Solving the encoding bottleneck: of the HHL algorithm, by the HHL algorithm

TL;DR

This paper tackles the encoding bottleneck of the HHL algorithm, where efficiently preparing the initial state $|b\rangle$ is crucial for preserving the exponential speedup. It reframes state preparation as a QSPP using a block-embedding matrix $B$ and a vector $h$ with $b=Bh$, then leverages a modified HHL routine to approximate $|b\rangle$ in $O(\mathrm{poly}(\log N))$ time. The authors present a concrete state-preparation circuit that introduces two modifications to HHL: initializing the $n_b$ target qubits in $|h\rangle$ and a redesigned rotation step to keep $|C\tilde{\lambda}_j|\le 1$, achieving polylogarithmic resource costs. Under a well-conditioned $B$ (i.e., $\kappa(B)$ not growing with $N$), the overall runtime remains $O(\log N)$, thereby preserving the exponential speedup for solving $A x=b$ and offering a standalone fast state-preparation method for other quantum applications such as amplitude encoding.

Abstract

The Harrow-Hassidim-Lloyd (HHL) algorithm offers exponential speedup for solving the quantum linear-system problem. But some caveats for the speedup could be hard to met. One of the difficulties is the encoding bottleneck, i.e., the efficient preparation of the initial quantum state. To prepare an arbitrary $N$-dimensional state exactly, existing state-preparation approaches generally require a runtime of $O(N)$, which will ruin the speedup of the HHL algorithm. Here we show that the states can be prepared approximately with a runtime of $O(poly(\log N))$ by employing a slightly modified version of the HHL algorithm itself. Thus, applying this approach to prepare the initial state of the original HHL algorithm can preserve the exponential speedup advantage. It can also serve as a standalone solution for other applications demanding fast state preparation.

Figures (1)

• Figure 1: Diagram of the quantum circuit of our state-preparation algorithm, following Fig.1 of ml91. Our modifications are marked by the two red dashed boxes.