This paper assesses the numerical accuracy of the Harrow-Hassidim-Lloyd (HHL) algorithm in solving a finite-difference approximation of the Poisson equation, with the Thomas algorithm used as a comparative benchmark. In the one-dimensional setting with homogeneous boundary conditions, the HHL algorithm exhibits numerical accuracy comparable to the Thomas algorithm; however, the HHL algorithm exhibits increased sensitivity to variations in input parameters for simulations with nonhomogeneous boundary conditions. While the current implementation of the HHL algorithm is limited to solving the one-dimensional Poisson equation, we propose a hybrid framework to extend the applicability of the HHL algorithm to solving the two-dimensional Poisson equation. Our results show that although both algorithms perform similarly for small systems of 8 × 8 mesh nodes, the HHL algorithm shows increased input-parameter sensitivity for systems of 16 × 16 mesh nodes, indicating that further development is needed to scale the hybrid approach effectively.
Although the Harrow-Hassidim-Lloyd (HHL) algorithm offers an exponential speedup in system size for treating linear equations of the form A x ⃗ = b ⃗ on quantum computers when compared to their traditional counterparts, it faces a challenge related to the condition number ( κ ) scaling of the A matrix. In this work, we address the issue by introducing the postselection-improved HHL (Psi-HHL) framework that operates on a simple yet effective premise: subtracting mixed and wrong signals to extract correct signals while providing the benefit of optimal scaling in the condition number of A (denoted as κ ) for large κ scenarios. This approach, which leads to minimal increase in circuit depth, has the important practical implication of having to use substantially fewer shots relative to the traditional HHL algorithm. The term “signal” refers to a feature of | x 〉 . We design circuits for overlap and expectation value estimation in the Psi-HHL framework. We demonstrate performance of Psi-HHL via numerical simulations. We carry out two sets of computations, where we go up to 26-qubit calculations, to demonstrate the ability of Psi-HHL to handle situations involving large- κ matrices via (a) a set of toy matrices, for which we go up to size 64 × 64 and κ values of up to ≈ 1 × 10 6 , and (b) application to quantum chemistry, where we consider matrices up to size 256 × 256 that reach κ of about 393. The molecular systems that we consider are Li 2 , KH, RbH, and CsH.
The Harrow-Hassidim-Lloyd quantum algorithm was proposed to solve linear systems of equations $A\stackrel{\ensuremath{\rightarrow}}{x}=\stackrel{\ensuremath{\rightarrow}}{b}$ and it is the core of various applications. However, there is no explicit quantum circuit for the subroutine that maps the inverse of the problem matrix
A into an ancillary qubit. This makes implementation in current quantum devices challenging, forcing us to use hybrid approaches. Here, we propose a systematic method to implement this subroutine, which can be adapted to other functions
f(A) of matrix
A, we present a co-designed quantum processor that reduces the depth of the algorithm, and we introduce its digital-analog implementation. The depth of our proposal scales with precision
ϵ as $\mathcal{O}({ϵ}^{\ensuremath{-}1})$, which is bounded by the number of samples allowed for a certain experiment. The co-design of the Harrow-Hassidim-Lloyd algorithm leads to a ``kitelike'' architecture, which allows us to reduce the number of required swap gates. Finally, merging a co-design quantum processor architecture with a digital-analog implementation contributes to the reduction of noise sources during the experimental realization of the algorithm.