A stochastic model and an adaptive method for estimating the dominant states of a quantum system have been developed. The proposed approach demonstrates a stable advantage over the frequency-based method under the influence of noise and fluctuations in the measurement process. The dominant state is defined as the most probable basis state into which the wave function of the quantum system collapses after the act of measurement. Determining the dominant state is essential for improving the accuracy of quantum measurements and quantum state reconstruction. The measurement process is modeled on an ensemble of identical quantum systems prepared in the same superposition of basis states. This approach allows the continuation of a series of measurements after the collapse of individual ensemble elements, enabling the accumulation of statistical data and the formation of a reliable estimate of the observation probabilities for each state. The obtained data are used for adaptive updating of the algorithm’s parameters and for constructing a more accurate estimate of the system’s dominant state. The adaptive method is based on a Markovian mechanism of stochastic amplification of the probability of correctly predicting the outcome of measurements of discrete quantum states. The deviation between the predicted and actual results is treated as a penalty function, which is used for dynamic adjustment of the prior probabilities. This approach enables gradual refinement of the dominant state prediction through iterative learning. Based on the proposed method, an adaptive algorithm for predicting the dominant state of a quantum system has been developed and implemented as a computational procedure. The operating principle of the algorithm lies in the multi-stage updating of prediction probabilities for the dominant state, depending on the current value of the penalty function, with the goal of minimizing the expected loss. The convergence of the adaptive process to the actual dominant state of the system has been confirmed by computer simulations. Studies were conducted on the influence of algorithm parameters and quantum system characteristics on accuracy and convergence speed. It was found that the average convergence rate of the adaptive method depends on the intensity of noise affecting the measurement channel, the probabilistic gap between the quantum states, and the learning step parameters. Increasing the variance of white noise or decreasing the probabilistic gap between states leads to slower algorithm convergence, which is consistent with analytical estimates from stochastic optimization theory. The obtained results make it possible to define the necessary constraints on learning parameters to ensure stable convergence under uncertainty. Practical recommendations have been formulated for applying the developed method to problems of quantum computing and quantum information processing.
A stochastic model and an adaptive method for estimating the dominant states of a quantum system have been developed. The proposed approach demonstrates a stable advantage over the frequency-based method under the influence of noise and fluctuations in the measurement process. The dominant state is defined as the most probable basis state into which the wave function of the quantum system collapses after the act of measurement. Determining the dominant state is essential for improving the accuracy of quantum measurements and quantum state reconstruction. The measurement process is modeled on an ensemble of identical quantum systems prepared in the same superposition of basis states. This approach allows the continuation of a series of measurements after the collapse of individual ensemble elements, enabling the accumulation of statistical data and the formation of a reliable estimate of the observation probabilities for each state. The obtained data are used for adaptive updating of the algorithm’s parameters and for constructing a more accurate estimate of the system’s dominant state. The adaptive method is based on a Markovian mechanism of stochastic amplification of the probability of correctly predicting the outcome of measurements of discrete quantum states. The deviation between the predicted and actual results is treated as a penalty function, which is used for dynamic adjustment of the prior probabilities. This approach enables gradual refinement of the dominant state prediction through iterative learning. Based on the proposed method, an adaptive algorithm for predicting the dominant state of a quantum system has been developed and implemented as a computational procedure. The operating principle of the algorithm lies in the multi-stage updating of prediction probabilities for the dominant state, depending on the current value of the penalty function, with the goal of minimizing the expected loss. The convergence of the adaptive process to the actual dominant state of the system has been confirmed by computer simulations. Studies were conducted on the influence of algorithm parameters and quantum system characteristics on accuracy and convergence speed. It was found that the average convergence rate of the adaptive method depends on the intensity of noise affecting the measurement channel, the probabilistic gap between the quantum states, and the learning step parameters. Increasing the variance of white noise or decreasing the probabilistic gap between states leads to slower algorithm convergence, which is consistent with analytical estimates from stochastic optimization theory. The obtained results make it possible to define the necessary constraints on learning parameters to ensure stable convergence under uncertainty. Practical recommendations have been formulated for applying the developed method to problems of quantum computing and quantum information processing.
Quantum state estimation is one of the most extensively studied topics in the field of quantum theory and quantum information. This process is essential in quantum protocols to ensure that the estimated state matches the true state. Typically, the quantum state of identical systems can be estimated from the statistics of measurement outcomes across different measurement bases. In this research, we study adaptive-measurement strategies for quantum state estimation used to identify a pure state of two-level systems and compare the obtained estimated state to the state obtained when the measurement bases are chosen randomly. The adaptive measurements in this study include the maximum-information-gain strategy, and the confirmation strategy. In this work, we simulate the outcome of the measurement on a system and use the Bayesian estimation method to estimate the obtained state. We then adaptively change the basis of the next measurement in accordance with the estimated state and the used adaptive strategy. We find that the maximum-information-gain strategy provides the fastest convergence and the most accurate state estimation compared to the confirmation strategy and the randomly-chosen-measurement-basis strategy, especially in the presence of measurement noise. Keywords: Quantum state estimation, State tomography, Adaptive measurements
This paper presents a hybrid quantum logistic regression model for binary data classification that combines classical logistic regression methods with quantum computing to improve efficiency and speed up the learning process. Quantum computing is used to represent and process feature and weight vectors as quantum states and to efficiently compute the dot product underlying the logistic function. The use of normalized amplitude coding allows for compact representation of feature vectors and weights, reducing the number of qubits required and potentially speeding up computations for high-dimensional classification problems where classical methods may encounter scalability limitations. The quantum dot product computation is performed using the Hadamard test, which provides a statistical estimate of the probability of measuring an auxiliary qubit. The auxiliary qubit is first put into a superposition state, and after performing controlled unitary operations, amplitude interference is formed, which depends on the scalar product of the feature vector and weight coefficients. The obtained measurement probability is used to estimate the scalar product of the feature vectors and weights. The classical part of the model provides for the execution of operations that are difficult to implement on modern quantum devices, in particular, vector normalization, application of the sigmoid function to estimate the probability of an object belonging to a certain class, calculation of the loss function and iterative update of weight coefficients by the gradient descent method. The purpose of classification is to predict the binary class for an unknown sample based on the training sample. To demonstrate the model, a numerical example is given with a step-by-step calculation of one iteration of logistic regression training. In addition, computer modeling of the hybrid algorithm training process was performed, taking into account the influence of noise on the results of quantum state measurements. A quantum noise model was developed based on the probability of confidence in the measurement result, which is interpreted as an assessment of its reliability in a Bernoulli experiment. Comparison of the obtained results with the classical approach confirms the practical suitability of the hybrid algorithm for binary classification problems. The hybrid model preserves the key properties of classical logistic regression – parameter interpretability and optimization stability, while allowing for the use of quantum advantages for compact data encoding and scalar product computation. This makes the study of hybrid logistic regression relevant for the development of machine learning algorithms on modern quantum platforms.
Quantum state measurement is a fundamental process in quantum mechanics, where a quantum system is observed, collapsing its superposition into a definite state. This process inherently introduces uncertainty, and classical measurement techniques often struggle to accurately determine the precise outcome. This paper explores a novel approach to quantum state measurement – the "Adaptive Quantum Measurement" – leveraging the inherent self-adjusting properties of quantum states. We propose a method that dynamically adjusts measurement parameters during the measurement process, leading to significantly improved accuracy and the potential for more robust information extraction. This work introduces a new theoretical framework and preliminary experimental demonstrations to illustrate the feasibility and potential impact of this approach.
Purpose of work is to analyze classical, quantum and quantum-inspired optimization methods in relation to the problem of ensuring the stability of blockchain systems, to develop a QUBO model of cumulative risk, taking into account the probabilities of attacks and resource constraints, to adapt the discrete algorithm of simulated dSB bifurcation and its modification HdSB to the problem of ensuring the stability of systems by minimizing risk. Research methods: comparative analysis of the performance of optimization algorithms on problems of various structures and dimensions, construction of a QUBO risk model with the incorporation of resource constraints through a quadratic penalty function, adaptation of dSB and HdSB algorithms by converting the QUBO formulation into an Ising model and adjusting the evolution parameters. Results of the study: a strictly defined QUBO formulation of the problem of ensuring the resilience of blockchain systems to a variety of classical and quantum attacks has been formed. In the model, the combined probabilities of various attack scenarios are reduced to a system of linear coefficients reflecting individual risks and quadratic coefficients describing the interactions between vulnerabilities. Resource constraints on the number of simultaneously implemented protective measures are incorporated by means of a quadratic penalty function, which allows us to consider the choice of the optimal configuration of protective measures as a single task of discrete global optimization. The dSB and HdSB algorithms were adapted to the task of minimizing the QUBO function of cumulative risk. The adaptation includes the transformation of binary variables to a spin representation and the determination of the parameters of the evolution of a system of oscillators with nonlinear interactions. A series of computational experiments on synthetic problems modeling the structural properties of vulnerability graphs of real blockchain platforms is presented, confirming the effectiveness of adapted algorithms. Based on the analysis of the structure of the QUBO risk model, the prospects for further improvement of optimization quality are substantiated by integrating game theory methods for local configuration adjustment after the evolution stage according to the dynamics of simulated bifurcation. It is shown that the QUBO problem can be interpreted as a potential game using Nash equilibrium search procedures using the best response method to systematically improve solutions. Scientific novelty: for the first time, a specialized QUBO model of cumulative risk has been proposed, adapted to the challenges of ensuring the stability of blockchain systems while taking into account complex interactions between vulnerabilities and limited resources, and a discrete simulated bifurcation algorithm and its modifications have been adapted to solve this problem.
Quantum phase estimation (QPE) is a fundamental technique for determining the phase of a quantum state, crucial for various quantum algorithms and quantum measurements. This paper introduces a novel adaptive optimization algorithm, dubbed the "Adaptive Quantum Phase Estimation" (AQPE) algorithm, designed to enhance the efficiency of QPE by dynamically adjusting parameters during the state estimation process. We demonstrate the effectiveness of AQPE through rigorous theoretical analysis and extensive numerical simulations, achieving significantly improved convergence rates compared to traditional optimization techniques. The algorithm leverages a feedback mechanism based on the observed phase error during the estimation, allowing it to iteratively refine the parameter set and ultimately locate the optimal phase estimation. We present preliminary results demonstrating the algorithm's capability to mitigate the challenges associated with conventional QPE, particularly in the presence of noisy data. The implications of this research extend to broader advancements in quantum algorithm design and computational power.
Wave function collapse is a cornerstone of quantum mechanics, representing the irreversible process by which a quantum system's state is determined. This paper introduces Adaptive Wave Function Collapse (AWC), a dynamic algorithm designed to optimize information extraction and minimize measurement errors during this collapse process. The core of AWC lies in a novel "wave function feedback" loop that continuously adjusts the collapse probability, guided by a "collapse rate" parameter. This dynamic adjustment ensures a more efficient and stable collapse, addressing limitations inherent in traditional collapse methods. The paper details the algorithm's implementation, explores its performance characteristics, and provides a conceptual analysis of its potential impact on quantum information processing.