Quantum Logspace Algorithm for Powering Matrices with Bounded Norm
Abstract
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 deferredmeasurement 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 nontrivial classical simulations of quantum logspace learning algorithms.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.04880
 Bibcode:
 2020arXiv200604880G
 Keywords:

 Computer Science  Computational Complexity;
 Quantum Physics