计算矩阵行列式与逆矩阵及求解线性代数方程组的量子算法
Quantum algorithms for calculating determinant and inverse of matrix and solving linear algebraic systems
AI总结：
本文提出纯量子算法计算矩阵行列式与逆矩阵并求解线性代数方程组，通过将矩阵行编码为量子纯态并利用代数余子式表示逆矩阵元素，给出了电路设计及复杂度估计。
摘要：
我们提出了纯粹量子性质的量子算法，用于计算 $(N-1)\times (N-1)$ 矩阵的行列式和逆矩阵（深度为 $O(N^2\log N)$），该算法是计算 $N\times N$ 矩阵行列式算法（深度为 $O(N\log^2 N)$）的简单修改。基本思想是将矩阵的每一行编码为某个量子系统的纯态。此外，我们利用代数余子式来表示逆矩阵的元素。该算法与早先提出的矩阵乘法算法结合，产生了求解线性代数方程组的算法（深度为 $O(N\log^2 N)$）。对输出为 1 的辅助态进行测量（概率为 $\sim 2^{-O(N\log N)}$）可移除计算过程中产生的垃圾信息。我们给出了所有三种算法的相应电路，且它们在空间（电路中的量子比特数）上具有相同的估计 $O(N\log N)$。
英文摘要：
We propose quantum algorithms, purely quantum in nature, for calculating the determinant and inverse of an $(N-1)\times (N-1)$ matrix (depth is $O(N^2\log N)$) which is a simple modification of the algorithm for calculating the determinant of an $N\times N$ matrix (depth is $O(N\log^2 N)$. The basic idea is to encode each row of the matrix into a pure state of some quantum system. In addition, we use the representation of the elements of the inverse matrix in terms of algebraic complements. This algorithm together with that for matrix multiplication { proposed earlier} yields the algorithm for solving systems of linear algebraic equations (depth is $O(N\log^2 N)$. Measurement of the ancilla state with output 1 (probability is $\sim 2^{-O(N\log N)}$) removes the garbage acquired during calculation. Appropriate circuits for all three algorithms are presented and have the same estimation $O(N\log N)$ for the space (number of qubits in the circuit).
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