Quantum Machine Learning (QML) is one of promising applications for near-term quantum computing. Two popular methods in QML are kernel methods and variational methods. Variational methods, which consist of parametrized quantum circuits (PQCs) to encode data and define classifiers, work faster in theory (i.e., O(N) to learn from$N$training examples) than kernel methods, which use quantum circuits to compute O(N2) elements of kernel matrices. However, in practice when dealing with large N, it is necessary to speed up variational methods due to the slow quantum gates. In this work, we propose a parallelization of training variational quantum classifiers to utilize the availability of many quantum devices with dozens of qubits. In contrast to existing parallelization of variational methods with gradient-based algorithms, we develop a novel distributed coordinate descent algorithm to optimize parametrized gates of variational quantum circuits. There are several gradient-free methods to optimize PQCs that have been shown to converge faster. Here, by focusing on the so-called Free-axis selection (Fraxis) method, we show how the gradient-free methods can be parallelized, and demonstrate their efficacies by running the algorithm on both simulators and IBM Quantum devices. We confirm the proposed algorithm achieves high classification accuracy and gains almost-linear speedup with the degree of parallelization.
Quantum Machine Learning (QML) is one of promising applications for near-term quantum computing. Two popular methods in QML are kernel methods and variational methods. Variational methods, which consist of parametrized quantum circuits (PQCs) to encode data and define classifiers, work faster in theory (i.e., O(N) to learn fromNtraining examples) than kernel methods, which use quantum circuits to compute O(N2) elements of kernel matrices. However, in practice when dealing with large N, it is necessary to speed up variational methods due to the slow quantum gates. In this work, we propose a parallelization of training variational quantum classifiers to utilize the availability of many quantum devices with dozens of qubits. In contrast to existing parallelization of variational methods with gradient-based algorithms, we develop a novel distributed coordinate descent algorithm to optimize parametrized gates of variational quantum circuits. There are several gradient-free methods to optimize PQCs that have been shown to converge faster. Here, by focusing on the so-called Free-axis selection (Fraxis) method, we show how the gradient-free methods can be parallelized, and demonstrate their efficacies by running the algorithm on both simulators and IBM Quantum devices. We confirm the proposed algorithm achieves high classification accuracy and gains almost-linear speedup with the degree of parallelization.
Quantum machine learning offers the ability to change the significant task of classifying high-dimensions data by taking advantage of infinite-dimensional, enriched state spaces granting a net benefit of quantum computation. We offer a variational quantum algorithm that accommodates many use cases in quantum computing to classify large-scale and high-dimensions data in real time and in cloud-all computing environments. We created our unique quantum circuits; we optimized the parameterized gates using either gradient methods or gradients-free methods and we investigated quantum kernels that provide the capability to identify embeddings in Hilbert space. Our experimental results using both customized synthetic data and benchmarks have shown that the variational quantum classifier was able to achieve comparability in accuracy to classical deep learning accuracy, while improving scale and robustness to noise. In addition, after we deployed the application in a cloud environment, we assessed latency, throughput and metrics from each cloud vendor, and we discovered that if the latency of the quantum accelerators can be controlled by a pipeline of optimized data and workloads, then they are eligible to satisfy the real-time requirement. Based on a theoretical and practical rapidly emerging basis, these results have indicated that variation quantum classifiers are a pathway for future dataintensive applications deployed in the quantum-cloud ecosystem.
Parameterized quantum circuits (PQCs) play an essential role in the application of variational quantum algorithms (VQAs) in noisy intermediate-scale quantum (NISQ) devices. The PQCs are a leading candidate to achieve a quantum advantage in NISQ devices and have already been applied in various domains such as quantum chemistry, quantum machine learning, combinatorial optimization, and many others. There is no single definitive way to optimize PQCs. The most commonly used methods are based on computing the gradient via the parameter-shift rule to use classical gradient descent (GD) optimizers like Adam, stochastic GD, and others. In addition, sequential single-qubit optimizers have been proposed, such as Rotosolve, Free-Axis Selection (Fraxis), Free-Quaternion Selection (FQS), and hybrid algorithms from the aforementioned optimizers. We further develop hybrid algorithms than those represented in the previous work by drawing inspiration from the early stopping method used in classical machine learning. The switch between the optimizers depends on the previous cost function values compared to the previous ones. We introduce two new hybrid algorithms that are more robust and scalable, and they outperform previous hybrid methods in terms of convergence towards the global minima across various cost functions. In addition, we find that they are feasible for NISQ devices with different noise profiles.
Parameterized quantum circuits (PQCs) are fundamental to many hybrid quantum-classical algorithms. However, existing structure-based optimizers for the training of PQCs, such as Rotosolve and sequential minimal optimization, rely on heuristic node selection or ignore statistical noise, which limits their robustness and accuracy. To address this issue, we propose an interpolation-based coordinate descent (ICD) method as a unified framework for all structure-based optimizers. ICD approximates the cost function through interpolation, recovers its trigonometric structure, and performs global one-dimensional updates on individual parameters. Unlike previous methods, ICD derives optimal interpolation nodes that minimize statistical errors from measurements. For the common case of r equidistant frequencies, we prove that equidistant nodes with spacing 2π/(2r + 1) jointly minimize the mean squared error of Fourier coefficient estimates, the condition number of the interpolation matrix, and the average variance of the approximated cost function. Numerical experiments confirm the superior robustness and efficiency of ICD over gradient-based methods. Parameterized quantum circuits are a common tool in variational quantum algorithms and quantum machine learning. The authors design an interpolation-based coordinate descent method that reconstructs the cost landscape from a few circuit runs and achieves more efficient training than standard gradient and coordinate descent methods in our numerical tests.
The paradigm of variational quantum classifiers (VQCs) encodes classical information as quantum states, followed by quantum processing and then measurements to generate classical predictions. VQCs are promising candidates for efficient utilizations of noisy intermediate scale quantum (NISQ) devices: classifiers involving M-dimensional datasets can be implemented with only [Formula: see text] qubits by using an amplitude encoding. A general framework for designing and training VQCs, however, is lacking. An encouraging specific embodiment of VQCs, quantum circuit learning (QCL), utilizes an ansatz: a circuit with a predetermined circuit geometry and parametrized gates expressing a time-evolution unitary operator; training involves learning the gate parameters through a gradient-descent algorithm where the gradients themselves can be efficiently estimated by the quantum circuit. The representational power of QCL, however, depends strongly on the choice of the ansatz, as it limits the range of possible unitary operators that a VQC can search over. Equally importantly, the landscape of the optimization problem may have challenging properties such as barren plateaus and the associated gradient-descent algorithm may not find good local minima. Thus, it is critically important to estimate (i) the price of ansatz; that is, the gap between the performance of QCL and the performance of ansatz-independent VQCs, and (ii) the price of using quantum circuits as classical classifiers: that is, the performance gap between VQCs and equivalent classical classifiers. This paper develops a computational framework to address both these open problems. First, it shows that VQCs, including QCL, fit inside the well-known kernel method. Next it introduces a framework for efficiently designing ansatz-independent VQCs, which we call the unitary kernel method (UKM). The UKM framework enables one to estimate the first known computationally-determined bounds on both the price of ansatz and the price of any speedup advantages of VQCs: numerical results with datatsets of various dimensions, ranging from 4 to 256, show that the ansatz-induced gap can vary between 10 and 20[Formula: see text], while the VQC-induced gap (between VQC and kernel method) can vary between 10 and 16[Formula: see text]. To further understand the role of ansatz in VQCs, we also propose a method of decomposing a given unitary operator into a quantum circuit, which we call the variational circuit realization (VCR): given any parameterized circuit block (as for example, used in QCL), it finds optimal parameters and the number of layers of the circuit block required to approximate any target unitary operator with a given precision.
Parameterized quantum circuits (PQCs) are pivotal components of variational quantum algorithms (VQAs), which represent a promising pathway to quantum advantage in noisy intermediate-scale quantum (NISQ) devices. PQCs enable flexible encoding of quantum information through tunable quantum gates and have been successfully applied across domains such as quantum chemistry, combinatorial optimization, and quantum machine learning. Despite their potential, PQC performance on NISQ hardware is hindered by noise, decoherence, and the presence of barren plateaus, which can impede gradient-based optimization. To address these limitations, we propose novel methods for improving gradient-free optimizers Rotosolve, Fraxis, and FQS, incorporating information from previous parameter iterations. Our approach conserves computational resources by reallocating optimization efforts toward poorly optimized gates, leading to improved convergence. The experimental results demonstrate that our techniques consistently improve the performance of various optimizers, contributing to more robust and efficient PQC optimization.
Recently, Variational Quantum Circuits (VQCs) are attracting considerable attention among quantum algorithms thanks to their robustness to the noise characterizing the current quantum devices. In detail, VQCs involve parameterized quan-tum circuits to be trained by means of a classical optimizer that makes queries to the quantum device. VQCs play a key role in several applications including quantum classifiers where the Hilbert space is used as feature space. Currently, the most used classical optimizer to learn V QCs is the gradient descent method. However, the so-called barren plateaus issue causes gradients of cost functions to become exceedingly small as the dimension of the classification problem is increased. As consequence, gradient descent method could be not efficient in real-world classification problems. This paper proposes to apply Genetic Algorithms (GAs) to train VQCs used as quantum classifiers. As shown in the experiments, the application of GAs results in accurate solutions obtained with a reduced number of queries to quantum devices.