AI 中文总结
研究针对矩阵链乘法问题，经典操作数随链长度和矩阵维度增长。提出“双塔矩阵乘法”量子子程序，能将K个矩阵乘积编码到量子态，电路深度与K无关，量子比特数有优势，证明其正确性并提供两种实现，适用于多种下游量子算法。
AI 中文摘要
矩阵链乘法——计算$\mathcal{W}=M^{(0)}\cdots M^{(K - 1)}$，其中$M^{(k)}\in\mathbb{R}^{P_k\times P_{k + 1}}$——出现在科学计算、机器学习和图分析中。对于不同矩阵链，经典操作数随链长度$K$线性增长且与矩阵维度成多项式关系。我们提出“双塔矩阵乘法”，这是一个量子子程序，能将$K$个矩阵的乘积$\mathcal{W}$编码到量子态中，电路深度为$\mathcal{O}(\max_{k}\mathrm{polylog}(P_kP_{k + 1}))$，在基于QRAM的态制备模型中与$K$无关，量子比特数为$\mathcal{O}(\sum_{k}\log P_k)$；总门数仍与$K$成线性关系，优势在于电路深度。其构造在两层间交错态制备算子，每层内算子作用于不相交寄存器并并行执行。该子程序可专门用于链向量情况。我们证明了子程序对所有$K$的正确性，并提供了使用Qiskit和QCLAB框架的两种实现。该子程序适用于任何对态矢中编码矩阵进行操作的下游量子算法，包括范数估计、图矩阵幂、线性系统求解和量子机器学习核。
英文摘要
Matrix chain multiplication -- computing $\mathcal{W} = M^{(0)}\cdots M^{(K-1)}$ where $M^{(k)} \in \mathbb{R}^{P_k \times P_{k+1}}$ -- arises in scientific computing, machine learning, and graph analysis. Despite the importance of this problem, for chains of distinct matrices, the classical number of operations grows linearly with the chain length $K$ and polynomially in the matrix dimensions. We present \emph{Two-Tower Matrix Multiplication}, a quantum subroutine that encodes the product $\mathcal{W}$ of the $K$ matrices into a quantum state in circuit depth $\mathcal{O}(\max_{k} \mathrm{polylog} (P_k P_{k+1}))$, which is independent of~$K$ within the QRAM-based state-preparation model, whereas the qubit count is $\mathcal{O}\bigl(\sum_{k} \log P_k \bigr)$; the total gate count remains linear in $K$, so the gain is in the circuit depth. The construction interleaves state-preparation operators across two layers; within each layer, all operators act on disjoint registers and execute in parallel. This subroutine can be specialized for the chain-vector case, which computes the product of $K-1$ matrices applied to a vector. We prove the correctness of the subroutine for all $K$ and provide two implementations using the Qiskit and QCLAB frameworks. The subroutine is applicable to any downstream quantum algorithm that operates on a matrix encoded in the statevector, including norm estimation, graph-matrix powers, linear system solving, and quantum machine learning kernels.
