Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

IEEE International Conference on Quantum Computing and Engineering (QCE)

Practical Scalability of LuGo: Benchmarking the HHL Algorithm Using an Enhanced QPE Algorithm

Chao Lu, Muralikrishnan Gopalakrishnan Meena, Antigoni Georgiadou, Kalyana C. Gottiparthi, Michael Sandoval, E. A. Coello Pérez, Paul T. Lin, In‐Saeng Suh, Seongmin Kim, Alessandro Baroni

Published Aug 30, 2025• 0 citations
Publisher

Abstract

The HHL algorithm is a prominent quantum algorithm that offers exponential speedup over its classical counterparts for solving a system of linear equations. However, synthesizing and executing HHL circuits demand significant computational resources from both classical and quantum systems. In this paper, we benchmark the HHL algorithm using the optimized Quantum Phase Estimation (QPE) generation algorithm, LuGo [1], to enhance its scalability and efficiency. We leverage various highperformance computing (HPC) systems–the National Energy Research Scientific Computing Center's (NERSC's) Perlmutter supercomputer and Oak Ridge Leadership Computing Facility's (OLCF's) Andes system and Frontier supercomputer–to evaluate the scalability of generating HHL circuits and to measure the time to simulate the generated circuits. Additionally, enabled by the Quantum Computing User Program (QCUP) of OLCF, we provide a comprehensive analysis of the algorithm's performance on various state-of-the-art superconducting and trappedion quantum devices, including studies on qubit connectivity, fidelity comparisons, and hardware compatibility and robustness. Our results offer preliminary insights into potential practical applications of the HHL algorithm enabled by LuGo and the performance of various types of quantum hardware.11Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231.

DOI
10.1109/qce65121.2025.00029

Related papers

Quantum Speedup for Linear Systems: An Analysis of the HHL Algorithm Using IBM Qiskit

V Shwetha, Abinaya Selvarajan, A Aarthi +1 more

One of the most significant developments in quantum computing is the Harrow-Hassidim-Lloyd (HHL) method, which can solve linear equation systems at exponential speedup. Because linear systems are essential to many scientific fields, including physics, engineering, and machine learning, this approach has great potential to revolutionize computational paradigms. The HHL algorithm is thoroughly examined in this work, with particular attention paid to its theoretical framework, real-world application utilizing IBM's Qiskit platform, and the difficulties in simulating quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. Using parameters like fidelity, time complexity, and scalability, the research further evaluates the HHL algorithm's performance in comparison to traditional methods. According to the findings, quantum simulations work well for small-scale matrices like 2x2 and 4x4, but expanding the approach to bigger systems is still difficult because of hardware and software constraints. Finally, the research emphasizes the key directions for advancing quantum hardware and algorithms to overcome current scalability challenges, enabling broader applicability of the HHL algorithm in solving complex linear systems.

Journal of Electronics and Informatics•Jan 9, 2025•0 citations

The HHL algorithm: Implementation and research directions

Varsha Sambhaje, Anju Chaurasia

Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed “NP-completeness” as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm’s potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.

International Journal of Quantum Information•May 31, 2024•2 citations

Benchmarking Highly Precise Quantum Linear Systems Algorithms

Adrian Harkness, Kate Saltovets, Mohammadhossein Mohammadisiahroudi +1 more

Systems of linear equations are ubiquitous across science, engineering, machine learning, and even finance. While classical methods can be prohibitively slow for large-scale problems, quantum linear systems algorithms offer the potential for exponential speedup in certain parameter regimes. However, a significant gap persists between this theoretical promise and practical implementation, as the advantages are often obscured by the substantial quantum resources and high sensitivity to noise inherent in current quantum hardware. One way to bridge this gap is through the use of Iterative Refinement, a classical post-processing scheme that can exponentially improve the accuracy to which a linear system of equations can be solved using low-precision arithmetic. In the context of quantum linear systems algorithms, such as the HHL algorithm proposed by Harrow, Hassidim, and Loyd, Iterative Refinement can greatly reduce the quantum resources required to calculate an accurate solution in terms of tomography cost, circuit volume, and fault-tolerant overhead. Here, we compute and benchmark highly precise solutions to linear systems of equations of up to eight variables by running HHL with Iterative Refinement on NISQ quantum computers. We also present our open-source implementation, emphasizing that our circuit is not tailored to specific problem instances, as most available implementations are.

IEEE International Conference on Quantum Computing and Engineering (QCE)•Aug 30, 2025•0 citations

Enabling Scalable VQE Simulation on Leading HPC Systems

Meng Wang, Fei Hua, Chenxu Liu +6 more

Large-scale simulations of quantum circuits pose significant challenges, especially in quantum chemistry, due to the number of qubits, circuit depth, and the number of circuits needed per problem. High-performance computing (HPC) systems offer massive computational capabilities that could help overcome these obstacles. We developed a high-performance quantum circuit simulator called NWQ-Sim, and demonstrated its capability to simulate large quantum chemistry problems on NERSC’s Perlmutter supercomputer. Integrating NWQ-Sim with XACC, an open-source programming framework for quantum-classical applications, we have executed quantum phase estimation (QPE) and variational quantum eigensolver (VQE) algorithms for downfolded quantum chemistry systems at unprecedented scales. Our work demonstrates the potential of leveraging HPC resources and optimized simulators to advance quantum chemistry and other applications of near-term quantum devices. By scaling to larger qubit counts and circuit depths, high-performance simulators like NWQ-Sim will be critical for characterizing and validating quantum algorithms before their deployment on actual quantum hardware.

International Conference for High Performance Computing, Networking, Storage and Analysis (SC)•Nov 10, 2023•9 citations

Identifying Bottlenecks of NISQ-Friendly HHL Algorithms

Marc Andreu Marfany, Alona Sakhnenko, Jeanette Miriam Lorenz

Quantum computing promises enabling solving large problem instances, e.g. large linear equation systems with HHL algorithm, once the hardware stack matures. For the foreseeable future quantum computing will remain in the so-called NISQ era, in which the algorithms need to account for the flaws of the hardware such as noise. In this work, we perform an empirical study to test scaling properties and directly related noise resilience of the the most resources-intense component of the HHL algorithm, namely QPE and its NISQ-adaptation Iterative QPE. We explore the effectiveness of noise mitigation techniques for these algorithms and investigate whether we can keep the gate number low by enforcing sparsity constraints on the input or using circuit optimization techniques provided by Qiskit package. Our results indicate that currently available noise mitigation techniques, such as Qiskit readout and Mthree readout packages, are insufficient for enabling results recovery even in the small instances tested here. Moreover, our results indicate that the scaling of these algorithms with increase in precision seems to be the most substantial obstacle. These insights allowed us to deduce an approximate bottleneck for algorithms that consider a similar time evolution as QPE. Such observations provide evidence of weaknesses of such algorithms on NISQ devices and help us formulate meaningful future research directions.

arXiv:2406.06288•IEEE International Conference on Quantum Computing and Engineering (QCE)•Sep 15, 2024•5 citations

An Early Investigation of the HHL Quantum Linear Solver for Scientific Applications

Muqing Zheng, Chenxu Liu, Samuel Stein +4 more

In this paper, we explore using the Harrow–Hassidim–Lloyd (HHL) algorithm to address scientific and engineering problems through quantum computing, utilizing the NWQSim simulation package on a high-performance computing platform. Focusing on domains such as power-grid management and climate projection, we demonstrate the correlations of the accuracy of quantum phase estimation, along with various properties of coefficient matrices, on the final solution and quantum resource cost in iterative and non-iterative numerical methods such as the Newton–Raphson method and finite difference method, as well as their impacts on quantum error correction costs using the Microsoft Azure Quantum resource estimator. We summarize the exponential resource cost from quantum phase estimation before and after quantum error correction and illustrate a potential way to reduce the demands on physical qubits. This work lays down a preliminary step for future investigations, urging a closer examination of quantum algorithms’ scalability and efficiency in domain applications.

arXiv:2404.19067•Algorithms•Aug 6, 2025•8 citations
The HHL algorithm is a prominent quantum algorithm that offers exponential speedup over its classical counterparts for solving a system of linear equations. However, synthesizing and executing HHL circuits demand significant computational resources from both classical and quantum systems. In this paper, we benchmark the HHL algorithm using the optimized Quantum Phase Estimation (QPE) generation algorithm, LuGo [1], to enhance its scalability and efficiency. We leverage various highperformance computing (HPC) systems–the National Energy Research Scientific Computing Center's (NERSC's) Perlmutter supercomputer and Oak Ridge Leadership Computing Facility's (OLCF's) Andes system and Frontier supercomputer–to evaluate the scalability of generating HHL circuits and to measure the time to simulate the generated circuits. Additionally, enabled by the Quantum Computing User Program (QCUP) of OLCF, we provide a comprehensive analysis of the algorithm's performance on various state-of-the-art superconducting and trappedion quantum devices, including studies on qubit connectivity, fidelity comparisons, and hardware compatibility and robustness. Our results offer preliminary insights into potential practical applications of the HHL algorithm enabled by LuGo and the performance of various types of quantum hardware.11Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231.
One of the most significant developments in quantum computing is the Harrow-Hassidim-Lloyd (HHL) method, which can solve linear equation systems at exponential speedup. Because linear systems are essential to many scientific fields, including physics, engineering, and machine learning, this approach has great potential to revolutionize computational paradigms. The HHL algorithm is thoroughly examined in this work, with particular attention paid to its theoretical framework, real-world application utilizing IBM's Qiskit platform, and the difficulties in simulating quantum algorithms on noisy intermediate-scale quantum (NISQ) devices. Using parameters like fidelity, time complexity, and scalability, the research further evaluates the HHL algorithm's performance in comparison to traditional methods. According to the findings, quantum simulations work well for small-scale matrices like 2x2 and 4x4, but expanding the approach to bigger systems is still difficult because of hardware and software constraints. Finally, the research emphasizes the key directions for advancing quantum hardware and algorithms to overcome current scalability challenges, enabling broader applicability of the HHL algorithm in solving complex linear systems.
Linear systems of equations lie at the heart of numerous scientific and engineering challenges. In cutting-edge arena like artificial intelligence, machine learning and neuro-computation, these systems serve as a fundamental tool for mathematical modeling. Classical algorithms for solving linear systems have been extensively developed and forms the backbone of diverse applications across various scientific disciplines. While classical algorithms exist for solving linear systems, they often encounter limitations termed “NP-completeness” as data complexity increases. The emerging field of quantum computing offers a revolutionary approach to deal with these kinds of problems. The Harrow–Hassidim–Lloyd (HHL) algorithm tackles these challenges and opens new avenues for research. This study delves into the contemporary effectiveness of the HHL algorithm to address systems of linear equations. By examining recent research in quantum machine learning, we aim to assess the HHL algorithm’s potential to revolutionize the process of optimizing hyperparameters for machine learning models, resulting in increased efficiency and cost savings. This paper meticulously analyzes the HHL algorithm and explores its evolution from conception to the latest advancements. A comprehensive examination of the HHL algorithm, including its evolution over time, is thoroughly explored. The investigation delves into the potential challenges and limitations that might hinder the practical deployment of the HHL algorithm. Identifying these roadblocks will pave the way for future research and development efforts.
Systems of linear equations are ubiquitous across science, engineering, machine learning, and even finance. While classical methods can be prohibitively slow for large-scale problems, quantum linear systems algorithms offer the potential for exponential speedup in certain parameter regimes. However, a significant gap persists between this theoretical promise and practical implementation, as the advantages are often obscured by the substantial quantum resources and high sensitivity to noise inherent in current quantum hardware. One way to bridge this gap is through the use of Iterative Refinement, a classical post-processing scheme that can exponentially improve the accuracy to which a linear system of equations can be solved using low-precision arithmetic. In the context of quantum linear systems algorithms, such as the HHL algorithm proposed by Harrow, Hassidim, and Loyd, Iterative Refinement can greatly reduce the quantum resources required to calculate an accurate solution in terms of tomography cost, circuit volume, and fault-tolerant overhead. Here, we compute and benchmark highly precise solutions to linear systems of equations of up to eight variables by running HHL with Iterative Refinement on NISQ quantum computers. We also present our open-source implementation, emphasizing that our circuit is not tailored to specific problem instances, as most available implementations are.
Large-scale simulations of quantum circuits pose significant challenges, especially in quantum chemistry, due to the number of qubits, circuit depth, and the number of circuits needed per problem. High-performance computing (HPC) systems offer massive computational capabilities that could help overcome these obstacles. We developed a high-performance quantum circuit simulator called NWQ-Sim, and demonstrated its capability to simulate large quantum chemistry problems on NERSC’s Perlmutter supercomputer. Integrating NWQ-Sim with XACC, an open-source programming framework for quantum-classical applications, we have executed quantum phase estimation (QPE) and variational quantum eigensolver (VQE) algorithms for downfolded quantum chemistry systems at unprecedented scales. Our work demonstrates the potential of leveraging HPC resources and optimized simulators to advance quantum chemistry and other applications of near-term quantum devices. By scaling to larger qubit counts and circuit depths, high-performance simulators like NWQ-Sim will be critical for characterizing and validating quantum algorithms before their deployment on actual quantum hardware.
Quantum computing promises enabling solving large problem instances, e.g. large linear equation systems with HHL algorithm, once the hardware stack matures. For the foreseeable future quantum computing will remain in the so-called NISQ era, in which the algorithms need to account for the flaws of the hardware such as noise. In this work, we perform an empirical study to test scaling properties and directly related noise resilience of the the most resources-intense component of the HHL algorithm, namely QPE and its NISQ-adaptation Iterative QPE. We explore the effectiveness of noise mitigation techniques for these algorithms and investigate whether we can keep the gate number low by enforcing sparsity constraints on the input or using circuit optimization techniques provided by Qiskit package. Our results indicate that currently available noise mitigation techniques, such as Qiskit readout and Mthree readout packages, are insufficient for enabling results recovery even in the small instances tested here. Moreover, our results indicate that the scaling of these algorithms with increase in precision seems to be the most substantial obstacle. These insights allowed us to deduce an approximate bottleneck for algorithms that consider a similar time evolution as QPE. Such observations provide evidence of weaknesses of such algorithms on NISQ devices and help us formulate meaningful future research directions.
In this paper, we explore using the Harrow–Hassidim–Lloyd (HHL) algorithm to address scientific and engineering problems through quantum computing, utilizing the NWQSim simulation package on a high-performance computing platform. Focusing on domains such as power-grid management and climate projection, we demonstrate the correlations of the accuracy of quantum phase estimation, along with various properties of coefficient matrices, on the final solution and quantum resource cost in iterative and non-iterative numerical methods such as the Newton–Raphson method and finite difference method, as well as their impacts on quantum error correction costs using the Microsoft Azure Quantum resource estimator. We summarize the exponential resource cost from quantum phase estimation before and after quantum error correction and illustrate a potential way to reduce the demands on physical qubits. This work lays down a preliminary step for future investigations, urging a closer examination of quantum algorithms’ scalability and efficiency in domain applications.