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.
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.
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
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.
Variational quantum algorithms (VQAs) rely on parameterized quantum circuits (PQCs), whose performance is governed by expressibility and trainability. Existing studies typically evaluate these properties at the logical circuit level, implicitly assuming that designed PQCs remain unchanged during hardware execution. In practice, however, hardware-aware transpilation modifies circuit structure through qubit mapping, routing, and basis decomposition, potentially altering PQC behavior. In this paper, we perform a systematic hardware-aware analysis of expressibility and trainability by comparing logical and transpiled PQCs across multiple ansatz families, qubit counts, and circuit depths. Expressibility is measured using fidelity-based KL divergence, while trainability is quantified through gradient variance. Our results show that transpilation acts as an implicit architectural perturbation, producing strongly ansatz-dependent effects. Expressibility deviations exceed upto 125% in some cases, while trainability variations reach up to 25%. Structured ansatzes are generally more robust, whereas highly entangled architectures are more sensitive to transpilation-induced transformations. We further show that transpilation can alter the commonly assumed expressibility-trainability trade-off, demonstrating that logical-level analyses may not reliably predict hardware-level behavior. These findings highlight the importance of hardware-aware evaluation for accurate characterization of VQAs.
Abstract To execute quantum circuits on a quantum processor, they must be modified to meet the physical constraints of the quantum device. This process, called quantum circuit mapping , results in a gate/circuit depth overhead that depends on both the circuit properties and the hardware constraints, being the limited qubit connectivity a crucial restriction. In this paper, we propose to extend the characterization of quantum circuits by including qubit interaction graph properties using graph theory-based metrics in addition to previously used circuit-describing parameters. This approach allows for an in-depth analysis and clustering of quantum circuits and a comparison of performance when run on different quantum processors, aiding in developing better mapping techniques. Our study reveals a correlation between interaction graph-based parameters and mapping performance metrics for various existing configurations of quantum devices. We also provide a comprehensive collection of quantum circuits and algorithms for benchmarking future compilation techniques and quantum devices.
Quantum computing offers the potential for revolutionary advancements across diverse fields, demanding efficient algorithms and hardware architectures. Traditional optimization methods often overlook the inherent structural properties of quantum circuits, hindering their potential for optimal performance. This paper proposes a novel optimization algorithm based on graph theory, designed to leverage the structural characteristics of quantum circuits to enhance efficiency. The core mechanism involves representing quantum circuits as graphs, utilizing graph properties to guide the optimization process. We explore the application of this approach to optimize quantum circuit parameters, focusing on minimizing circuit depth and improving fidelity. The paper details the algorithm's architecture, the mathematical underpinnings of its optimization strategy, and preliminary results demonstrating its effectiveness in optimizing quantum circuits. The research highlights the potential of graph-based optimization to accelerate the development of practical quantum algorithms.