Benchmarking Improved Tomography and Implementation of Quantum Linear Systems Algorithms

Systems of linear equations are ubiquitous across science, engineering, machine learning, and 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 by running HHL with Iterative Refinement on noisy emulators, Nvidia CUDA-Q, and quantum hardware. We also present our open-source implementation, emphasizing that our circuit is not tailored to specific problem instances, as most available implementations are.