Enhancing the Harrow-Hassidim-Lloyd (HHL) algorithm in systems with large condition numbers

TL;DR

This paper introduces Psi-HHL, a quantum algorithm enhancement that improves the HHL algorithm's performance for large condition number matrices by reducing shot count and circuit depth, demonstrated through simulations on large matrices and quantum chemistry applications.

Contribution

The paper proposes the Psi-HHL framework, which effectively mitigates condition number scaling issues in the HHL algorithm using post-selection, with minimal circuit depth increase and fewer measurements.

Findings

Psi-HHL handles matrices with condition numbers up to 1 million.

Demonstrated effectiveness on matrices up to 256x256 in quantum chemistry.

Requires fewer shots compared to traditional HHL for large condition numbers.

Abstract

Although the Harrow-Hassidim-Lloyd (HHL) algorithm offers an exponential speedup in system size for treating linear equations of the form $A\vec{x}=\vec{b}$ on quantum computers when compared to their traditional counterparts, it faces a challenge related to the condition number ($\mathcal{\kappa}$) scaling of the $A$ matrix. In this work, we address the issue by introducing the post-selection-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 $\mathcal{\kappa}$) for large $\mathcal{\kappa}$ 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 $|x\rangle$. 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 $\mathcal{\kappa}$ matrices via: (a) a set of toy matrices, for which we go up to size $64 \times 64$ and $\mathcal{\kappa}$ values of up to $\approx$ 1 million, and (b) application to quantum chemistry, where we consider matrices up to size $256 \times 256$ that reach $\mathcal{\kappa}$ of about 393. The molecular systems that we consider are Li$_{\mathrm{2}}$, KH, RbH, and CsH.