Learning Fourier series with parametrized quantum circuits

Variational quantum algorithms (VQAs) can potentially solve practical problems using contemporary Noisy Intermediate Scale Quantum (NISQ) computers. VQAs find near-optimal solutions in the presence of qubit errors by classically optimizing a loss function computed by parameterized quantum circuits. However, developing and testing VQAs is challenging due to the limited availability of quantum hardware, their high error rates, and the significant overhead of classical simulations. Furthermore, VQA researchers must pick the right initialization for circuit parameters, utilize suitable classical optimizer configurations, and deploy appropriate error mitigation methods. Unfortunately, these tasks are done in an ad-hoc manner today, as there are no software tools to configure and tune the VQA hyperparameters.

The recent advent of noisy intermediate-scale quantum devices, especially near-term quantum computers, has sparked extensive research efforts concerned with their possible applications. At the forefront of the considered approaches are variational methods that use parametrized quantum circuits. The classical and quantum Fisher information are firmly rooted in the field of quantum sensing and have proven to be versatile tools to study such parametrized quantum systems. Their utility in the study of other applications of noisy intermediate-scale quantum devices, however, has only been discovered recently. Hoping to stimulate more such applications, this article aims to further popularize classical and quantum Fisher information as useful tools for near-term applications beyond quantum sensing. We start with a tutorial that builds an intuitive understanding of classical and quantum Fisher information and outlines how both quantities can be calculated on near-term devices. We also elucidate their relationship and how they are influenced by noise processes. Next, we give an overview of the core results of the quantum sensing literature and proceed to a comprehensive review of recent applications in variational quantum algorithms and quantum machine learning.

Variational quantum algorithms, which combine highly expressive parameterized quantum circuits (PQCs) and optimization techniques in machine learning, are one of the most promising applications of a near-term quantum computer. Despite their huge potential, the utility of variational quantum algorithms beyond tens of qubits is still questioned. One of the central problems is the trainability of PQCs. The cost function landscape of a randomly initialized PQC is often too flat, asking for an exponential amount of quantum resources to find a solution. This problem, dubbed barren plateaus, has gained lots of attention recently, but a general solution is still not available. In this paper, we solve this problem for the Hamiltonian variational ansatz (HVA), which is widely studied for solving quantum many-body problems. After showing that a circuit described by a time-evolution operator generated by a local Hamiltonian does not have exponentially small gradients, we derive parameter conditions for which the HVA is well approximated by such an operator. Based on this result, we propose an initialization scheme for the variational quantum algorithms and a parameter-constrained ansatz free from barren plateaus.

In variational quantum algorithms, it is important to balance conflicting requirements of expressibility and trainability of a parameterized quantum circuit (PQC). However, appropriate PQC designs are not necessarily trivial. Here, we propose an algorithm for optimizing the PQC structure, where single-qubit gates are sequentially replaced by the optimal ones via diagonalization of a matrix whose elements are evaluated on slightly modified circuits. This replacement leads to a better approximation of target states with limited circuit depth. Furthermore, we clarify the existence of a barren plateau in the sequential optimization in terms of the spectrum concentration of the matrix, which defines the cost landscape with respect to changes in the target gate. Then, we rigorously show the concentration is no faster than polynomials in the number of qubits when an n-qubit PQC depth is O(log⁡n) using local observables. Finally, numerical experiments are provided to show the convergence of our method which is faster than classical optimizers on both simulators and a real device. Our results provide evidences for sequential optimizers as better alternatives to optimize PQCs on near-term quantum devices.

Variational quantum algorithms (VQAs) are promising hybrid quantum-classical methods designed to leverage the computational advantages of quantum computing while mitigating the limitations of current noisy intermediate-scale quantum (NISQ) hardware. Although VQAs have been demonstrated as proofs of concept, their practical utility in solving real-world problems—and whether quantum-inspired classical algorithms can match their performance—remains an open question. We present a novel application of the variational quantum linear solver (VQLS) and its classical neural quantum states-based counterpart, the variational neural linear solver (VNLS), as key components within a minimum map Newton solver for a complementarity-based rigid-body contact model. We demonstrate using the VNLS that our solver accurately simulates the dynamics of rigid spherical bodies during collision events. These results suggest that quantum and quantum-inspired linear algebra algorithms can serve as viable alternatives to standard linear algebra solvers for modelling certain physical systems.This article is part of the theme issue ‘Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms’.

As a mainstream approach in the quantum machine learning field, variational quantum algorithms (VQAs) are frequently mentioned among the most promising applications for quantum computing. However, VQAs suffer from inefficient training methods. Here, we propose a pretraining strategy named near Clifford circuits warm start (NCC-WS) to find the initialization for parameterized quantum circuits (PQCs) in VQAs. We explored the expressibility of NCCs and the correlation between the expressibility and acceleration. The achieved results suggest that NCC-WS can find the correct initialization for the training of VQAs to achieve acceleration.

Variational quantum algorithms (VQAs) have been successfully applied to quantum approximate optimization algorithms, variational quantum compiling, and quantum machine learning models. The performance of VQAs is significantly influenced by the architecture of parameterized quantum circuits (PQCs). Quantum architecture search aims to automatically discover high-performance quantum circuits for specific VQA tasks. Quantum architecture search algorithms that utilize both SuperCircuit training and a parameter-sharing approach can save computational resources. If we directly follow the parameter-sharing approach, the SuperCircuit has to be trained to compensate for the worse search space. To address the challenges posed by the worse search space, we introduce an optimization strategy known as the efficient continuous evolutionary approach using Non-dominated Sorting Genetic Algorithm-II (NSGA-II). Then, we leverage prior information (symmetric property) designing Structure Symmetric Pruning for removing redundant gates of the searched ansatz. Experiments show that the efficient continuous evolutionary approach can search for more quantum architectures with better performance; the number of high-performance ansatzes obtained by our method is 10% higher than that in the literature (Du et al. in npj Quantum Inf. 8:62, 2022). The application of Structure Symmetric Pruning effectively reduces the number of parameters in quantum circuits without compromising their performance significantly. In binary classification tasks, the pruned quantum circuits exhibit an average accuracy reduction of 0.044 compared to their unpruned counterparts.

Even a minor boost in solving combinatorial optimization problems can greatly benefit multiple industries. Quantum computers, with their unique information processing capabilities, hold promise for delivering such enhancements. The filtering variational quantum eigensolver (F-VQE) is a variational hybrid quantum algorithm designed to solve combinatorial optimization problems on existing quantum computers with limited qubit number, connectivity, and fidelity. In this work we employ instantaneous quantum polynomial circuits as our parameterized quantum circuits. We propose a hardware-efficient implementation that respects limited qubit connectivity and show that they halve the number of circuits necessary to evaluate the gradient with the parameter-shift rule. To assess the potential of this protocol in the context of combinatorial optimization, we conduct extensive numerical analysis. We compare the performance against three classical baseline algorithms on weighted MaxCut and the asymmetric traveling salesperson problem (ATSP). We employ noiseless simulators for problems encoded on 13–29 qubits, and up to 37 qubits on the IBMQ real quantum devices. The ATSP encoding employed reduces the number of qubits and avoids the need of constraints compared to the standard quadratic unconstrained binary optimization/Ising model. Despite some observed positive signs, we conclude that significant development is necessary for a practical advantage with F-VQE.

Quantum computers have proven to be effective in simulating many quantum systems. Simulating nuclear processes and state preparation poses significant challenges, even for traditional supercomputers. This study demonstrates the feasibility of a complete simulation of a nuclear transition, including the preparation of both ground and first excited states. To tackle the complexity of strong interactions between two and three nucleons, the states are modeled on the tritium nucleus. Both the initial and final states are represented using quantum circuits with variational quantum algorithms and inductive biases. Describing the spin‐isospin states requires four qubits, and a parameterized quantum circuit that exploits a total of 16 parameters is initialized. The estimated energy has a relative error of 2% for the ground state and 10% for the first excited state of the system. The simulation estimates the transition probability between the two states as a function of the dipole polarization angle. This work marks a first step toward leveraging digital quantum computers to simulate nuclear physics.

Quantum machine learning stands out as one of the most promising applications of quantum computing, widely believed to possess potential quantum advantages. In the era of noisy intermediate-scale quantum, the scale and quality of quantum computers are limited, and quantum algorithms based on fault-tolerant quantum computing paradigms cannot be experimentally verified in the short term. The variational quantum algorithm design paradigm can better adapt to the practical characteristics of noisy quantum hardware and is currently one of the most promising solutions. However, variational quantum algorithms, due to their highly entangled nature, encounter the phenomenon known as the “barren plateau” during the optimization and training processes, making effective optimization challenging. This paper addresses this challenging issue by researching a variational quantum neural network optimization method based on collective intelligence algorithms. The aim is to overcome optimization difficulties encountered by traditional methods such as gradient descent. We study two typical applications of using quantum neural networks: random 2D Hamiltonian ground state solving and quantum phase recognition. We find that the collective intelligence algorithm shows a better optimization compared to gradient descent. The solution accuracy of ground energy and phase classification is enhanced, and the optimization iterations are also reduced. We highlight that the collective intelligence algorithm has great potential in tackling the optimization of variational quantum algorithms.

In the current noisy intermediate-scale quantum (NISQ)era, quantum machine learning is emerging as a dominant paradigm to program gate-based quantum computers. In quantum machine learning, the gates of a quantum circuit are parameterized, and the parameters are tuned via classical optimization based on data and on measurements of the outputs of the circuit. Parameterized quantum circuits (PQCs) can efficiently address combinatorial optimization problems, implement probabilistic generative models, and carry out inference (classification and regression). This monograph provides a self-contained introduction to quantum machine learning for an audience of engineers with a back groundin probability and linear algebra. It first describes the necessary background, concepts, and tools necessary to describe quantum operations and measurements. Then,it covers parameterized quantum circuits, the variation alquantum eigensolver, as well as unsupervised and supervised quantum machine learning formulations.

This research explores the integration of quantum computing with classical machine learning to enhance data classification tasks using Quantum Neural Networks (QNN) and Parameterized Quantum Circuits (PQC). The hybrid approach leverages the advantages of both quantum and classical systems to improve the efficiency and accuracy of data processing. In this model, data is encoded into qubits using amplitude encoding, representing input vectors as amplitudes of quantum states. The QNN is initialized by placing the qubits in superposition using Hadamard gates, followed by data encoding with parameterized rotational gates that map classical data to quantum states using rotation angles. PQC plays a central role by applying layers of parameterized quantum operations to process data in the quantum space. These parameters are optimized during the training process, where a quadratic loss function minimizes the error between the predicted quantum states and the true class labels using gradient descent. Experiments conducted on the MNIST dataset show that the hybrid the hybrid quantum-classical neural network (QCNN) with PQC achieves a classification accuracy of over 95%, highlighting its potential in machine learning applications. The results demonstrate that integrating quantum computing with classical machine learning enhances performance in complex data analysis tasks due to the exponential growth of quantum state space and the parallelism of quantum systems, making hybrid models promising for computer vision and classification tasks.

Quantum data processing requires classical data to be encoded into quantum states. Current noisy intermediate-scale quantum devices have a limited number of qubits that are stable only briefly. Encoding classical data into qubits is the initial step in Quantum Machine Learning (QML), and effective encoding is crucial for quantum processing. This algorithms for data processing are still emerging, and compact data representations are essential for their success. This research proposes a novel data encoding technique using uniformly controlled rotation gates, achieving high storage density by encoding real-valued time series data as qubit rotations. The model uses a binary representation for computations on time series data, reducing the number of quantum measurements needed. The research explores quantum forward propagation in simulations to improve prediction accuracy for time series signals using parameterized quantum circuits, handling trends, noise, and sinusoidal components. The efficiency of the encoding process depends on data volume and chosen encoding, with potential infinite loading time in the worst case. This study presents a Forward Selection Time Series Data Pro-cessing and Feature Extraction Model for Qubits generation with Parameterized Quantum Gates (FSDPFEM-PQG), demonstrating superior performance in quantum representations compared to existing models.

TensorCircuit is an open source quantum circuit simulator based on tensor network contraction, designed for speed, flexibility and code efficiency. Written purely in Python, and built on top of industry-standard machine learning frameworks, TensorCircuit supports automatic differentiation, just-in-time compilation, vectorized parallelism and hardware acceleration. These features allow TensorCircuit to simulate larger and more complex quantum circuits than existing simulators, and are especially suited to variational algorithms based on parameterized quantum circuits. TensorCircuit enables orders of magnitude speedup for various quantum simulation tasks compared to other common quantum software, and can simulate up to 600 qubits with moderate circuit depth and low-dimensional connectivity. With its time and space efficiency, flexible and extensible architecture and compact, user-friendly API, TensorCircuit has been built to facilitate the design, simulation and analysis of quantum algorithms in the Noisy Intermediate-Scale Quantum (NISQ) era.

In the era of noisy intermediate-scale quantum devices, variational quantum algorithms (VQAs) stand as a prominent strategy for constructing quantum machine learning models. These models comprise both a quantum and a classical component. The quantum facet is characterized by a parametrization U , typically derived from the composition of various quantum gates. On the other hand, the classical component involves an optimizer that adjusts the parameters of U to minimize a cost function C. Despite the extensive applications of VQAs, several critical questions persist, such as determining the optimal gate sequence, devising efficient parameter optimization strategies, selecting appropriate cost functions, and understanding the influence of quantum chip architectures on the final results. This article aims to address the last question, emphasizing that, in general, the cost function tends to converge towards an average value as the utilized parameterization approaches a 2-design. Consequently, when the parameterization closely aligns with a 2-design, the quantum neural network model's outcome becomes less dependent on the specific parametrization. This insight leads to the possibility of leveraging the inherent architecture of quantum chips to define the parametrization for VQAs. By doing so, the need for additional swap gates is mitigated, consequently reducing the depth of VQAs and minimizing associated errors.