# An Early Investigation of the HHL Quantum Linear Solver for Scientific Applications
## 1 Introduction  
Starting with the Deutsch-Jozsa algorithm and [PERSON]'s discrete logarithm algorithm [1, 2], the potential of quantum computing algorithms has extended beyond merely simulating quantum systems. The potential speedup of quantum algorithms over their classical counterparts has gathered tremendous attention, including a fundamental demand in science and engineering: solving linear systems. [PERSON], [PERSON], and [PERSON] (HHL) first developed a quantum linear solver with an exponential speedup in problem dimensions in [3]. Built upon the exponential speedup of quantum linear system algorithms (QLSAs), many works have explored theoretical quantum advantages in various applications. These fields include portfolio optimization [4], machine learning [5, 6], differential equation solving [7], linear optimization [8, 9, 10, 11], and semi-definite optimization [12, 13]. However, the HHL algorithm proposed in [3] has a quadratic dependency on matrix condition number and matrix sparsity, worse than classical linear solvers such as factorization methods and conjugate gradient, where condition number is the product of the norm of the coefficient matrix and the norm of the inverse matrix. Several works have been proposed to reduce the dependency on the condition number of coefficient matrices and the accuracy of the solution state [14, 15, 16, 17, 18, 19, 20, 21, 22]. Specifically, based on adiabatic theorems,the state of the art has a linear or quasi-linear dependency on the condition number and a logarithmic dependency on the inverse of the solution accuracy [18; 19; 20; 22]. The HHL algorithm has been demonstrated in experiments to solve linear algebra problems. The largest linear systems demonstrated on real gate-based quantum machines are up to \(4\times 4\) systems with variants of the HHL algorithm [23; 24; 25] and an \(8\times 8\) system with the linear solver based on adiabatic quantum computing [26]. However, testing QLSAs on real quantum devices to demonstrate a quantum advantage still suffers from multiple obstacles, such as the large number of required quantum gates and the high noise level of current quantum devices [27]. With the current development of quantum hardware and exploration of quantum error correction (QEC) codes, a large-scale fault-tolerant quantum computer is expected to be demonstrated in the foreseeable future [28; 29; 30; 31; 32; 33; 34]. QEC codes, such as surface codes, are expected to detect and correct Pauli errors, as well as any linear combinations of them, provided the errors occur below a certain threshold probability [35]. Although the gap between algorithm requirements and hardware specifications is shrinking, the gap still exists, which necessitates the analysis of the resource costs involved [36]. Resource estimations have been performed for chemistry [37], [PERSON]'s algorithm on the Advanced Encryption Standard [38], [PERSON]'s discrete logarithm algorithm for the RSA cryptosystem [39], and the computation of elliptic curve discrete logarithms [40]. However, despite this being essential for understanding the disparity between hardware capabilities and practical applications, there is limited work on non-asymptotic resource estimation for QLSAs [41]. In this paper, we focus on resource estimation and experiment with the HHL algorithm on several applications selected from domain science, such as power grid and climate projection. Different from the previous works about asymptotic and non-asymptotic resource analysis [3; 14; 15; 16; 17; 18; 19; 20; 21; 22; 41], we investigate the factors affecting the final accuracy, resource cost, and fault-tolerant hardware requirements. Our experiments show the effectiveness of the HHL algorithm in scientific applications with a low accuracy in quantum phase estimation.