An Enhanced Hybrid HHL Algorithm

We present a classical enhancement to improve the accuracy of the Hybrid variant (Hybrid HHL) of the quantum algorithm for solving liner 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 an enhanced classical processing step to guide the eigenvalue inversion part of Hybrid HHL. We show that eigenvalue estimates with just two extra bits of precision results in tighter error bounds for our Enhanced Hybrid HHL compared to HHL. We also show that 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 Hanoi and IonQ Aria-1 hardware, we see that the error of Enhanced Hybrid HHL algorithm is on average 13 percent and 20 percent (respecitvely) less than that of HHL for a similar set of 2x2 systems. Finally, we use simulated eigenvalue estimates to perform an inversion of a 4x4 matrix on IonQ Aria-1 with a fidelity of 0.61. To our knowledge this is the largest HHL implementation with a fidelity greater than 0.5.