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.
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.
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.
In contrast to a classical bit, which can only take the value 0 or 1, its quantum counterpart -- a qubit -- can exist in a superposition of 0 and 1. This is a superposition of data values, naturally raising the question of whether one can superpose not only data but also programs. For example, a particular superposition of programs, known as the quantum SWITCH, has attracted much attention, and its implementations and computational advantages have been studied extensively within the physics community.
A naive way to control a program by a qubit is by means of a controlled operation. Given an operation F, this amounts to considering an operation that behaves as F when the control qubit is ∣1⟩, and as the identity operation when the control qubit is ∣0⟩. This idea works well when F is a unitary operation, but it is not well-defined for a general quantum channel. By contrast, the quantum SWITCH is free from the well-definedness issue. This contrast leads to the key insight of this paper: controlled operations and the quantum SWITCH should be regarded as different kinds of quantum control mechanisms.
Building on this insight, we develop a novel quantum programming language with quantum control and measurement that can express the quantum SWITCH over quantum channels. Using a semantic analysis based on program transformations, we identify the source of the ill-behavedness of controlled operations as the \emph{correspondence problem}: a lack of coordination between the measurements performed in the then- and else-branches of quantum conditional branching. We address this problem with a linear type system that enforces alignment of the quantum operations used in the two branches, yielding a well-behaved language capable of expressing the quantum SWITCH.
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.
Quantum computers promise to solve computational problems that are inaccessible to classical machines. The reliable execution of large-scale quantum algorithms requires robust operations in the presence of noise. Quantum error correction addresses this challenge by encoding quantum information redundantly across several physical qubits, such that errors can be detected and corrected during a computation. However, no single error-correcting code intrinsically supports an inherently robust, fault-tolerant implementation of all gates needed for universal quantum computing. Moreover, most modern approaches to both quantum error correction and the implementation of universal logical operations on encoded qubits rely on mid-circuit measurements and feed-forward control. These types of operations are technologically demanding, often slow, and prone to high error rates, and they limit the overall success probability of quantum algorithms. In this thesis, we develop fault-tolerant protocols and circuits for quantum error correction and universal logical computation, with an emphasis on implementations compatible with current hardware systems. We first focus on fault-tolerant quantum error correction on small codes. By comparing different protocols, we identify trade-offs between compact circuit constructions and their robustness to noise. We then assess their practical performance by comparing numerical simulations of noisy circuits with experimental data. Beyond the preservation of encoded information, we study how universal logic can be implemented in a fault-tolerant manner on encoded qubits. A central approach considered in this thesis is code switching, in which logical information is transferred between two distinct error-correcting codes that support complementary sets of inherently fault-tolerant logical gates. In combination, these codes give access to a complete universal fault-tolerant logical gate set. In this thesis, we determine conditions that allow us to construct fault-tolerant protocols for code switching. We adapt these protocols to the capabilities of current hardware, thereby enabling the first experimental realization of universal fault-tolerant logic by means of code switching. However, the overall performance of these schemes is limited by slow and noisy mid-circuit measurements. To address this bottleneck, we construct a suite of measurement-free fault-tolerant universal operations that do not require any measurements during the execution of an algorithm. In this framework, error correction and logical operations are implemented coherently as part of the quantum circuit, without mid-circuit measurements or in-sequence classical control. We introduce measurement-free code-switching schemes that can be scaled to arbitrary code distances through concatenation, and which thereby provide, in principle, arbitrarily strong error suppression. We further tailor our approach to the capabilities of current trapped-ion quantum processors. New circuit constructions support the first experimental implementation of a logical quantum algorithm without mid-circuit measurements, demonstrated through Grover's search on three encoded qubits. Together, the results of this thesis expand the toolbox for early fault-tolerant quantum computation. They show how protocols for quantum error correction and universal logical gates can be designed without mid-circuit measurements with realistic hardware constraints in mind, opening new pathways towards universal quantum logic.
Abstract In quantum computational logics meaning of sentences are repre-sented by quantum information quantities. Any n-qubit sequence is identified with a unit vector of the tensor product of the complex Hilbert space ⊗nC2. Let us split the set of the vectors of its computational basis in the two sets: Bn1: = {i: ‖ i À= | x1,.., xn 〉 and xn = 1}; Bn0: = {i: ‖ i À= | x1,.., xn 〉 and xn = 0}. Any unit vector | ψ 〉 can be expressed in the form: | ψ 〉 = ∑i∈B0 ai ‖ iÀ +∑j∈B1 bj ‖ j À where by the Born rule,∑ i∈B0 | ai |2 + j∈B1 | bj |2 = 1 A key role in the definition of semantics is played by the probability-value of | ψ〉, defined as follows: p( | ψ〉): = ∑j∈B1 | bj |2 Moreover a function g: [0, 1]2 7 → [0, 1] is said quantum simulable iff ∃n ≥ 1, a unitary operator Ug: ⊗n+2C2 7 → ⊗n+2C2 and a quregister | χ 〉 ∈ ⊗nC2 such