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Classifying quantum circuits with qubit functional configurations.

Zixuan Hu, Sabre Kais

Published Oct 5, 2021• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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 present a deterministic framework for preparing an arbitrary three-qubit pure state. To leverage entanglement structure in the state-preparation task, we classify three-qubit pure states into five types with respect to a 1∣21|21∣2 bipartition. Given a target state specified by its amplitudes, we provide concrete criteria and concurrence-based tests that determine its type. For each type, we derive an explicit circuit template composed of elementary single-qubit rotations and CNOT gates, with gate parameters determined systematically from the Schmidt decomposition. The full construction is described step by step from the target amplitudes, with no procedural ambiguity. As an application, we further group frequently encountered three-qubit pure states in quantum information into four classes and provide an explicit circuit for each class. Compared with prior approaches, our circuits are designed for practical use: they admit a direct algorithmic instantiation, use only CNOT gates between adjacent qubits, and for certain classes achieve smaller gate counts and circuit depth.
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.
In order to demonstrate nontrivial quantum computations experimentally, such as the synthesis of arbitrary entangled states, it will be useful to understand how to decompose a desired quantum computation into the shortest possible sequence of one-qubit and two-qubit gates. We contribute to this effort by providing a method to construct an optimal quantum circuit for a general two-qubit gate that requires at most 3 controlled-NOT (CNOT) gates and 15 elementary one-qubit gates. Moreover, if the desired two-qubit gate corresponds to a purely real unitary transformation, we provide a construction that requries at most 2 CNOT and 12 one-qubit gates. We then prove that these constructions are optimal with respect to the family of CNOT, yyy-rotation, zzz-rotation, and phase gates.