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ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА

Единственность оператора диффузии Гровера для квантового алгоритма поиска Гровера: Аннотации

M. T. Dima, M. Dima, M. Mihailescu, М. Т. Дима, М. Т. Дима, М. Михаилеску

Published Sep 24, 2025• 0 citations
OpenAlex

Abstract

The repetition of Grover-diffusion operator of the order of $\sqrt{N}$ times is the essence of the Grover quantum selection algorithm. For large N the coherence time $T \,2$ of the qubits limits the application of the algorithm. We explore what other operators could be devised in its place to accelerate convergence. We present a $C++ SU(2)$ model of the Grover-diffusion operator implemented using our $SU2$ package.

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The repetition of Grover-diffusion operator of the order of N\sqrt{N}N​ times is the essence of the Grover quantum selection algorithm. For large N the coherence time T 2T \,2T2 of the qubits limits the application of the algorithm. We explore what other operators could be devised in its place to accelerate convergence. We present a C++SU(2)C++ SU(2)C++SU(2) model of the Grover-diffusion operator implemented using our SU2SU2SU2 package.
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.
Grover's search algorithm drives a quantum system from an initial state $|s\ensuremath{\rangle}$ to a desired final state $|t\ensuremath{\rangle}$ by using selective phase inversions of these two states. Earlier, we studied a generalization of Grover's algorithm that relaxes the assumption of the efficient implementation of Is{I}_{s}Is​, the selective phase inversion of the initial state, also known as a diffusion operator. This assumption is known to become a serious handicap in cases of physical interest. Our general search algorithm works with almost any diffusion operator Ds{D}_{s}Ds​ with the only restriction of having $|s\ensuremath{\rangle}$ as one of its eigenstates. The price that we pay for using any operator is an increase in the number of oracle queries by a factor of O(B)O(B)O(B), where BBB is a characteristic of the eigenspectrum of Ds{D}_{s}Ds​ and can be large in some situations. Here we show that by using a quantum Fourier transform, we can regain the optimal query complexity of Grover's algorithm without losing the freedom of using any diffusion operator for quantum searching. However, the total number of operators required by the algorithm is still O(B)O(B)O(B) times more than that of Grover's algorithm. So our algorithm offers an advantage only if the oracle operator is computationally more expensive than the diffusion operator, which is true in most search problems.
The difference between classical and quantum algorithms (QA) is following: problem solved by QA is coded in the structure of the quantum operators.Input to QA in this case is always the same.Output of QA says which problem coded.In some sense, give a function to QA to analyze and QA returns its property as an answer without quantitative computing.QA studies qualitative properties of the functions.The core of any QA is a set of unitary quantum operators or quantum gates.In practical representation, quantum gate is a unitary matrix with particular structure.The size of this matrix grows exponentially with an increase in the number of inputs, which significantly limits the QA simulation on a classical computer with von Neumann architecture.Quantum search algorithm (QSA)models apply for the solution of computer science problems as searching in unstructured data base, quantum cryptography, engineering tasks, control system design, robotics, smart controllers, etc. Grover's algorithm is explained in details along with implementations on a local computer simulator.The presented article describes a practical approach to modeling one of the most famous QA on classical computers, the Grover algorithm.
We have developed a non-unitary extension of Grover's search algorithm by changing the hidden geometry of Hilbert space carried by diffusion operator. Our algorithm finds the solution for search problem by performing a unique bigger rotation rather than small rotations in order polynomial times in the size NNN of search space. We analyze the complexity of implementing the non-unitary operation and we observed that the price paid by performing this rotation is due the normalization. In Kraus operator approach we need O(N)O(N)O(N) repetition of the algorithm to have a chance of measuring a solution in a post-selection, this is no better than the classical solution. However, the quantum singular value transform in addition with block encoding and Chebyshev polynomial approximation, we got complexity O(N)O(\sqrt{N})O(N​) and reach the Grover's bound with an extra resource of one single qubit, compared with the standard Grover's algorithm.
Целью данной работы, является практическое исследование возможности формирования квантовых схем для машинного обучения нейронных сетей. За основу демонстрируемых квантовых схем были взяты принципы работы квантового алгоритма Гровера (Grover’s Search Algorithm). За архитектуру примера обучаемой нейронной сети был взят персептрон. Многослойный персептрон является достаточно популярной архитектурой нейронной сети за счет масштабируемости и возможности применения для решения широкого спектра задач. The purpose of this work is a practical study of the possibility of forming quantum schemes for machine learning neural networks. The principles of operation of the Grovers Search Algorithm (Grover’s Search Algorithm) were taken as the basis of the quantum schemes. For the architecture of an example of a trained neural network was taken by Perseceptron. The multilayer perceptron is a fairly popular neural network architecture due to scalability and the possibility of applying a wide range of tasks.