Linear systems of equations can be found in various mathematical domains, as
well as in the field of machine learning. By employing noisy intermediate-scale
quantum devices, variational solvers promise to accelerate finding solutions
for large systems. Although there is a wealth of theoretical research on these
algorithms, only fragmentary implementations exist. To fill this gap, we have
developed the variational-lse-solver framework, which realizes existing
approaches in literature, and introduces several enhancements. The
user-friendly interface is designed for researchers that work at the
abstraction level of identifying and developing end-to-end applications.

通过使用噪声中尺度量子设备，变分求解器可以加速大规模线性方程组的求解，本研究提出了可实现现有方法的变分 - 线性系统方程求解器框架，并引入了几个增强措施，其用户友好的界面适用于研究人员在识别和开发端到端应用程序的抽象层面工作。

全面的变分 LSE 求解器库

Comprehensive Library of Variational LSE Solvers

We study the practical performance of quantum-inspired algorithms for
recommendation systems and linear systems of equations. These algorithms were
shown to have an exponential asymptotic speedup compared to previously known
classical methods for problems involving low-rank matrices, but with complexity
bounds that exhibit a hefty polynomial overhead compared to quantum algorithms.
This raised the question of whether these methods were actually useful in
practice. We conduct a theoretical analysis aimed at identifying their
computational bottlenecks, then implement and benchmark the algorithms on a
variety of problems, including applications to portfolio optimization and movie
recommendations. On the one hand, our analysis reveals that the performance of
these algorithms is better than the theoretical complexity bounds would
suggest. On the other hand, their performance as seen in our implementation
degrades noticeably as the rank and condition number of the input matrix are
increased. Overall, our results indicate that quantum-inspired algorithms can
perform well in practice provided that stringent conditions are met: low rank,
low condition number, and very large dimension of the input matrix. By
contrast, practical datasets are often sparse and high-rank, precisely the type
that can be handled by quantum algorithms.

研究量子启发式算法在推荐系统和线性方程组方面的实际性能优化，证明这些算法可以应用于低秩矩阵问题的实际情境下表现良好，但当输入矩阵的秩或条件数增加时，他们的性能会明显降低。

量子启发式算法实践

Quantum-inspired algorithms in practice

Solving linear systems of equations is a frequently encountered problem in
machine learning and optimisation. Given a matrix $A$ and a vector $\mathbf b$
the task is to find the vector $\mathbf x$ such that $A \mathbf x = \mathbf b$.
We describe a quantum algorithm that achieves a sparsity-independent runtime
scaling of $\mathcal{O}(\kappa^2 \|A\|_F \text{polylog}(n)/\epsilon)$, where
$n\times n$ is the dimensionality of $A$ with Frobenius norm $\|A\|_F$,
$\kappa$ denotes the condition number of $A$, and $\epsilon$ is the desired
precision parameter. When applied to a dense matrix with spectral norm bounded
by a constant, the runtime of the proposed algorithm is bounded by
$\mathcal{O}(\kappa^2\sqrt{n} \text{polylog}(n)/\epsilon)$, which is a
quadratic improvement over known quantum linear system algorithms. Our
algorithm is built upon a singular value estimation subroutine, which makes use
of a memory architecture that allows for efficient preparation of quantum
states that correspond to the rows and row Frobenius norms of $A$.

介绍了一个基于奇异值估算子程序的量子算法，可解决线性方程组问题，其运行时间与矩阵 A 的条件数、Frobenius 范数和精度参数有关。当应用于具有范数受到限制的密集矩阵时，所提出的算法的运行时间受到限制，其运行时间比已知的量子线性系统算法提高了平方级别。