The Harrow-Hassidim-Lloyd (HHL) algorithm remains a paradigmatic quantum routine for solving linear systems and a widely used benchmark in theoretical and experimental studies. Several circuit-level variants have been proposed to simplify its implementation, including constructions in which the clock register is factorized via Hadamard gates to reduce circuit depth and entanglement. In this work, we show that this commonly deployed modification can fundamentally alter the asymptotic behavior of the algorithm and may fail to converge in the limit of large clock size due to persistent phase interference effects. This phenomenon is not a finite-size artifact and compromises the correctness guarantees of the factorized-clock implementation. We provide an analytic characterization of the underlying mechanism and demonstrate that a minimal and experimentally feasible modification suffices to restore convergence while preserving the practical advantages of the simplified circuit. Our results clarify an overlooked correctness issue in a widely used HHL implementation pattern and delineate the conditions under which simplified realizations faithfully reproduce the intended algorithmic behavior.
The Harrow-Hassidim-Lloyd (HHL) algorithm remains a paradigmatic quantum routine for solving linear systems and a widely used benchmark in theoretical and experimental studies. Several circuit-level variants have been proposed to simplify its implementation, including constructions in which the clock register is factorized via Hadamard gates to reduce circuit depth and entanglement. In this work, we show that this commonly deployed modification can fundamentally alter the asymptotic behavior of the algorithm and may fail to converge in the limit of large clock size due to persistent phase interference effects. This phenomenon is not a finite-size artifact and compromises the correctness guarantees of the factorized-clock implementation. We provide an analytic characterization of the underlying mechanism and demonstrate that a minimal and experimentally feasible modification suffices to restore convergence while preserving the practical advantages of the simplified circuit. Our results clarify an overlooked correctness issue in a widely used HHL implementation pattern and delineate the conditions under which simplified realizations faithfully reproduce the intended algorithmic behavior.
We analyze the performance of the Harrow-Hassidim-Lloyd algorithm (HHL algorithm) for solving linear problems and of a variant of this algorithm (HHL variant) commonly encountered in literature. This variant relieves the algorithm of preparing an entangled initial state of an auxiliary register. We prove that the computational error of the variant algorithm does not always converge to zero when the number of qubits is increased, unlike the original HHL algorithm. Both algorithms rely upon two fundamental quantum algorithms, the quantum phase estimation and the amplitude amplification. In particular, the error of the HHL variant oscillates due to the presence of undesired phases in the amplitude to be amplified, while these oscillations are suppressed in the original HHL algorithm. Then, we propose a modification of the HHL variant, by amplifying an amplitude of the state vector that does not exhibit the above destructive interference. We also study the complexity of these algorithms in the light of recent results on simulation of unitaries used in the quantum phase estimation step, and show that the modified algorithm has smaller error and lower complexity than the HHL variant. We supported our findings with numerical simulations.
In this paper, we study the error propagation and generation in the Harrow-Hassidim-Lloyd (HHL) quantum algorithm runs on IBM-Q hardware with the help of a MATLAB simulator. HHL is a quantum algorithm that can provide exponential speedup over the fastest classical algorithm (conjugate gradient method) in solving systems of linear equations (SLE). However, without error correction, it cannot give correct results even in a 2-variable system due to its complexity. In this study, an HHL quantum circuit for a 2-variable SLE is implemented in IBM-Q and the error is extracted after each stage of the circuit and compared to a MATLAB simulator. We identified three major sources of errors, namely single-qubit flipping, gate infidelity, and error propagation. We also found that at the ancillary bit rotation stage, the error becomes large but the encoded solution still has high fidelity. However, the solution is mostly lost after the inverse quantum phase estimation which is necessary to extract the solution efficiently. Therefore, it is suggested that error correction resources, if limited, should be added to the second half of the circuit.
The limited capabilities of current quantum hardware significantly constrain the scale of experimental demonstrations of most quantum algorithmic primitives. This makes it challenging to perform benchmarking of the current hardware using useful quantum algorithms, i.e., application-oriented benchmarking. In particular, the Harrow-Hassidim-Lloyd (HHL) algorithm is a critical quantum linear algebra primitive, but the majority of the components of HHL are far out of the reach of noisy intermediate-scale quantum devices, which has led to the proposal of hybrid classical-quantum variants. The goal of this work is to further bridge the gap between proposed near-term friendly implementations of HHL and the kinds of quantum circuits that can be executed on noisy hardware. Our proposal adds to the existing literature of hybrid quantum algorithms for linear algebra that are more compatible with the current scale of quantum devices. Specifically, we propose two modifications to the Hybrid HHL algorithm proposed by Lee et al., leading to our algorithm Hybrid HHL++ : (1) propose a novel algorithm for determining a scaling factor for the linear system matrix that maximizes the utility of the amount of ancillary qubits allocated to the phase estimation component of HHL, and (2) introduce a heuristic for compressing the HHL circuit. We demonstrate the efficacy of our work by running our modified Hybrid HHL on Quantinuum System Model H-series trapped-ion quantum computers to solve different problem instances of small-scale portfolio optimization problems, leading to the largest experimental demonstrations of HHL for an application to date.
We propose a hybrid quantum-classical implementation of the Harrow-Hassidim-Lloyd (HHL) algorithm for the matrix inversion (linear-systems) problem, a key subroutine in quantum computing. Rather than claiming a new complexity result, we provide an assumption-explicit proof sketch, aligned with established literature, that contextualizes matrix inversion as Bounded-Error Quantum Polynomial Time (BQP)-complete and offers reduction-based intuition via Grover-type amplitude amplification. We implement HHL for 2 × 2, 4 × 4, and 8 × 8 Hermitian matrices using Qiskit and evaluate performance using simulator-based experiments with calibrated noise models and executions on IBM Quantum and IonQ backends. To enhance robustness on Noisy Intermediate-Scale Quantum (NISQ) hardware, we integrate measurement-error mitigation, zero-noise extrapolation, and post-selection, and we quantify performance using fidelity and mean squared error. Results from noiseless simulation, calibrated-noise simulation, and real-device execution show that mitigation improves reconstruction fidelity for small Hermitian systems, particularly for deeper circuits. These findings support mitigation-aware feasibility on current NISQ devices; however, they do not establish present-day runtime advantage over optimized classical solvers. The proposed workflow is relevant to quantum optimization, quantum machine learning, and quantum chemistry, where linear-system primitives arise frequently. Future work will focus on reducing phase-estimation depth (e.g., iterative or semiclassical variants), improving compilation and benchmarking transparency, and re-evaluating larger instances on higher-fidelity hardware.
In 2019, Yonghae Lee et al. combined the circuit implementation of the Harrow-Hassidim-Lloyd (HHL) algorithm with a classical computer, and designed a hybrid HHL algorithm to reduce experimental errors caused by decoherence and so on. However, the improvement is achieved only in the auxiliary quantum coding phase, and no quantum resource reduction is done on the quantum phase estimation and inverse quantum phase estimation stages. At the same time, the circuit improvement illustration on a [Formula: see text] linear system just has the result and no specific process. In this paper, based on the idea of the hybrid HHL algorithm and a generic circuit of HHL algorithm, an improved circuit implementation of the HHL algorithm is proposed. The feasibility of the improved circuit implementation of the HHL algorithm is verified by IBM's qiskit. The improved circuit illustrations on a [Formula: see text] linear system show that the improved circuit implementation of the HHL algorithm can effectively reduce quantum resources without losing the fidelity of the results. Thus the improved circuit implementation of the HHL algorithm can further avoid some result errors than the existing implementation methods.
The development of quantum processors for practical fluid flow problems is a promising yet distant goal. Recent advances in quantum linear solvers have highlighted their potential for classical fluid dynamics. In this study, we evaluate the Harrow–Hassidim–Lloyd (HHL) quantum linear systems algorithm (QLSA) for solving the idealized Hele–Shaw flow. Our focus is on the accuracy and computational cost of the HHL solver, which we find to be sensitive to the condition number, scaling exponentially with problem size. This emphasizes the need for preconditioning to enhance the practical use of QLSAs in fluid flow applications. Moreover, we perform shots-based simulations on quantum simulators and test the HHL solver on superconducting quantum devices, where noise, large circuit depths, and gate errors limit performance. Error suppression and mitigation techniques improve accuracy, suggesting that such fluid flow problems can benchmark noise mitigation efforts. Our findings provide a foundation for future, more complex application of QLSAs in fluid flow simulations.