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.

中文翻译：

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用有界范数驱动矩阵的量子对数空间算法

我们给出了一个量子对数空间算法来为收缩矩阵供电，即频谱范数最多为~1的矩阵。该算法将任意 $n\times n$ 收缩矩阵 $A$ 和参数 $T \leq \mathrm{poly}(n)$ 作为输入，并输出 $A^T$ 的条目，直到 (任意）多项式小附加误差。该算法仅应用幺正算子，没有中间测量。我们展示了这个结果的各种含义和应用：首先，我们使用这个算法来证明只有量子存储器和中间测量的量子对数空间算法类等效于只有量子存储器没有中间测量的量子对数空间算法类。这表明延迟测量原理，量子计算的基本原理，也适用于量子对数空间算法（无经典记忆）。更一般地说，我们给出了一个空间为 $O(S + \log T)$ 的量子算法，该算法将具有量子空间 $S$ 和时间 $T$ 的量子算法的描述作为输入，具有中间测量值（没有经典记忆） )，并以多项式小误差对其进行单一模拟，无需中间测量。由于酉变换是可逆的（而测量是不可逆的）这个结果的一个有趣的方面是它表明任何量子对数空间算法（没有经典记忆）都可以通过可逆量子对数空间算法来模拟。这证明了 Lange、McKenzie 和 Tapp 的结果的量子模拟，即确定性对数空间等于可逆对数空间 [LMT00]。最后，