Enabling High Performance Debugging for Variational Quantum Algorithms using Compressed Sensing

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’.

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, 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.

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) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether noise on NISQ devices places fundamental limitations on VQA performance. We rigorously prove a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau (i.e., vanishing gradient). Specifically, for the local Pauli noise considered, we prove that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n. These noise-induced barren plateaus (NIBPs) are conceptually different from noise-free barren plateaus, which are linked to random parameter initialization. Our result is formulated for a generic ansatz that includes as special cases the Quantum Alternating Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the former, our numerical heuristics demonstrate the NIBP phenomenon for a realistic hardware noise model.

Variational quantum algorithms (VQAs) and their applications in the field of quantum machine learning through parametrized quantum circuits (PQCs) are thought to be one major way of leveraging noisy intermediate-scale quantum computing devices. However, differences in the performance of certain VQA architectures are often unclear since established best practices, as well as detailed studies, are missing. In this paper, we build upon the work by Schuld [] and Vidal [] by comparing how well popular ansatz for PQCs learn different one-dimensional truncated Fourier series. We also examine dissipative quantum neural networks (dQNN) as introduced by Beer [] and propose a data reupload structure for dQNNs to increase their capability for this regression task. By comparing the results for different PQC architectures, we can provide guidelines for designing efficient PQCs. Published by the American Physical Society 2025

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.

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.

In the era of noisy intermediate-scale quantum (NISQ) computing, the synergistic collaboration between quantum and classical computing models has emerged as a promising solution for tackling complex computational challenges. Long short-term memory (LSTM), as a popular network for modeling sequential data, has been widely acknowledged for its effectiveness. However, with the increasing demand for data and spatial feature extraction, the training cost of LSTM exhibits exponential growth. In this study, we propose the quantum convolutional long short-term memory (QConvLSTM) model. By ingeniously integrating classical convolutional LSTM (ConvLSTM) networks and quantum variational algorithms, we leverage the variational quantum properties and the accelerating characteristics of quantum states to optimize the model training process. Experimental validation demonstrates that, compared to various LSTM variants, our proposed QConvLSTM model outperforms in terms of performance. Additionally, we adopt a hierarchical tree-like circuit design philosophy to enhance the model’s parallel computing capabilities while reducing dependence on quantum bit counts and circuit depth. Moreover, the inherent noise resilience in variational quantum algorithms makes this model more suitable for spatiotemporal sequence modeling tasks on NISQ devices.

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.

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.

Variational quantum algorithms (VQAs) offer a promising approach to solving computationally demanding problems by combining parameterized quantum circuits with classical optimization. Estimating probabilistic outcomes on quantum hardware requires repeated measurements (shots). However, in practice, the limited shot budget introduces significant noise in the evaluation of the objective function. Gradient estimation in VQAs often relies on the finite-difference, which evaluates the noisy objective function at perturbed circuit parameter values. The accuracy of this estimation is highly dependent on the choice of step size for these perturbations. An inappropriate step size can exacerbate the impact of noise, causing inaccurate gradient estimates and hindering the classical optimization in VQAs. This paper proposes QuGStep, an algorithm that addresses the challenge of determining the appropriate step size for finite-difference gradient estimation under a shot budget. QuGStep is grounded in a theorem that proves the optimal step size, which accounts for the shot budget, minimizes the error bound in gradient estimation using finite differences. Numerical experiments approximating the ground state energy of several molecules demonstrate that QuGStep can identify the appropriate step size for the given shot budget to obtain effective gradient estimation. Notably, the step size identified by QuGStep achieved convergence to the ground state energy with over 94% fewer shots compared to using a default step size (i.e., step size of 0.01). These findings highlight the potential of QuGStep to improve the practical deployment and scalability of quantum computing technologies.

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.

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.