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 $$\hbox {HHL}^{++}$$ : (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.
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.
We propose a hybrid quantum algorithm based on the Harrow-Hassidim-Lloyd (HHL) algorithm for solving a system of linear equations. In this paper, we show that our hybrid algorithm can reduce a circuit depth from the original HHL algorithm by means of a classical information feed-forward after the quantum phase estimation algorithm, and the results of the hybrid algorithm are identical to those of the HHL algorithm. In addition, it is experimentally examined with four qubits in the IBM Quantum Experience setups, and the experimental results of our algorithm show higher accurate performance on specific systems of linear equations than that of the HHL algorithm.
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.
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.
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.
We present a classical enhancement to improve the accuracy of the Hybrid variant (Hybrid HHL) of the quantum algorithm for solving linear systems of equations proposed by Harrow, Hassidim, and Lloyd (HHL). We achieve this by using higher precision quantum estimates of the eigenvalues relevant to the linear system, and a new classical step to guide the eigenvalue inversion part of Hybrid HHL. We show that eigenvalue estimates with just two extra bits of precision result in tighter error bounds for our Enhanced Hybrid HHL compared to HHL. Our enhancement reduces the error of Hybrid HHL by an average of 57 percent on an ideal quantum processor for a representative sample of 2x2 systems. On IBM Torino and IonQ Aria-1 hardware, we see that the error of Enhanced Hybrid HHL is on average 13 percent and 20 percent (respectively) less than that of HHL for the same set of systems.