论文标题
线性回归的改进的量子启发算法
An improved quantum-inspired algorithm for linear regression
论文作者
论文摘要
We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09, arXiv:0811.3171] for low-rank matrices [Wossnig, Zhao, and Prakash, Physical Review Letters'18, arXiv:1704.06174], when the input矩阵$ a $存储在适用于基于QRAM的状态准备的数据结构中。 也就是说，假设我们得到了\ Mathbb {C}^{M \ times n} $的$ a \，最小非零值$σ$，它支持某些有效的$ \ ell_2 $ \ ell_2 $ - norm重要性采样查询，以及a $ b \ in \ In \ in \ n \ in \ mathbb {c} c}^m $。然后，对于某些$ x \ in \ mathbb {c}^n $满足$ \ | x -a^+b \ | \ leq \ varepsilon \ | a^+b \ | $，我们可以在计算基础上输出$ | x \ rangle $的测量，并输出带有经典算法的$ x $的条目$ \ tilde {\ Mathcal {o}} \ big（\ frac {\ | a \ | _ {\ Mathrm {f}}^6 \ | a \ | a \ |^6} {σ^{σ^{12} {12} {12} \ varepsilon^4} \ big）$和和$ \ tilde {\ Mathcal {o}} \ big（\ frac {\ | a \ | _ {\ Mathrm {f}}^6 \ | a \ | a \ |^2} {σ^8 \ varepsilon^4} 4} \ big）分别分别。这在这一研究中的先前“量子启发”算法上的改进至少为$ \ frac {\ | a \ |^{16}}} {σ^{16} \ varepsilon^2} $ [结果，我们表明，在此QRAM数据结构设置和相关设置中，量子计算机最多可以实现12个线性回归的速度。我们的工作应用了从素描算法和优化到量子启发的文献的技术。与较早的作品不同，这是一个有希望的途径，可以导致在量子启发的设置中可行的经典回归实现，以与未来的量子计算机进行比较。
We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09, arXiv:0811.3171] for low-rank matrices [Wossnig, Zhao, and Prakash, Physical Review Letters'18, arXiv:1704.06174], when the input matrix $A$ is stored in a data structure applicable for QRAM-based state preparation. Namely, suppose we are given an $A \in \mathbb{C}^{m\times n}$ with minimum non-zero singular value $σ$ which supports certain efficient $\ell_2$-norm importance sampling queries, along with a $b \in \mathbb{C}^m$. Then, for some $x \in \mathbb{C}^n$ satisfying $\|x - A^+b\| \leq \varepsilon\|A^+b\|$, we can output a measurement of $|x\rangle$ in the computational basis and output an entry of $x$ with classical algorithms that run in $\tilde{\mathcal{O}}\big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^6}{σ^{12}\varepsilon^4}\big)$ and $\tilde{\mathcal{O}}\big(\frac{\|A\|_{\mathrm{F}}^6\|A\|^2}{σ^8\varepsilon^4}\big)$ time, respectively. This improves on previous "quantum-inspired" algorithms in this line of research by at least a factor of $\frac{\|A\|^{16}}{σ^{16}\varepsilon^2}$ [Chia, Gilyén, Li, Lin, Tang, and Wang, STOC'20, arXiv:1910.06151]. As a consequence, we show that quantum computers can achieve at most a factor-of-12 speedup for linear regression in this QRAM data structure setting and related settings. Our work applies techniques from sketching algorithms and optimization to the quantum-inspired literature. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired settings, for comparison against future quantum computers.
