Control is a fundamental concept in quantum and reversible computational models. It enables the conditional application of a transformation to a system, depending on the state of another system. We introduce a general framework for diagrammatic reasoning featuring control as a constructor. To this end, we provide an elementary axiomatization of control functors, extending the standard formalism of PROPs to controlled PROPs. As an application, we show that controlled PROPs facilitate diagrammatic reasoning for quantum circuits. In particular, they admit a simple and complete axiomatization involving at most three qubits, whereas in the standard PROP setting, completeness necessarily requires axioms involving arbitrarily many qubits.
Control is a fundamental concept in quantum and reversible computational models. It enables the conditional application of a transformation to a system, depending on the state of another system. We introduce a general framework for diagrammatic reasoning featuring control as a constructor. To this end, we provide an elementary axiomatization of control functors, extending the standard formalism of PROPs to controlled PROPs. As an application, we show that controlled PROPs facilitate diagrammatic reasoning for quantum circuits. In particular, they admit a simple and complete axiomatization involving at most three qubits, whereas in the standard PROP setting, completeness necessarily requires axioms involving arbitrarily many qubits.
High-dimensional quantum computation needs a native circuit-level equational theory for qudits. We give the first finite schematic equational theory that is sound and complete for exact unitary qudit circuits in every finite dimension at least two. Circuits are built from local gates, sequential and parallel composition, and value-controls; equality is derivable exactly when the standard unitary denotations agree. For each dimension, a finite list of local bounded-arity axiom schemata presents the theory, and the diagrammatic shapes do not depend on d. Primitive value-control makes control on a chosen basis value part of the language, so local rules generate the internal algebra of controlled operations within the circuit PROP. This gives a finite, dimension-uniform basis for exact equational reasoning about qudit circuits.
In the models of quantum computing usually considered, some quantum data is manipulated by means of operations which are controlled in an essentially classical way. Controlling these operations in a quantum way is actually possible, but has been much less studied. In particular, quantum control misses a formalism in which one could represent it in a simple way in order to efficiently reason on processes involving it. The first contribution of this thesis is to lay the foundations of a formal framework dedicated to quantum control, in the form of a graphical language. Our main result about this language is the introduction of a complete equational theory, that is, a set of equations that makes it possible, by successive local rewriting, to transform a given diagram into any other diagram representing the same program or physical process. A second contribution is to apply this formalism, on the one hand, to the problem of resource optimisation of processes involving quantum control, and on the other hand, to the characterisation of the observational equivalence of quantum communication channels. A third contribution of this thesis is to introduce a language for linear optical circuits. We equip this language with a complete equational theory, together with a simple normal form, reachable via a strongly normalising and confluent rewriting system. The last contribution of this thesis, maybe the most significant one, is to introduce a complete equational theory for the language of quantum circuits. We obtain this result by exploiting a correspondence between quantum circuits and optical circuits, which allows us to transfer the equational theory already obtained for optical circuits.
Quantum circuits form the standard framework of quantum computing, and have started playing the same role for some parts of quantum foundations. However, they cannot satisfactorily capture some possibilities offered by quantum theory, in particular coherent control and indefinite causal order. In this thesis, we therefore propose an extension to this framework, given by routed quantum circuits. Our extension is based on the addition of sectorial constraints, which specify the basic sectorial structure of the scenarios at hand, on top of the factorial structure described by the connectivity of circuits. This yields a consistent and scalable framework, applicable to pure and mixed quantum theory. It admits a sound and intuitive diagrammatic notation. We show that this framework allows to properly model all forms of coherent control of the application of a channel, for instance the superposition of a quantum information carrier's trajectory, or the 'extended circuit diagrams' recently introduced for the study of causal decompositions. We demonstrate how this leads to a redefinition of the task of coherently controlling a channel, and study in detail the necessary resources for this task. We further show that the scenarios featuring indefinite causal order also fit within our framework once feedback loops are introduced. The connectivity and sectorial constraints are then sufficient to specify the core behaviour of a process, and in particular to check that it is valid, i.e. that its cycles do not lead to any logical inconsistency. We display how several standard examples of exotic processes, including ones that violate causal inequalities, are among the class of processes that can be generated in this way; we conjecture that this class in fact includes all unitarily extendible processes.
In this report is considered the generalization of the principles, which manage the formalization of controlled qubit operators. The research builds on the standard external data sources, for development of a generalized system for conditional operators. Certain controlled operators are viewed as linear combinations of universal operators. This system provides a generalized scheme for the type of the controlled operators, which could be encountered in a circuit, composed entirely of elementary operators. In this report is presented a formalized generalization of single qubit and controlled operators, based on formalized single qubit operator. This work is part of the developed by the author formalized system for designing of algorithmic models for quantum circuits, based on phase encoding, decod- ing and parameterization of primitive quantum operators. In previous publications of the author (6, 7, 8) were defined sev- eral sets of operators on the n qubit, which generalize certain classical characteristics: identity and logical negation. Moreo- ver, some ways were explored in which can be constructed operators as linear combinations of elements from those sets. Such combinations capture the partial application of an opera- tor together with another operator, which in a broader sense is its logical negation.
Qubits are a great way to build a quantum computer, but a limited way to program one. We replace the usual "states and gates" formalism with a "props and ops" (propositions and operators) model in which (a) the C*-algebra of observables supplies the syntax; (b) states, viewed as linear functionals, give the semantics; and (c) a novel diagrammatic calculus unifies the two. The first part develops the basic objects of the framework, encoding consistent patterns of operator correlation, recovering Hilbert space via the GNS construction, and re-deriving the Bloch sphere as the set of all consistent correlations of operators in the Pauli algebra. We then turn to intervention, showing how measurement modifies state, proving an operator-algebraic version of the Knill-Laflamme conditions, and expressing stabilizer codes with the same diagrammatic machinery. This provides a concise, representation-agnostic account of quantum error correction. The result is a self-contained foundation in which C*-algebras, and their dual Hilbert spaces, offer a rich and universal substrate for quantum programming; forthcoming papers will build a high-level language and quantum software applications on top of this substrate.
This report introduces a novel class of reasoning architectures, termed Quantum Circuit Reasoning Models (QCRM), which extend the concept of Variational Quantum Circuits (VQC) from energy minimization and classification tasks to structured logical inference and reasoning. We posit that fundamental quantum mechanical operations, superposition, entanglement, interference, and measurement, naturally map to essential reasoning primitives such as hypothesis branching, constraint propagation, consistency enforcement, and decision making. The resulting framework combines quantum-inspired computation with differentiable optimization, enabling reasoning to emerge as a process of amplitude evolution and interference-driven selection of self-consistent states. We develop the mathematical foundation of QCRM, define its parameterized circuit architecture, and show how logical rules can be encoded as unitary transformations over proposition-qubit states. We further formalize a training objective grounded in classical gradient descent over circuit parameters and discuss simulation-based implementations on classical hardware. Finally, we propose the Quantum Reasoning Layer (QRL) as a differentiable hybrid component for composable reasoning models applicable to scientific, biomedical, and chemical inference domains.