Nikolajs Nahimovs

Search by Quantum Walks on Two-Dimensional Grid without Amplitude Amplification

We study search by quantum walk on a finite two dimensional grid. The algorithm of Ambainis, Kempe, Rivosh [AKR05] uses \(O(\sqrt{N \log{N}})\) steps and finds a marked location with probability O(1 / logN) for grid of size \(\sqrt{N} \times \sqrt{N}\). This probability is small, thus [AKR05] needs amplitude amplification to get Θ(1) probability. The amplitude amplification adds an additional \(O(\sqrt{\log{N}})\) factor to the number of steps, making it \(O(\sqrt{N} \log{N})\).

Nonlocal quantum XOR games for large number of players

Nonlocal games are used to display differences between classical and quantum world In this paper, we study nonlocal games with a large number of players We give simple methods for calculating the classical and the quantum values for symmetric XOR games with one-bit input per player, a subclass of nonlocal games We illustrate those methods on the example of the N-player game (due to Ardehali [Ard92]) that provides the maximum quantum-over-classical advantage.

Improved Constructions of Quantum Automata

We present a simple construction of quantum automata which achieve an exponential advantage over classical finite automata. Our automata use

Adjacent Vertices Can Be Hard to Find by Quantum Walks

\frac{4}{\epsilon} \log 2p + O(1)

Grover’s Algorithm with Errors

states to recognize a language that requires p states classically. The construction is both substantially simpler and achieves a better constant in the front of logp than the previously known construction of [2]. Similarly to [2], our construction is by a probabilistic argument. We consider the possibility to derandomize it and present some preliminary results in this direction.

Grover's Search with Faults on Some Marked Elements

Quantum walks have been useful for designing quantum algorithms that outperform their classical versions for a variety of search problems. Most of the papers, however, consider a search space containing a single marked element only. We show that if the search space contains more than one marked element, their placement may drastically affect the performance of the search. More specifically, we study search by quantum walks on general graphs and show a wide class of configurations of marked vertices, for which search by quantum walk needs \(\varOmega (N)\) steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or more adjacent marked vertices. The analysis is done for the two-dimensional grid and hypercube, and then is generalized for any graph.

Improved constructions of quantum automata

Grover’s algorithm is a quantum search algorithm solving the unstructured search problem of size n in \(O(\sqrt{n})\) queries, while any classical algorithm needs O(n) queries [3].

Exceptional Configurations of Quantum Walks with Grover's Coin

Grovers algorithm is a quantum query algorithm solving the unstructured search problem of size N using

On the Probability of Finding Marked Connected Components Using Quantum Walks

O\sqrt{N}