We describe sets of mutually unbiased bases (MUBs) for quantum states defined over the p-adic numbers Q_p, i.e. the states that can be described as elements of the (rigged) Hilbert space L2(Q_p). We find that for every prime p>2 there are at least p+
1 MUBs, which is in contrast with the situation for quantum states defined over the real line R for which only 3 MUBs are known. We comment on the possible reason for the difference regarding MUBs between these two infinite dimensional Hilbert spaces.
The problem of memory checking considers storing files on an unreliable public server whose memory can be modified by a malicious party. The main task is to design an online memory checker with the capability to verify that the information on the ser
ver has not been corrupted. To store n bits of public information, the memory checker has s private reliable bits for verification purpose; while to retrieve each bit of public information the checker communicates t bits with the public memory. Earlier work showed that, for classical memory checkers, the lower bound s*t in Omega(n) holds. In this article we study quantum memory checkers that have s private qubits and that are allowed to quantum query the public memory using t qubits. We prove an exponential improvement over the classical setting by showing the existence of a quantum checker that, using quantum fingerprints, requires only s in O(log n) qubits of local memory and t in O(polylog n) qubits of communication with the public memory.
Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical com
puters. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor.
