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Sovremennye informacionnye tehnologii i IT-obrazovanie.

Квантовые схемы и квантовое машинное обучение

C. B. Pronin

Published Jul 20, 2026• 0 citations
OpenAlex

Abstract

A consolidated review of studies, forming a single line of work on constructing and applying quantum circuits for machine learning: From the Deutsch-Jozsa Algorithm to Training Quantum Neural Networks Based on Grover’s Algorithm. The matrix formalism of transformations in quantum circuits is summarised, and the Deutsch-Jozsa algorithm is implemented step by step by manual matrix calculation, matching the Quirk simulation. Reversible quantum analogues of classical logic elements (NOT, AND, OR, XOR) and circuits — a full adder and a 2-bit multiplier — are constructed; a reversible multiplier is used as an oracle function of Grover's algorithm. A principle for building mathematical oracle functions that rotate only the amplitude of the solution while uncomputing auxiliary operations is formulated, and the gain of Grover iterations over brute-force search is estimated (about 326 times for a 16-qubit register). A concept of quantum circuits for perceptron training with an oracle built from the activation function is proposed; training of a single-weight quantum perceptron is demonstrated. Circuit optimisation by composing gate sequences into a single operation and debugging Grover's algorithm on IBM Q prototypes, which revealed noise growth with circuit depth, are reported. Limitations (2-3-qubit registers, simulators and NISQ devices, arithmetic bit width) and further research directions are discussed.

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A consolidated review of studies, forming a single line of work on constructing and applying quantum circuits for machine learning: From the Deutsch-Jozsa Algorithm to Training Quantum Neural Networks Based on Grover’s Algorithm. The matrix formalism of transformations in quantum circuits is summarised, and the Deutsch-Jozsa algorithm is implemented step by step by manual matrix calculation, matching the Quirk simulation. Reversible quantum analogues of classical logic elements (NOT, AND, OR, XOR) and circuits — a full adder and a 2-bit multiplier — are constructed; a reversible multiplier is used as an oracle function of Grover's algorithm. A principle for building mathematical oracle functions that rotate only the amplitude of the solution while uncomputing auxiliary operations is formulated, and the gain of Grover iterations over brute-force search is estimated (about 326 times for a 16-qubit register). A concept of quantum circuits for perceptron training with an oracle built from the activation function is proposed; training of a single-weight quantum perceptron is demonstrated. Circuit optimisation by composing gate sequences into a single operation and debugging Grover's algorithm on IBM Q prototypes, which revealed noise growth with circuit depth, are reported. Limitations (2-3-qubit registers, simulators and NISQ devices, arithmetic bit width) and further research directions are discussed.
This work contains the analysis of results received after running synthesized quantum circuits for training perceptron neural networks. The training is performed by creating a Grover’s algorithm with a custom oracle function. The concept of synthesizing quantum circuits was showcased in the process of generating training circuits for three perceptron topologies, which were designed to test the accuracy of the synthesis process. The test circuits serve to prove that the proposed synthesis approach could be scaled to utilize more complex quantum computing systems and to solve more practical tasks. IBM’s 100-qubit cloud quantum simulator was used as the debugging environment. Quantum circuits for described algorithms are generated by the "Naginata" quantum synthesizer, its source code is published and further documented on GitHub along with the code for the provided example algorithms. The article describes the processes behind the algorithm for synthesizing quantum circuits that perform the training process of single-layer perceptrons by finding their weights by filtering all possible input values through a predefined accuracy criterion. Since quantum computing is still in its early development phase, quantum circuits are created mainly by manual placement of logic elements. Implementing quantum algorithms, especially more use-case specific ones, directly on the quantum circuit level could lead to the circuit easily becoming too complex for human comprehension. Quantum Circuit Synthesizer "Naginata" was created to simplify the development and debugging process of quantum algorithms, by adding better clarity to their development process. In our case, better clarity for the development process is achieved by composing functions for commonly used operations performed in the implemented quantum algorithm. The programmer could now implement the quantum algorithm as a set of functions, instead of manually creating a circuit from single logic elements. After this, the synthesizer would handle the task of creating the data for placing logic elements on the circuit. This enables an opportunity of implementing quantum algorithms with higher-level commands. In the scope of this work, parametrically generated generic blocks for frequently used operations such as: the adder, multiplier and digital comparator were created and utilized to form the training circuits. The test results, proved that with the help of the proposed quantum synthesizer, these compositions could be used efficiently as building blocks for implementing quantum algorithms. And by visually comparing sizes of both code and circuit representations of the synthesized circuits, to the code examples used to synthesize these circuits, it is determined that the proposed approach for implementing quantum circuits greatly simplifies the processes of development and debugging a quantum algorithm.
This paper showcases a method of parametric synthesis of quantum circuits for training perceptron neural networks. Synapse weights are found using Grover's algorithm with a modified oracle function. The results of running these parametrically synthesized circuits for training perceptrons of three different topologies are described. The circuits were run on a 100-qubit IBM quantum simulator. The synthesis of quantum circuits is carried out using quantum synthesizer "Naginata", which was developed in the scope of this work, the source code of which is published and further documented on GitHub. The article describes the quantum circuit synthesis algorithm for training single-layer perceptrons. At the moment, quantum circuits are created mainly by manually placing logic elements on lines that symbolize quantum bits. The purpose of creating Quantum Circuit Synthesizer "Naginata" was due to the fact that even with a slight increase in the number of operations in a quantum algorithm, leads to the significant increase in size of the corresponding quantum circuit. This causes serious difficulties both in creating and debugging these quantum circuits. The purpose of our quantum synthesizer is enabling users an opportunity to implement quantum algorithms using higher-level commands. This is achieved by creating generic blocks for frequently used operations such as: the adder, multiplier, digital comparator (comparison operator), etc. Thus, the user could implement a quantum algorithm by using these generic blocks, and the quantum synthesizer would create a suitable circuit for this algorithm, in a format that is supported by the chosen quantum computation environment. This approach greatly simplifies the processes of development and debugging a quantum algorithm.
This paper highlights a practical research of the possibility of forming quantum circuits for training neural networks. The demonstrated quantum circuits were based on the principles of Grover's Search Algorithm. The perceptron was chosen as the architecture for the example neural network. The multilayer perceptron is a popular neural network architecture due to its scalability and applicability for solving a wide range of problems. Large scale neural networks can require significant computing resources for training purposes, therefore Grover’s quantum search algorithm could be used as a way to speed up this process, thanks to its ability of simultaneously processing multiple input values of a given function.
This paper highlights the possibility of creating quantum neural networks that are trained by Grover's Search Algorithm. The purpose of this work is to propose the concept of combining the training process of a neural network, which is performed on the principles of Grover's algorithm, with the functional structure of that neural network, interpreted as a quantum circuit. As a simple example of a neural network, to showcase the concept, a perceptron with one trainable parameter - the weight of a synapse connected to a hidden neuron.
Perceptron model is a fundamental linear classifier in machine learning and also the building block of artificial neural networks. Recently, Wiebe et al. (arXiv:1602.04799) proposed that the training of a perceptron can be quadratically speeded using Grover search with a quantum computer, which has potentially important big-data applications. In this paper, we design a quantum circuit for implementing this algorithm. The Grover oracle, the central part of the circuit, is realized by Quantum-Fourier-Transform based arithmetics that specifies whether an input weight vector can correctly classify all training data samples. We also analyze the required number of qubits and universal gates for the algorithm, as well as the success probability using uniform sampling, showing that it has higher possibility than spherical Gaussian distribution N(0,1)N(0,1)N(0,1). The feasibility of the circuit is demonstrated by a testing example using the IBM-Q cloud quantum computer, where 16 qubits are used to classify four data samples.
В данной работе рассматривается возможность создания квантовых нейронных сетей, обучаемых с помощью алгоритма Гровера. Цель данной работы заключается в совмещении процесса обучения нейронной сети, производимого с помощью алгоритма Гровера, и функциональной структуры нейронной сети, интерпретированной в виде квантовой схемы. В качестве простого примера нейронной сети, для рассмотрения концепции, был предложен персептрон с одним обучаемым параметром – весом синапса, входящего в скрытый нейрон. This paper discusses the possibility of creating quantum neural networks trained using Grover’s algorithm. The purpose of this work is to combine the learning process of the neural network, produced using Grover’s algorithm, and the functional structure of the neural network, interpreted as a quantum circuit. As a simple example of a neural network, to consider the concept, a perceptron was proposed with one learning parameter – the weight of a synapse entering a hidden neuron.