From Quantum Query Complexity to State Complexity

State complexity of quantum finite automata is one of the interesting topics in studying the power of quantum finite automata. It is therefore of importance to develop general methods how to show state succinctness results for quantum finite automata. One such method is presented and demonstrated in this paper. In particular, we show that state succinctness results can be derived out of query complexity results.Comment: Some typos in references were fixed. To appear in Gruska Festschrift (2014). Comments are welcome. arXiv admin note: substantial text overlap with arXiv:1402.7254, arXiv:1309.773

Optimal lower bounds for quantum automata and random access codes

Consider the finite regular language L_n = {w0 : w \in {0,1}^*, |w| \le n}. It was shown by Ambainis, Nayak, Ta-Shma and Vazirani that while this language is accepted by a deterministic finite automaton of size O(n), any one-way quantum finite automaton (QFA) for it has size 2^{Omega(n/log n)}. This was based on the fact that the evolution of a QFA is required to be reversible. When arbitrary intermediate measurements are allowed, this intuition breaks down. Nonetheless, we show a 2^{Omega(n)} lower bound for such QFA for L_n, thus also improving the previous bound. The improved bound is obtained by simple entropy arguments based on Holevo's theorem. This method also allows us to obtain an asymptotically optimal (1-H(p))n bound for the dense quantum codes (random access codes) introduced by Ambainis et al. We then turn to Holevo's theorem, and show that in typical situations, it may be replaced by a tighter and more transparent in-probability bound.Comment: 8 pages, 1 figure, Latex2e. Extensive modifications have been made to increase clarity. To appear in FOCS'9

Non-classical computing: feasible versus infeasible

Physics sets certain limits on what is and is not computable. These limits are very far from having been reached by current technologies. Whilst proposals for hypercomputation are almost certainly infeasible, there are a number of non classical approaches that do hold considerable promise. There are a range of possible architectures that could be implemented on silicon that are distinctly different from the von Neumann model. Beyond this, quantum simulators, which are the quantum equivalent of analogue computers, may be constructable in the near future

Lower Bounds on Quantum Query Complexity

Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.Comment: survey, 23 page