesis will be structured as follows: in Chapter 2 we will review some basics of universal algebra and functional analysis. In Chapters 3 through 6 the fundamentals of quantum gate theory will be produced. In Chapter 7 we will introduce quasi-MV algebras, a formal study of a suitable selection of algebraic operations associated with quantum gates. In Chapter 8 quasi-MV algebras will be expanded by a unary operation hereby dubbed square root of the inverse, formalising a quantum gate which allows to induce entanglement states. In Chapter 9 we will investigate some categorial dualities for the classes of algebras introduced in Chapters 7 and 8. In Chapter 10 the discriminator variety of linear Heyting quantum computational structures, an algebraic counterpart of the strong quantum computational logic, will be considered. In Chapter 11, we will list some open problems and, at the same time, draw some tentative conclusions. Lastly, in Chapter 12 we will provide a few examples of the previously investigated structures.
The main motivation for this thesis is given by the open problems regarding the axiomatisation of quantum computational logics. This thesis will be structured as follows: in Chapter 2 we will review some basics of universal algebra and functional analysis. In Chapters 3 through 6 the fundamentals of quantum gate theory will be produced. In Chapter 7 we will introduce quasi-MV algebras, a formal study of a suitable selection of algebraic operations associated with quantum gates. In Chapter 8 quasi-MV algebras will be expanded by a unary operation hereby dubbed square root of the inverse, formalising a quantum gate which allows to induce entanglement states. In Chapter 9 we will investigate some categorial dualities for the classes of algebras introduced in Chapters 7 and 8. In Chapter 10 the discriminator variety of linear Heyting quantum computational structures, an algebraic counterpart of the strong quantum computational logic, will be considered. In Chapter 11, we will list some open problems and, at the same time, draw some tentative conclusions. Lastly, in Chapter 12 we will provide a few examples of the previously investigated structures.
This thesis is a study of the construction and representation of typed models of quantum mechanics for use in quantum computation. We introduce logical and graphical syntax for quantum mechanical processes and prove that these formal systems provide sound and complete representations of abstract quantum mechanics. In addition, we demonstrate how these representations may be used to reason about the behaviour of quantum computational processes. Quantum computation is presently mired in low-level formalisms, mostly derived directly from matrices over Hilbert spaces. These formalisms are an obstacle to the full understanding and exploitation of quantum effects in informatics since they obscure the essential structure of quantum states and processes. The aim of this work is to introduce higher level tools for quantum mechanics which will be better suited to computation than those presently employed in the field. Inessential details of Hilbert space representations are removed and the informatic structures are presented directly. Entangled states are particularly important in this treatment, as is appropriate, since entanglement is a fundamental driver of quantum computation. The benefits two-fold: as well as producing foundational tools for the study of quantum computation this work also connects quantum mechanics to mainstream areas of computer science such as categorical logic, type theory, program language semantics, and rewriting. We describe, following Abramsky and Coecke, how quantum mechanics may be carried out without reference to Hilbert space, in a strongly compact closed category. In particular we show how to freely construct a categorical model of abstract quantum mechanics from an arbitrary category. We introduceMultiplicative Categorical Quantum Logic (mCQL), a sequent calculus whose proof rules capture the structure of compact closed categories. This sequent calculus is interpreted in a freely generated compact closed category, and its semantics is sound with respect to cut elimination. We define an equivalent graphical syntax, similar to linear logic’s proof-nets, and prove that these proof-nets provide a full and faithful representation of any freely generated compact closed category. Further analysis of the structure of quantum states which correspond to mCQL proofs using multiplicative linear logic shows that the linear type system describes the quantum entanglement found in such states. We show that the entanglement present in these states is always of a particularly simple form: collections of entangled pairs. In order to tackle arbitrary entanglement, we generalise the work of Kelly and Laplaza to give a representation theorem for the free compact closed category by a polycategory. Such categories are shown to be equivalent to a generalised system of proof-nets whose axioms may have more than one premise
Arising from a special session held at the 2010 North American Annual Meeting of the Association for Symbolic Logic, this volume is an international cross-disciplinary collaboration with contributions from leading experts exploring connections across their respective fields. Themes range from philosophical examination of the foundations of physics and quantum logic, to exploitations of the methods and structures of operator theory, category theory, and knot theory in an effort to gain insight into the fundamental questions in quantum theory and logic. The book will appeal to researchers and students working in related fields, including logicians, mathematicians, computer scientists, and physicists. A brief introduction provides essential background on quantum mechanics and category theory, which, together with a thematic selection of articles, may also serve as the basic material for a graduate course or seminar.
A quantum computer is similar to an ordinary computer, except that instead of bits it uses two-level quantum systems (qubits), and the usual logical operations on bits are replaced with noiseless quantum dynamics in the form of unitary gates or Hamiltonian interactions. Many sets of resources that are universal for quantum computation (that is, capable of implementing any quantum computation) have been found, but general conditions for universality are still only partially understood. Improving our understanding of these conditions contributes to two broad goals. On the one hand, understanding what class of physical systems is suitable for quantum computation allows us choose the systems that are most easily controlled or built in the laboratory. On the other hand, identifying the unifying characteristics of such a class provides insight into the fundamental physics responsible for the much greater computational power of quantum computers. The original results presented in this thesis are a contribution to our understanding of the physical requirements for universal quantum computation. These results lead to the identification of a new, large class of universal dynamics. More specifically, the main problem that is addressed is the following: consider a physical system that is made up of a finite number of qubits that are subject to a natural, fixed interaction described by a two-body Hamiltonian. Suppose we have the ability to control the system only by applying fast one-qubit operations. Under what conditions is this system capable of universal quantum computation? The first original result of the thesis is to prove that universal quantum computation is possible under these conditions given the single extra constraint that the Hamiltonian must entangle, or connect, all of the qubits. In other words, since a two-body Hamiltonian is expressible as a sum of coupling terms each of which acts nontrivially on a pair of qubits, consider the graph whose vertices correspond to qubits and whose edges connect qubits that are coupled by a term in the Hamiltonian. Then this Hamiltonian, together with fast one-qubit unitaries, is universal for quantum computation precisely when the corresponding graph is connected. This result suggests many potential generalizations, three of which are explored in this thesis. The first generalization is to show that the same result holds for a two-body Hamiltonian that acts on D-dimensional quantum systems (qudits), given fast one-qudit unitaries. A second extension is to the case of a many-body Hamiltonian, that is, a Hamiltonian that has coupling terms that act on more than two qubits. Again, we find that under mild restrictions, such Hamiltonians are universal provided they are connected and that arbitrary one-qubit unitaries are available. We also discover and explore some surprising structure in many-body Hamiltonians that is associated with the parity of the couplings. A third related problem that is addressed is the question of what two-qubit unitaries are universal for quantum computation. This problem was originally solved by J.-L. and R. Brylinski [BB02], who gave a technical proof of the universality of any two-qudit gate that is not the swap gate or a product of one-qudit unitaries, given one-qudit unitaries as a free resource. We develop an elementary and constructive proof for the case of qubits, and prove that our construction is very close to optimal in the number of uses of the two-qubit gate. Given a qualitative understanding of what dynamical resources are universal for quantum computation, it is natural to consider developing a quantitative measure of the usefulness of quantum dynamics for quantum information processing. This is done in the final chapter of the thesis, in which the beginnings of a quantitative theory of quantum dynamics are developed.
Quantum computing represents one of the most significant paradigms shifts in the history of computation, promising to solve classes of problems that remain intractable for classical machines. Unlike classical computing, which is built on Boolean algebra and binary logic, quantum computing is fundamentally a mathematical discipline rooted in linear algebra, complex vector spaces, and probability theory. This article examines the mathematical structures that underpin quantum computation, including qubits, superposition, entanglement, unitary transformations, and quantum gates, and explores how these structures enable algorithms such as Shor's and Grover's to outperform their classical counterparts. The discussion further extends to quantum error correction, the challenges of decoherence, and the interdisciplinary convergence of mathematics and computer science that is shaping next-generation computing architectures. The article concludes with a reflection on open mathematical problems and the future trajectory of quantum information science.
Quantum computational logics provide a fertile common ground for a unied treatment of vagueness and uncertainty. In this paper we describe an approach to the logic of quantum computation that has been recently taken up and developed by the present authors. Special attention will be devoted to a generalisation of Chang's MV algebras (called quasi-MV alge- bra) which abstracts over the algebra whose universe is the set of qumixes of the 2-dimensional complex Hilbert space, as well as to its expansions by additional quantum connectives. We fur- thermore explore some future research perspectives, also in the light of some recent limitative results whose general signicance will be duly assessed.
In this dissertation, we investigate algebraic structures in finite-dimensional Hilbert spaces, as concerns quantum computation and quantum information, as well as these structures' applications to lattices. On the quantum computation side, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Subsequently, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, using the algebraic framework developed. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many algebraic identities, including a novel algebraic proof of a Yang-Baxter equation. We also derive a new identity for the braid elements, which is key to our proofs. We then use the Yang-Baxter equation proof to resolve an open question of Cobanera and Ortiz. We demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. In addition, we show how to explicitly compute various vector states in an efficient manner using algebraic methods. On the quantum information side, we introduce a new decomposition of quantum channels acting on group algebras, which we term Kraus-like operator decompositions (Kraus-like decompositions for short). An important motivation for this new decomposition is a general nonexistence result that we show for Kraus operator decompositions for quantum channels in this setting. We show that the notion of convex Kraus-like operator decompositions (in which the coefficients in the sum decomposition are nonnegative and satisfy a sum rule) that are induced by the irreducible characters of a finite group is equivalent to the notion of a conditionally negative-definite length when the length is a class function. For a general finite group G, we prove a stability condition which shows that if the semigroup associated with a length has a convex Kraus-like operator decomposition for all t>0 small enough, then it has a convex Kraus-like operator decomposition for all time t>0. Using the stability condition, we show that for a general finite group, conditional negativity of the length function is equivalent to a set of semidefinite linear constraints on the length function. By a result of Schoenberg, our result implies that in the group algebra setting, a semigroup P_t induced by a length function which is a class function is a quantum channel for all t greater than or equal to 0 if and only if it possesses a convex Kraus-like operator decomposition for all t>0. Finally, motivated by the importance of lattice problems in quantum cryptography, we extend the algebraic framework for multi-qudit representations of generalized Clifford algebras to d-dimensional lattices mod P. We show that under suitable number-theoretic conditions, the subalgebra induced by a lattice has trivial center. Under the trivial center constraint, we construct for pairs of lattice vectors satisfying an algebraic constraint a unitary operator based on the product of generalized Clifford algebra generators associated to each lattice vector.