In this paper we address the problem of using one probability space for estimating parameters and predicting future data when the observed data come from multiple contexts and thus from distinct spaces. We explain that a set-based probabilistic space might be suboptimal in the case of multiple contexts. To overcome suboptimality and reconcile multiple contexts in one space, the paper introduces the Quantum Probability Space (QPS). We also present an algorithm to calculate the QPS for data observed from multiple contexts and provide a web application that implements the algorithm. The QPS has application in Information Retrieval (IR), Machine Learning (ML) and in any domain where items should optimally be ranked and classified by some properties but under conditions of uncertainty due to context.
We introduce a new approach for quantum linear algebra based on quantum subspace states and present three new quantum machine learning algorithms. The first is a quantum determinant sampling algorithm that samples from the distribution
Pr[S]=det(XSXST) for
∣S∣=d using
O(nd) gates and with circuit depth
O(dlogn). The state of art classical algorithm for the task requires
O(d3) operations \cite{derezinski2019minimax}. The second is a quantum singular value estimation algorithm for compound matrices
Ak, the speedup for this algorithm is potentially exponential. It decomposes a
(kn) dimensional vector of order-
k correlations into a linear combination of subspace states corresponding to
k-tuples of singular vectors of
A. The third algorithm reduces exponentially the depth of circuits used in quantum topological data analysis from
O(n) to
O(logn). Our basic tool are quantum subspace states, defined as
∣Col(X)⟩=∑S⊂[n],∣S∣=ddet(XS)∣S⟩ for matrices
X∈Rn×d such that
XTX=Id, that encode
d-dimensional subspaces of
Rn. We develop two efficient state preparation techniques, the first using Givens circuits uses the representation of a subspace as a sequence of Givens rotations, while the second uses efficient implementations of unitaries
Γ(x)=∑ixiZ⊗(i−1)⊗X⊗In−i with
O(logn) depth circuits that we term Clifford loaders.
A central task in the field of quantum computing is to find applications where quantum computer could provide exponential speedup over any classical computer. Machine learning represents an important field with broad applications where quantum computer may offer significant speedup. Several quantum algorithms for discriminative machine learning have been found based on efficient solving of linear algebraic problems, with potential exponential speedup in runtime under the assumption of effective input from a quantum random access memory. In machine learning, generative models represent another large class which is widely used for both supervised and unsupervised learning. Here, we propose an efficient quantum algorithm for machine learning based on a quantum generative model. We prove that our proposed model is exponentially more powerful to represent probability distributions compared with classical generative models and has exponential speedup in training and inference at least for some instances under a reasonable assumption in computational complexity theory. Our result opens a new direction for quantum machine learning and offers a remarkable example in which a quantum algorithm shows exponential improvement over any classical algorithm in an important application field.
The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm to solve binary-variable optimization problems. Due to its expected robustness to systematic errors and the short circuit depth, it is one of the promising candidates likely to run on near-term quantum devices. We project the performance of QAOA applied to the Max-Cut problem and compare it with some of the best classical alternatives, both for exact or approximate solution. When comparing approximate solvers, their performance is characterized by the computational time taken to achieve a given quality of solution. Since QAOA is based on sampling, we introduce performance metrics based on the probability of observing a sample above a certain quality. In addition, we show that the QAOA performance varies significantly with the graph type. By selecting a suitable optimizer for the variational parameters and reducing the number of function evaluations, QAOA performance improves by up to 2 orders of magnitude compared to previous estimates. Especially for 3-regular random graphs, this setting decreases the performance gap with classical alternatives.
Quantum information theory offers unprecedented opportunities for computational advancement, particularly in the realm of quantum computing. However, the inherent complexity of quantum systems necessitates sophisticated computational techniques. This paper explores the optimization of probabilistic space operations within quantum information theory, a critical aspect for enhancing computation efficiency and achieving more precise results. We introduce a novel optimization framework that leverages the unique properties of probability spaces, leading to algorithmic improvements and enhanced performance. The core mechanism focuses on dynamically adjusting parameters within the probability space to minimize error and maximize fidelity. This research addresses a long-standing challenge in quantum information processing, offering a pathway towards more robust and scalable quantum algorithms.
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.