Variational quantum recurrent neural networks for multi-feature forecasting

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’.

The classical distortion models similarly to temporal data analysis provide a way to predict the trends in the output of the algorithms and discard the anomalies in the output which may impact the accuracy of the method. In this paper, we introduce a quantum distortion modeling framework that enables variational quantum algorithms to operate effectively in the presence of errors without traditional error correction. Drawing inspirations from classical distortion-tolerant computing, we describe different distortions measures and formulation which can be used in variational quantum algorithms. In particular, we develop mathematical foundations for distortion metrics including energy progression, parameter stability, and state fidelity distortions, and demonstrate their applicability to variational quantum eigensolver, quantum approximate optimization algorithm, and quantum power method. We believe this paper will provide a milestone for distortion-aware quantum computing which can expand the practical applicability of the pre-fault-tolerant era devices to problems where approximate solutions provide sufficient value, representing a paradigm shift from exact error elimination to managed accuracy degradation.

Variational quantum algorithms have the potential for significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these algorithms is generally nonconvex, leading the algorithms to converge to local, rather than global, minima and the production of suboptimal solutions. In this work, we introduce a variational quantum algorithm that couples classical Markov chain Monte Carlo techniques with variational quantum algorithms, allowing the former to provably converge to global minima and thus assure solution quality. Due to the generality of our approach, it is suitable for a myriad of quantum minimization problems, including optimization and quantum state preparation. Specifically, we devise a Metropolis–Hastings method that is suitable for variational quantum devices and use it, in conjunction with quantum optimization, to construct quantum ensembles that converge to Gibbs states. These performance guarantees are derived from the ergodicity of our algorithm’s state space and enable us to place analytic bounds on its time-complexity. We demonstrate both the effectiveness of our technique and the validity of our analysis through quantum circuit simulations for MaxCut instances, solving these problems deterministically and with perfect accuracy, as well as large-scale quantum Ising and transverse field spin models of up to 50 qubits. Our technique stands to broadly enrich the field of variational quantum algorithms, improving and guaranteeing the performance of these promising, yet often heuristic, methods.

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.

Effects of data smoothing and recurrent neural network (RNN) algorithms for real-time forecasting of tunnel boring machine (TBM) performance

Variational hybrid quantum–classical algorithms are powerful tools to maximize the use of noisy intermediate-scale quantum devices. While past studies have developed powerful and expressive ansatze, their near-term applications have been limited by the difficulty of optimizing in the vast parameter space. In this work, we propose a heuristic optimization strategy for such ansatze used in variational quantum algorithms, which we call ‘parameter-efficient circuit training (PECT)’. Instead of optimizing all of the ansatz parameters at once, PECT launches a sequence of variational algorithms, in which each iteration of the algorithm activates and optimizes a subset of the total parameter set. To update the parameter subset between iterations, we adapt the Dynamic Sparse Reparameterization scheme which was originally proposed for training deep convolutional neural networks. We demonstrate PECT for the Variational Quantum Eigensolver, in which we benchmark unitary coupled-cluster ansatze including UCCSD and k-UpCCGSD, as well as the Low-Depth Circuit Ansatz (LDCA), to estimate ground state energies of molecular systems. We additionally use a layerwise variant of PECT to optimize a hardware-efficient circuit for the Sycamore processor to estimate the ground state energy densities of the one-dimensional Fermi-Hubbard model. From our numerical data, we find that PECT can enable optimizations of certain ansatze that were previously difficult to converge and more generally can improve the performance of variational algorithms by reducing the optimization runtime and/or the depth of circuits that encode the solution candidate(s).

Advances in quantum simulator technology are increasingly required because research on quantum algorithms is becoming more sophisticated and complex. State vector simulation utilizes CPU and memory resources in computing nodes exponentially with respect to the number of qubits; furthermore, in a variational quantum algorithm, the large number of repeated runs by classical optimization is also a heavy load. This problem has been addressed by preparing numerous computing nodes or simulation frameworks that operate effectively. This study aimed to accelerate quantum simulation using two newly proposed methods: to efficiently utilize limited computational resources by adjusting the ratio of the MPI and distributed processing parallelism corresponding to the target problem settings and to slim down the Hamiltonian by considering the effect of accuracy on the calculation result. Ground-state energy calculations of a fermionic model were performed using a variational quantum eigensolver (VQE) on an HPC cluster with up to 1024 FUJITSU Processor A64FX connected to each other by InfiniBand; the processor is also used on supercomputer Fugaku. We achieved 200 times higher speed over VQE simulations and demonstrated 32 qubits ground-state energy calculations in acceptable time. This result indicates that > 30 qubit state vector simulations can be realistically utilized for further research on variational quantum algorithms.

Variational Quantum Algorithms (VQAs) are among the most promising NISQ-era algorithms for harnessing quantum computing in diverse fields. However, the underlying optimization processes within these algorithms usually deal with local minima and barren plateau problems, preventing them from scaling efficiently. Our goal in this paper is to study alternative optimization methods that can avoid or reduce the effect of these problems. To this end, we propose to apply the Differential Evolution (DE) algorithm to VQAs optimizations. Our hypothesis is that DE is resilient to vanishing gradients and local minima for two main reasons: (1) it does not depend on gradients, and (2) its mutation and recombination schemes allow DE to continue evolving even in these cases. To demonstrate the performance of our approach, first, we use a robust local minima problem to compare state-of-the-art local optimizers (SLSQP, COBYLA, L-BFGS-B and SPSA) against DE using the Variational Quantum Eigensolver algorithm. Our results show that DE always outperforms local optimizers. In particular, in exact simulations of a 1D Ising chain with 14 qubits, DE achieves the ground state with a 100% success rate, while local optimizers only exhibit around 40%. We also show that combining DE with local optimizers increases the accuracy of the energy estimation once avoiding local minima. Finally, we demonstrate how our results can be extended to more complex problems by studying DE performance in a 1D Hubbard model.

Variational quantum algorithms (VQAs) have dominated literature as tools for demonstrating quantum utility on near-term quantum hardware, with applications in optimisation, quantum simulation, and machine learning. While researchers have studied how easy VQAs are to train, the effect of quantum noise on the classical optimisation process is still not well understood. Contrary to expectations, we find that twirling, which is commonly used in standard error-mitigation strategies to symmetrise noise, actually degrades performance in the variational setting, whereas preserving biased or non-unital noise can help classical optimisers find better solutions. Analytically, we study a universal quantum regression model and demonstrate that relatively uniform Pauli channels suppress gradient magnitudes and reduce expressivity, making optimisation more difficult. Conversely, asymmetric noise such as amplitude damping or biased Pauli channels introduces directional bias that can be exploited during optimisation. Numerical experiments on a variational eigensolver for the transverse-field Ising model confirm that non-unital noise yields lower-energy states compared to twirled noise. Finally, we show that coherent errors are fully mitigated by re-parameterisation. These findings challenge conventional noise-mitigation strategies and suggest that preserving noise biases may enhance VQA performance.

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.

Variational quantum algorithms (VQAs) are increasingly being applied in simulations of strongly bound (covalently bonded) systems using full molecular orbital basis representations. The application of quantum computers to the weakly bound intermolecular and noncovalently bonded regime, however, has remained largely unexplored. In this work, we develop a coarse-grained representation of the electronic response that is ideally suited for determining the ground state of weakly interacting molecules using a VQA. We require qubit numbers that grow linearly with the number of molecules and derive scaling behavior for the number of circuits and measurements required, which compare favorably to traditional variational quantum eigensolver methods. We demonstrate our method on IBM superconducting quantum processors and show its capability to resolve the dispersion energy as a function of separation for a pair of nonpolar molecules---thereby establishing a means by which quantum computers can model Van der Waals interactions directly from zero-point quantum fluctuations. Within this coarse-grained approximation, we conclude that current-generation quantum hardware is capable of probing energies in this weakly bound but nevertheless chemically ubiquitous and biologically important regime. Finally, we perform experiments on simulated and real quantum computers for systems of three, four, and five oscillators as well as oscillators with anharmonic onsite binding potentials; the consequences of the latter are unexamined in large systems using classical computational methods but can be incorporated here with low computational overhead.

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.

Differential equations (DEs) serve as the cornerstone for a wide range of scientific endeavors, their solutions weaving through the core of diverse fields such as structural engineering, fluid dynamics, and financial modeling. DEs are notoriously hard to solve, due to their intricate nature, and finding solutions to DEs often exceeds the capabilities of traditional computational approaches. Recent advances in quantum computing have triggered a growing interest from researchers for the design of quantum algorithms for solving DEs. In this work, we introduce two different architectures of a novel variational quantum algorithm (VQA) with Lagrange polynomial encoding in combination with derivative quantum circuits using the Hadamard test differentiation to approximate the solution of DEs. To demonstrate the potential of our new VQA, two well-known ordinary differential equations are used: the damped mass-spring system from a given initial condition and the Poisson equation for periodic, Dirichlet, and Neumann boundary conditions. It is shown that the proposed new VQA has a reduced gate complexity compared to previous variational quantum algorithms, for a similar or better quality of the solution.

Variational quantum algorithms (VQAs) are widely applied in the noisy intermediate-scale quantum era and are expected to demonstrate quantum advantage. However, training VQAs faces difficulties, one of which is the so-called barren plateaus (BPs) phenomenon, where gradients of cost functions vanish exponentially with the number of qubits. In this paper, inspired by transfer learning, where knowledge of pre-solved tasks could be further used in a different but related work with training efficiency improved, we report a parameter initialization method to mitigate BP. In the method, a small-sized task is solved with a VQA. Then the ansatz and its optimum parameters are transferred to tasks with larger sizes. Numerical simulations show that this method could mitigate BP and improve training efficiency. A brief discussion on how this method can work well is also provided. This work provides a reference for mitigating BP, and therefore, VQAs could be applied to more practical problems.

Variational quantum algorithms (VQAs) are promising methods to demonstrate quantum advantage on near-term devices as the required resources are divided between a quantum simulator and a classical optimizer. As such, designing a VQA which is resource-efficient and robust against noise is a key factor to achieve a potential advantage with the existing noisy quantum simulators. It turns out that a fixed VQA circuit design, such as the widely used hardware-efficient ansatz, is not necessarily robust against imperfections. In this work, we propose a genome-length-adjustable evolutionary algorithm to design a robust VQA circuit that is optimized over variations of both circuit ansatz and gate parameters, without any prior assumptions on circuit structure or depth. Remarkably, our method not only generates a noise-effect-minimized circuit with shallow depth, but also accelerates the classical optimization by substantially reducing the number of parameters. In this regard, the optimized circuit is far more resource-efficient with respect to both quantum and classical resources. As applications, based on two typical error models in VQA, we apply our method to calculate the ground energy of the hydrogen and the water molecules as well as the Heisenberg model. Simulations suggest that, compared with conventional hardware-efficient ansatz, our circuit-structure-tunable method can generate circuits apparently more robust against both coherent and incoherent noise and hence is more likely to be implemented on near-term devices.