Hybrid Quantum-classical Algorithms Research Articles

Meta-optimization of resources on quantum computers

The current state of quantum computing is commonly described as the Noisy Intermediate-Scale Quantum era. Available computers contain a few dozens of qubits and can perform a few dozens of operations before the inevitable noise erases all information encoded in the calculation. Even if the technology advances fast within the next years, any use of quantum computers will be limited to short and simple tasks, serving as subroutines of more complex classical procedures. Even for these applications the resource efficiency, measured in the number of quantum computer runs, will be a key parameter. Here we suggest a general meta-optimization procedure for hybrid quantum-classical algorithms that allows finding the optimal approach with limited quantum resources. This method optimizes the usage of resources of an existing method by testing its capabilities and setting the optimal resource utilization. We demonstrate this procedure on a specific example of variational quantum algorithm used to find the ground state energy of a hydrogen molecule.

Quantum-Informed Recursive Optimization Algorithms

We propose and implement a family of quantum-informed recursive optimization (QIRO) algorithms for combinatorial optimization problems. Our approach leverages quantum resources to obtain information that is used in problem-specific classical reduction steps that recursively simplify the problem. These reduction steps address the limitations of the quantum component (e.g., locality) and ensure solution feasibility in constrained optimization problems. Additionally, we use backtracking techniques to further improve the performance of the algorithm without increasing the requirements on the quantum hardware. We showcase the capabilities of our approach by informing QIRO with correlations from classical simulations of shallow circuits of the quantum approximate optimization algorithm, solving instances of maximum independent set and maximum satisfiability problems with hundreds of variables. We also demonstrate how QIRO can be deployed on a neutral atom quantum processor to find large independent sets of graphs. In summary, our scheme achieves results comparable to classical heuristics even with relatively weak quantum resources. Furthermore, enhancing the quality of these quantum resources improves the performance of the algorithms. Notably, the modular nature of QIRO offers various avenues for modifications, positioning our work as a template for a broader class of hybrid quantum-classical algorithms for combinatorial optimization. Published by the American Physical Society 2024

Comparative study on compact quantum circuits of hybrid quantum-classical algorithms for quantum impurity models

Predicting the properties of strongly correlated materials is a significant challenge in condensed matter theory. The widely used dynamical mean-field theory faces difficulty in solving quantum impurity models numerically. Hybrid quantum-classical algorithms such as variational quantum eigensolvers emerge as a potential solution for quantum impurity models. A common challenge in these algorithms is the rapid growth of the number of variational parameters with the number of spin-orbitals in the impurity. In our approach to this problem, we develop compact using a combination of two different strategies. First, we employ a compact physics-inspired , k-unitary cluster Jastrow , developed in the field of quantum chemistry. Second, we eliminate largely redundant variational parameters of physics-inspired associated with bath sites based on physical intuition. This is based on the fact that a quantum impurity model with a star-like geometry has no direct hopping between bath sites. We benchmark the accuracy of these for both ground-state energy and dynamic quantities by solving typical quantum impurity models with and without shot noise. The results suggest that we can maintain the accuracy of ground-state energy while we drop the number of variational parameters associated with bath sites. Furthermore, we demonstrate that a moment expansion, when combined with the proposed , can calculate the imaginary-time Green's functions under the influence of shot noise. This study demonstrates the potential for addressing complex impurity models in large-scale quantum simulations with fewer variational parameters without sacrificing accuracy. Published by the American Physical Society 2024

Hybrid Quantum-classical Search Algorithms

Search is one of the most commonly used primitives in quantum algorithm design. It is known that quadratic speedups provided by Grover’s algorithm are optimal, and no faster quantum algorithms for Search exist. While it is known that at least some quantum computation is required to achieve these speedups, the existing bounds do not rule out the possibility of an equally fast hybrid quantum-classical algorithm where most of the computation is classical. In this work, we study such hybrid algorithms, and we show that classical computation, unless it by itself can solve the Search problem, cannot assist quantum computation. In addition, we generalize this result to algorithms with subconstant success probabilities.

The effects of quantum hardware properties on the performances of variational quantum learning algorithms

In-depth theoretical and practical research is nowadays being performed on variational quantum algorithms (VQAs), which have the potential to surpass traditional, classical, algorithms on a variety of problems, in physics, chemistry, biology, and optimization. Because they are hybrid quantum-classical algorithms, it takes a certain set of optimal conditions for their full potential to be exploited. For VQAs, the construction of an appropriate ansatz in particular is crucial, since it lays the ground for efficiently solving the particular problem being addressed. To prevent severe negative effects that hamper quantum computation, the substantial noise, together with the structural limitations, characteristic of currently available devices must be also taken into consideration while building the ansatz. In this work the effect of the quantum hardware structure, namely the topological properties emerging from the couplings between the physical qubits and the basis gates of the device itself, on the performances of VQAs is addressed. Specifically, it is here experimentally shown that a complex connectivity in the ansatz, albeit being beneficial for exploring wider sets of solutions, introduces an overhead of gates during the transpilation on a quantum computer that increases the overall error rate, thus undermining the quality of the training. It is hence necessary, when implementing a variation quantum learning algorithm, to find the right balance between a sufficiently parametrized ansatz and a minimal cost in terms of resources during transpilation. Moreover, the experimental finding allows to construct a heuristic metric function, which aids the decision-making process on the best possible ansatz structure to be deployed on a given quantum hardware, thus fostering a more efficient application of VQAs in realistic situations. The experiments are performed on two widely used variational algorithms, the VQE (variational quantum eigensolver) and the VQC (variational quantum classifier), both tested on two different problems, the first on the Markowitz portfolio optimization using real-world financial data, and the latter on a classification task performed on the Iris dataset.

QPU integration in OpenCL for heterogeneous programming

The integration of quantum processing units (QPUs) in a heterogeneous high-performance computing environment requires solutions that facilitate hybrid classical–quantum programming. Standards such as OpenCL facilitate the programming of heterogeneous environments, consisting of CPUs and hardware accelerators. This study presents an innovative method that incorporates QPU functionality into OpenCL, standardizing quantum processes within classical environments. By leveraging QPUs within OpenCL, hybrid quantum–classical computations can be sped up, impacting domains like cryptography, optimization problems, and quantum chemistry simulations. Using Portable Computing Language (Jääskeläinen et al. in Int J Parallel Program 43(5):752–785, 2014. https://doi.org/10.1007/s10766-014-0320-y) and the Qulacs library (Suzuki et al. in Quantum 5:559, 2021. https://doi.org/10.22331/q-2021-10-06-559), results demonstrate, for instance, the successful execution of Shor’s algorithm (Nielsen and Chuang in Quantum computation and quantum information, 10th anniversary edn. Cambridge University Press, Cambridge, 2010), serving as a proof of concept for extending the approach to larger qubit systems and other hybrid quantum–classical algorithms. This integration approach bridges the gap between quantum and classical computing paradigms, paving the way for further optimization and application to a wide range of computational problems.

Towards Quantum Ray Tracing.

Rendering on conventional computers is capable of generating realistic imagery, but the computational complexity of these light transport algorithms is a limiting factor of image synthesis. Quantum computers have the potential to significantly improve rendering performance through reducing the underlying complexity of the algorithms behind light transport. This paper investigates hybrid quantum-classical algorithms for ray tracing, a core component of most rendering techniques. Through a practical implementation of quantum ray tracing in a 3D environment, we show quantum approaches provide a quadratic improvement in query complexity compared to the equivalent classical approach. Based on domain specific knowledge, we then propose algorithms to significantly reduce the computation required for quantum ray tracing through exploiting image space coherence and a principled termination criteria for quantum searching. We show results obtained using a simulator for both Whitted style ray tracing, and for accelerating ray tracing operations when performing classical Monte Carlo integration for area lights and indirect illumination.

Quantum computation of π → π* and n → π* excited states of aromatic heterocycles

The computation of excited electronic states is an important application for quantum computers. In this work, we simulate the excited state spectra of four aromatic heterocycles on IBM superconducting quantum computers, focussing on active spaces of π → π ∗ and n → π ∗ excitations. We approximate the ground state with the entanglement forging method, a qubit reduction technique that maps a spatial orbital to a single qubit, rather than two qubits. We then determine excited states using the quantum subspace expansion method. We showcase these algorithms on quantum hardware using up to 8 qubits and employing readout and gate error mitigation techniques. Our results demonstrate a successful application of quantum computing in the simulation of active-space electronic wavefunctions of substituted aromatic heterocycles, and outline challenges to be overcome in elucidating the optical properties of organic molecules with hybrid quantum-classical algorithms.

Learning to Learn Variational Quantum Algorithm.

Variational quantum algorithms (VQAs) use classical computers as the quantum outer loop optimizer and update the circuit parameters to obtain an approximate ground state. In this article, we present a meta-learning variational quantum algorithm (meta-VQA) by recurrent unit, which uses a technique called "meta-learner." Motivated by the hybrid quantum-classical algorithms, we train classical recurrent units to assist quantum computing, learning to find approximate optima in the parameter landscape. Here, aiming to reduce the sampling number more efficiently, we use the quantum stochastic gradient descent method and introduce the adaptive learning rate. Finally, we deploy on the TensorFlow Quantum processor within approximate quantum optimization for the Ising model and variational quantum eigensolver for molecular hydrogen (H2), lithium hydride (LiH), and helium hydride cation (HeH+). Our algorithm can be expanded to larger system sizes and problem instances, which have higher performance on near-term processors.

Noise-robust ground state energy estimates from deep quantum circuits

In the lead up to fault tolerance, the utility of quantum computing will be determined by how adequately the effects of noise can be circumvented in quantum algorithms. Hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) have been designed for the short-term regime. However, as problems scale, VQE results are generally scrambled by noise on present-day hardware. While error mitigation techniques alleviate these issues to some extent, there is a pressing need to develop algorithmic approaches with higher robustness to noise. Here, we explore the robustness properties of the recently introduced quantum computed moments (QCM) approach to ground state energy problems, and show through an analytic example how the underlying energy estimate explicitly filters out incoherent noise. Motivated by this observation, we implement QCM for a model of quantum magnetism on IBM Quantum hardware to examine the noise-filtering effect with increasing circuit depth. We find that QCM maintains a remarkably high degree of error robustness where VQE completely fails. On instances of the quantum magnetism model up to 20 qubits for ultra-deep trial state circuits of up to 500 CNOTs, QCM is still able to extract reasonable energy estimates. The observation is bolstered by an extensive set of experimental results. To match these results, VQE would need hardware improvement by some 2 orders of magnitude on error rates.

Advancing hybrid quantum-classical algorithms via mean operators

Advancing hybrid quantum-classical algorithms via mean operators

Practical advantage of quantum machine learning in ghost imaging

Demonstrating the practical advantage of quantum computation remains a long-standing challenge whereas quantum machine learning becomes a promising application that can be resorted to. In this work, we investigate the practical advantage of quantum machine learning in ghost imaging by overcoming the limitations of classical methods in blind object identification and imaging. We propose two hybrid quantum-classical machine learning algorithms and a physical-inspired patch strategy to allow distributed quantum learning with parallel variational circuits. In light of the algorithm, we conduct experiments for imaging-free object identification and blind ghost imaging under different physical sampling rates. We further quantitatively analyze the advantage through the lens of information geometry and generalization capability. The numerical results showcase that quantum machine learning can restore high-quality images but classical machine learning fails. The advantage of identification rate are up to 10% via fair comparison with the classical machine learning methods. Our work explores a physics-related application capable of practical quantum advantage, which highlights the prospect of quantum computation in the machine learning field.

Machine learning aided dimensionality reduction toward a resource efficient projective quantum eigensolver: Formal development and pilot applications.

In recent times, a variety of hybrid quantum-classical algorithms have been developed that aim to calculate the ground state energies of molecular systems on Noisy Intermediate-Scale Quantum (NISQ) devices. Albeit the utilization of shallow depth circuits in these algorithms, the optimization of ansatz parameters necessitates a substantial number of quantum measurements, leading to prolonged runtimes on the scantly available quantum hardware. Through our work, we lay the general foundation for an interdisciplinary approach that can be used to drastically reduce the dependency of these algorithms on quantum infrastructure. We showcase these pertinent concepts within the framework of the recently developed Projective Quantum Eigensolver (PQE) that involves iterative optimization of the nonlinearly coupled parameters through repeated residue measurements on quantum hardware. We demonstrate that one may perceive such a nonlinear optimization problem as a collective dynamic interplay of fast and slow relaxing modes. As such, the synergy among the parameters is exploited using an on-the-fly supervised machine learning protocol that dynamically casts the PQE optimization into a smaller subspace by reducing the effective degrees of freedom. We demonstrate analytically and numerically that our proposed methodology ensures a drastic reduction in the number of quantum measurements necessary for the parameter updates without compromising the characteristic accuracy. Furthermore, the machine learning model may be tuned to capture the noisy data of NISQ devices, and thus the predicted energy is shown to be resilient under a given noise model.

Efficient quantum feature extraction for CNN-based learning

Hybrid quantum-classical algorithms provide a promising way to harness the power of current quantum devices. In this framework, parametrized quantum circuits (PQCs) which consist of layers of parametrized unitaries can be considered as a kind of quantum neural networks. Recent works have begun to explore the potential of PQCs as general function approximators. In this work, we propose a quantum-classical deep network structure to enhance model discriminability of convolutional neural networks (CNNs). In CNNs, the convolutional layer uses linear filters to scan the input data followed by a nonlinear operation. Instead, we build PQCs, which are more potent function approximators, with more complex structures to capture the features within the receptive field. The feature maps are obtained by sliding the PQCs over the input in a similar way as CNN. We also give a training algorithm for the proposed model. Through numerical simulation, the proposed hybrid models demonstrate reasonable classification performance on MNIST and Fashion-MNIST (4-classes). In addition, we compare the performance of models in different settings. The results demonstrate that the model with high-expressibility ansaetze achieves lower cost and higher accuracy, but exhibits a “saturation” phenomenon.

Hybrid Quantum-Classical Algorithms for Loan-Collection Optimization with Loan-Loss Provisions

Hybrid Quantum-Classical Algorithms for Loan-Collection Optimization with Loan-Loss Provisions

Recent Advances in Quantum Computing for Drug Discovery and Development

Drug discovery and development is a time-consuming and cost-intensive process. Computer-aided drug design can speed up the timeline and reduce costs by decreasing the number of necessary biochemical experiments. The number of studies using quantum computing to solve problems in drug development has been increasing in recent years. In this review, we briefly introduce the main steps in drug discovery and development and how computers help to find potential drug candidates. Recent studies of quantum computing in drug development based on the structure of target proteins are listed chronologically. They include protein structure prediction, molecular docking, quantum simulation, and quantitative structure-activity relationship (QSAR) models. Current quantum devices are still susceptible to noise and error but are well suited for hybrid quantum-classical algorithms. The quantum advantage is demonstrated on hybrid systems and quantum-inspired devices such as quantum annealers. We hope to see more applications of quantum computing in the field of drug discovery and development.

Adaptive, problem-tailored variational quantum eigensolver mitigates rough parameter landscapes and barren plateaus

Variational quantum eigensolvers (VQEs) represent a powerful class of hybrid quantum-classical algorithms for computing molecular energies. Various numerical issues exist for these methods, however, including barren plateaus and large numbers of local minima. In this work, we consider the Adaptive, Problem-Tailored Variational Quantum Eiegensolver (ADAPT-VQE) ansätze, and examine how they are impacted by these local minima. We find that while ADAPT-VQE does not remove local minima, the gradient-informed, one-operator-at-a-time circuit construction accomplishes two things: First, it provides an initialization strategy that can yield solutions with over an order of magnitude smaller error compared to random initialization, and which is applicable in situations where chemical intuition cannot help with initialization, i.e., when Hartree-Fock is a poor approximation to the ground state. Second, even if an ADAPT-VQE iteration converges to a local trap at one step, it can still “burrow” toward the exact solution by adding more operators, which preferentially deepens the occupied trap. This same mechanism helps highlight a surprising feature of ADAPT-VQE: It should not suffer optimization problems due to barren plateaus and random initialization. Even if such barren plateaus appear in the parameter landscape, our analysis suggests that ADAPT-VQE avoids such regions by design.

Is quantum computing green? An estimate for an energy-efficiency quantum advantage

The quantum advantage threshold determines when a quantum processing unit (QPU) is more efficient with respect to classical computing hardware in terms of algorithmic complexity. The ‘green’ quantum advantage threshold—based on a comparison of energetic efficiency between the two—is going to play a fundamental role in the comparison between quantum and classical hardware. Indeed, its characterization would enable better decisions on energy-saving strategies, e.g. for distributing the workload in hybrid quantum–classical algorithms. Here, we show that the green quantum advantage threshold crucially depends on (a) the quality of the experimental quantum gates and (b) the entanglement generated in the QPU. Indeed, for noisy intermediate-scale quantum hardware and algorithms requiring a moderate amount of entanglement, a classical tensor network emulation can be more energy-efficient at equal final state fidelity than quantum computation. We compute the green quantum advantage threshold for a few paradigmatic examples in terms of algorithms and hardware platforms, and identify algorithms with a power-law decay of singular values of bipartitions—with power-law exponent —as the green quantum advantage threshold in the near future.

Variational Quantum Eigensolver with Mutual Variance-Hamiltonian Optimization

The zero-energy variance principle can be exploited in variational quantum eigensolvers for solving general eigenstates but its capacity for obtaining a specified eigenstate, such as ground state, is limited as all eigenstates are of zero energy variance. We propose a variance-based variational quantum eigensolver for solving the ground state by searching in an enlarged space of wavefunction and Hamiltonian. With a mutual variance-Hamiltonian optimization procedure, the Hamiltonian is iteratively updated to guild the state towards to the ground state of the target Hamiltonian by minimizing the energy variance in each iteration. We demonstrate the performance and properties of the algorithm with numeral simulations. Our work suggests an avenue for utilizing guided Hamiltonian in hybrid quantum-classical algorithms.

Avoiding barren plateaus via transferability of smooth solutions in a Hamiltonian variational ansatz

Parametrized quantum circuits inspired by adiabatic quantum computation often suffer from vanishing gradients, hindering the trainability of hybrid quantum-classical algorithms. Here, the authors put forward a strategy to overcome this by reusing parameters and iterating from small to large systems.