# An Early Investigation of the HHL Quantum Linear Solver for Scientific Applications
## 7 Conclusions  
In this paper, we investigate the practical applications and scalability of the HHL algorithm in solving quantum linear systems associated with scientific problems like power grids and heat transfer problems. Through the NWQSim package on high-performance computing platforms, we highlight the benefits of the utilization of low-accuracy QPE in HHL for both iterative and non-iterative methods in practice: low-accuracy QPE can exponentially reduce the gate counts and circuit depth in an HHL circuit, while keeping the same solution accuracy in iterative methods like the Newton-Raphson method and maintaining a similar level of accuracy in a non-iterative method like the finite difference method. Furthermore, with the Azure Quantum resource estimator, we evaluate the resource requirements of HHL circuits in our experiments under two settings that simulate superconducting and trapped-ion qubits. The correlations between QEC-related criteria and the input HHL circuits have been thoroughly studied. The runtime, number of logical cycles, and number of \(T\) states have exponential dependencies on the number of clock qubits in QPE. However, this relation is not necessarily inherited by the number of physical qubits demanded. In our experiments, we find that even as \(n_{c}\) increases and the error budget reduces, it is possible that \(T\) factory demand also decreases. More specifically, if the runtime growth is faster than the required amount of \(T\) states, the circuit needs fewer \(T\) factories and thus fewer physical qubits to prepare \(T\) factories. Since the growth of runtime is sensitive to the error budget, it is possible to reduce the physical qubit requirement if a low error budget is achievable on early fault-tolerant quantum devices. Our study provides pivotal insights into the operational requirements of quantum linear system algorithms, paving the way for further empirical studies. We propose future research on the applications of quantum linear system solvers and iterative refinement on high-fidelity quantum computers for small-scale experiments. For large-scale experiments, we suggest using noise-modeled simulators on high-performance platforms. In the context of QEC and early fault-tolerant quantum computing, we believe it is crucial to focus on controlling the resource cost of \(T\) factories by considering the runtime and error budget. These research directions hold promise for bridging the gap between theoretical potential and practical usability in quantum computing. All the code in this research will be hosted in a public repository ([[https://github.com/pnnl/nwqlib](https://github.com/pnnl/nwqlib)]([https://github.com/pnnl/nwqlib](https://github.com/pnnl/nwqlib)), accessed on 4 August 2025). Conceptualization, M.Z., C.L., S.S., X.L., J.M., Y.C. and A.L. ; methodology, M.Z., C.L., S.S. and A.L. ; software, M.Z. and A.L. ; validation, S.S., X.L., J.M., Y.C. and A.L. ; formal analysis, M.Z. and C.L. ; investigation, M.Z. ; resources, X.L., J.M., Y.C. and A.L. ; data curation, M.Z., X.L., J.M. and Y.C. ; writing--original draft preparation, M.Z., C.L., S.S., X.L., J.M., Y.C. and A.L. ; writing--review and editing, M.Z. and A.L. ; visualization, M.Z. ; supervision, S.S., J.M., Y.C. and A.L. ; project administration, A.L. ; funding acquisition, A.L. All authors have read and agreed to the published version of the manuscript. This research was supported by the Quantum Algorithms and Architecture for Domain Science Initiative (QuAADS), under the Laboratory Directed Research and Development (LDRD) Program at Pacific Northwest National Laboratory (PNNL). 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-AC05-00 OR22725.