We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most~1. The algorithm gets as an input an arbitrary $n\times n$ contraction matrix $A$, and a parameter $T \leq \mathrm{poly}(n)$ and outputs the entries of $A^T$, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements. We show various implications and applications of this result: First, we use this algorithm to show that the class of quantum logspace algorithms with only quantum memory and with intermediate measurements is equivalent to the class of quantum logspace algorithms with only quantum memory without intermediate measurements. This shows that the deferred-measurement principle, a fundamental principle of quantum computing, applies also for quantum logspace algorithms (without classical memory). More generally, we give a quantum algorithm with space $O(S + \log T)$ that takes as an input the description of a quantum algorithm with quantum space $S$ and time $T$, with intermediate measurements (without classical memory), and simulates it unitarily with polynomially small error, without intermediate measurements. Since unitary transformations are reversible (while measurements are irreversible) an interesting aspect of this result is that it shows that any quantum logspace algorithm (without classical memory) can be simulated by a reversible quantum logspace algorithm. This proves a quantum analogue of the result of Lange, McKenzie and Tapp that deterministic logspace is equal to reversible logspace [LMT00]. Finally, we use our results to show non-trivial classical simulations of quantum logspace learning algorithms.
翻译:我们给出用于增强缩缩矩阵的量子日志算法, 也就是说, 光谱标准矩阵最多为~ 1 。 算法作为一种输入, 是一个任意的 $n\timen n$ n$美元 缩缩缩矩阵 $A$, 和一个参数 $T\leq\ mathrm{poly}(n) 和输出输入 $A$T$, 直至( 任意的) 多元的微小添加错误。 算法只应用单一操作器, 而没有中间测量, 我们用光谱标准矩阵矩阵算法显示, 量子日志算算算算法只有量子内存和中间测量, 与量子日志算算算法等值等同, 没有中间存储存储存储存储存储, 量子计算法的中间级缩略图显示, 直流缩缩缩略图的缩略图显示, 最终的缩略图的缩略图是正缩略图。