Abstract Most existing quantum algorithms are discovered accidentally or adapted from classical algorithms, and there is the need for a systematic theory to understand and design quantum circuits. Here we develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. Compared to the conventional entanglement description of quantum circuits and states, the unitary dependence picture offers more practical information on the measurement and manipulation of qubits, easier generalization to many-qubit systems, and better robustness upon partitioning of the system. The unitary dependence theory can be applied to systematically understand existing quantum circuits and design new quantum algorithms.
Abstract Most existing quantum algorithms are discovered accidentally or adapted from classical algorithms, and there is the need for a systematic theory to understand and design quantum circuits. Here we develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. Compared to the conventional entanglement description of quantum circuits and states, the unitary dependence picture offers more practical information on the measurement and manipulation of qubits, easier generalization to many-qubit systems, and better robustness upon partitioning of the system. The unitary dependence theory can be applied to systematically understand existing quantum circuits and design new quantum algorithms.
We develop a unitary dependence theory to characterize the behaviors of quantum circuits and states in terms of how quantum gates manipulate qubits and determine their measurement probabilities. A qubit has dependence on a 1-qubit unitary gate if its measurement probabilities depend on the parameters of the gate. A 1-qubit unitary creates such a dependence onto the target qubit, and a CNOT gate copies all the dependences from the control qubit to the target qubit. The complete dependence picture of the output state details the connections qubits may have when being manipulated or measured. Compared to the conventional entanglement description of quantum circuits and states, the dependence picture offers more practical information, easier generalization to many-qubit systems, and better robustness upon partitioning of the system. The unitary dependence theory is a useful tool for understanding quantum circuits and states that is based on the practical ideas of how manipulations and measurements are affected by elementary gates.
We propose a theory of classifying quantum circuits with qubit functional configurations. Any quantum circuit can be decomposed into alternating sequences of 1-qubit unitary gates and CNOT gates. Each CNOT sequence prepares the current quantum state into a layer of qubit functional configuration to specify the rule for the next 1-qubit unitary sequence on how to collectively modify the state vector entries. The functional configuration layers on a quantum circuit define its circuit type. These functional configuration types characterize important properties of the corresponding quantum circuits such as the circuit complexity, providing systematic knowledge of large collections of quantum circuits. We demonstrate the theory by analyzing the quantum Fourier transform circuit with the functional configuration picture. For potential applications, the functional configuration picture may allow systematic understanding of quantum algorithms based on their quantum circuit types, which can benefit the development of new algorithms.
The paper is an exploration of a new approach to the systematic analysis and classificationof quantum circuits based on the functional configuration of qubits. The article examines in detailthe role of elementary gates in changing the elements of the state vector and highlights the importanceof functional configurations of qubits in the collective modification of quantum states.The main aspects covered in the article include the characterization of quantum circuits with functionalconfigurations of qubits, analysis of the impact of elementary gates on the state of a quantumvector, and determination of the number of possible types of functional configurations.The results of the study could have important implications for optimizing quantum circuits andimproving our understanding of their general properties. A qubit functional configuration is amathematical structure that can collectively classify the properties and behavior of quantum circuits.The development of quantum algorithms with efficient quantum circuits has been a centralpart of quantum computing, which has seen enormous progress both theoretically and experimentallyover the past 30 years. The paper makes a contribution to the field of quantum computing byproviding a systematic approach to classify and analyze quantum circuits based on their functionalqubit configurations. Quantum algorithms are an innovative class of algorithms based on theprinciples of quantum mechanics and using qubits instead of classical bits to process information.Unlike classical algorithms, which operate on bits that take on the values 0 or 1, quantum algorithmscan use the principles of quantum superposition and quantum interaction, which allowsthem to perform many calculations simultaneously. One of the key advantages of quantum algorithmsis their ability to solve certain problems much more efficiently than classical algorithms.However, the design and implementation of quantum algorithms pose significant technical andalgorithmic challenges, such as managing quantum states, minimizing errors, and creating robustquantum gates. Despite these challenges, quantum algorithms offer promising opportunities torevolutionize computing and solve problems that have traditionally been too complex for classicalcomputers
We develop a systematic framework for characterizing all quantum circuits with qubit functional configurations. The qubit functional configuration is a mathematical structure that can classify the properties and behaviors of quantum circuits collectively. Major benefits of classifying quantum circuits in this way include: 1. All quantum circuits can be classified into corresponding types; 2. Each type characterizes important properties (such as circuit complexity) of the quantum circuits belonging to it; 3. Each type contains a huge collection of possible quantum circuits allowing systematic investigation of their common properties. We demonstrate the theory's application to analyzing the hardware-efficient ansatzes of variational quantum algorithms. For potential applications, the functional configuration theory may allow systematic understanding and development of quantum algorithms based on their functional configuration types.
Abstract This practical guide provides a detailed introduction to quantum logic gates and unitary operators as implemented on the IBM Q platform. It focuses on the fundamental gate structures used in quantum programming, particularly within the Composer interface, and provides screenshots and detailed descriptions of each operator’s functionality. The guide emphasizes the operational role of gates in controlling qubit states and constructing quantum circuits. To enhance understanding, the behavior of gates is visualized on the Bloch sphere, demonstrating how single-qubit operations rotate qubit states around the XXX, YYY, and ZZZ axes, as well as how phase shifts and amplitude adjustments modify the qubit. These operations are essential for measurement, state preparation, and precise manipulation of quantum systems. The relevance of this study is underscored by the rapid development of quantum technologies and the increasing accessibility of quantum computing platforms like IBM Q. A thorough understanding of quantum gates and their implementations is critical for advancing quantum algorithm design and for training specialists capable of deploying quantum computational solutions in fields ranging from cryptography to complex physical system simulations. This guide provides both theoretical insights and practical exercises, bridging the gap between abstract quantum mechanics and applied quantum programming, thereby equipping readers with foundational skills for effective circuit construction and quantum computation.
Quantum computing is an emerging technology that has the potential to change the perspectives and applications of computing in general. A wide range of applications are enabled: from faster algorithmic solutions of classically still difficult problems to theoretically more secure communication protocols. A quantum computer uses the quantum mechanical effects of particles or particle-like systems, and a major similarity between quantum and classical computers consists of both being abstracted as information processing machines. Whereas a classical computer operates on classical digital information, the quantum computer processes quantum information, which shares similarities with analog signals. One of the central differences between the two types of information is that classical information is more fault-tolerant when compared to its quantum counterpart. Faults are the result of the quantum systems being interfered by external noise, but during the last decades quantum error correction codes (QECC) were proposed as methods to reduce the effect of noise. Reliable quantum circuits are the result of designing circuits that operate directly on encoded quantum information, but the circuit’s reliability is also increased by supplemental redundancies, such as sub-circuit repetitions. Reliable quantum circuits have not been widely used, and one of the major obstacles is their vast associated resource overhead, but recent quantum computing architectures show promising scalabilities. Consequently the number of particles used for computing can be more easily increased, and that the classical control hardware (inherent for quantum computation) is also more reliable. Reliable quantum circuits haev been investigated for almost as long as general quantum computing, but their limited adoption (until recently) has not generated enough interest into their systematic design. The continuously increasing practical relevance of reliability motivates the present thesis to investigate some of the first answers to questions related to the background and the methods forming a reliable quantum circuit design stack. The specifics of quantum circuits are analysed from two perspectives: their probabilistic behaviour and their topological properties when a particular class of QECCs are used. The quantum phenomena, such as entanglement and superposition, are the computational resources used for designing quantum circuits. The discrete nature of classical information is missing for quantum information. An arbitrary quantum system can be in an infinite number of states, which are linear combinations of an exponential number of basis states. Any nontrivial linear combination of more than one basis states is called a state superposition. The effect of superpositions becomes evident when the state of the system is inferred (measured), as measurements are probabilistic with respect to their output: a nontrivial state superposition will collapse to one of the component basis states, and the measurement result is known exactly only after the measurement. A quantum system is, in general, composed from identical subsystems, meaning that a quantum computer (the complete system) operates on multiple similar particles (subsystems). Entanglement expresses the impossibility of separating the state of the subsystems from the state of the complete system: the nontrivial interactions between the subsystems result into a single indivisible state. Entanglement is an additional source of probabilistic behaviour: by measuring the state of a subsystem, the states of the unmeasured subsystems will probabilistically collapse to states from a well defined set of possible states. Superposition and entanglement are the building blocks of quantum information teleportation protocols, which in turn are used in state-of-the-art fault-tolerant quantum computing architectures. Information teleportation implies that the state of a subsystem is moved to a second subsystem without copying any information during the process. The probabilistic approach towards the design of quantum circuits is initiated by the extension of classical test and diagnosis methods. Quantum circuits are modelled similarly to classical circuits by defining gate-lists, and missing quantum gates are modelled by the single missing gate fault. The probabilistic approaches towards quantum circuits are facilitated by comparing these to stochastic circuits, which are a particular type of classical digital circuits. Stochastic circuits can be considered an emulation of analogue computing using digital components. A first proposed design method, based on the direct comparison, is the simulation of quantum circuits using stochastic circuits by mapping each quantum gate to a stochastic computing sub-circuit. The resulting stochastic circuit is compiled and simulated on FPGAs. The obtained results are encouraging and illustrate the capabilities of the proposed simulation technique. However, the exponential number of possible quantum basis states was translated into an exponential number of stochastic computing elements. A second contribution of the thesis is the proposal of test and diagnosis methods for both stochastic and quantum circuits. Existing verification (tomographic) methods of quantum circuits were targeting the reconstruction of the gate-lists. The repeated execution of the quantum circuit was followed by different but specific measurement at the circuit outputs. The similarities between stochastic and quantum circuits motivated the proposal of test and diagnosis methods that use a restricted set of measurement types, which minimise the number of circuit executions. The obtained simulation results show that the proposed validation methods improve the feasibility of quantum circuit tomography for small and medium size circuits. A third contribution of the thesis is the algorithmic formalisation of a problem encountered in teleportation-based quantum computing architectures. The teleportation results are probabilistic and require corrections represented as quantum gates from a particular set. However, there are known commutation properties of these gates with the gates used in the circuit. The corrections are not applied as dynamic gate insertions (during the circuit’s execution) into the gate-lists, but their effect is tracked through the circuit, and the corrections are applied only at circuit outputs. The simulation results show that the algorithmic solution is applicable for very large quantum circuits. Topological quantum computing (TQC) is based on a class of fault-tolerant quantum circuits that use the surface code as the underlying QECC. Quantum information is encoded in lattice-like structures and error protection is enabled by the topological properties of the lattice. The 3D structure of the lattice allows TQC computations to be visualised similarly to knot diagrams. Logical information is abstracted as strands and strand interactions (braids) represent logical quantum gates. Therefore, TQC circuits are abstracted using a geometrical description, which allows circuit input-output transformations (correlations) to be represented as geometric sub-structures. TQC design methods were not investigated prior to this work, and the thesis introduces the topological computational model by first analysing the necessary concepts. The proposed TQC design stack follows a top-down approach: an arbitrary quantum circuit is decomposed into the TQC supported gate set; the resulting circuit is mapped to a lattice of appropriate dimensions; relevant resulting topological properties are extracted and expressed using graphs and Boolean formulas. Both circuit representations are novel and applicable to TQC circuit synthesis and validation. Moreover, the Boolean formalism is broadened into a formal mechanism for proving circuit correctness. The thesis introduces TQC circuit synthesis, which is based on a novel logical gate geometric description, whose formal correctness is demonstrated. Two synthesis methods are designed, and both use a general planar representation of the circuit. Initial simulation results demonstrate the practicality and performance of the methods. An additional group of proposed design methods solves the problem of automatic correlation construction. The methods use validity criteria which were introduced and analysed beforehand in the thesis. Input-output correlations existing in the circuit are inferred using both the graph and the Boolean representation. The thesis extends the TQC state-of-the-art by recognising the importance of correlations in the validation process: correlation construction is used as a sub-routine for TQC circuit validation. The presented cross-layer validation procedure is useful when investigating both the QECC and the circuit, while a second proposed method is QECC-independent. Both methods are scalable and applicable even to very large circuits. The thesis completes with the analysis of TQC circuit identities, where the developed Boolean formalism is used. The proofs of former known circuit identities were either missing or complex, and the presented approach reduces the length of the proofs and represents a first step towards standardising them. A new identity is developed and detailed during the process of illustrating the known circuit identities. Reliable quantum circuits are a necessity for quantum computing to become reality, and specialised design methods are required to support the quest for scalable quantum computers. This thesis used a twofold approach towards this target: firstly by focusing on the probabilistic behaviour of quantum circuits, and secondly by considering the requirements of a promising quantum computing architecture, namely TQC. Both approaches resulted in a set of design methods enabling the investigation of reliable quantum circuits. The thesis contributes with the proposal of a new quantum simulation technique, no