0000000001204499
AUTHOR
Nikolajs Nahimovs
Exceptional Configurations of Quantum Walks with Grover’s Coin
We study search by quantum walk on a two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. We show what the most natural coin transformation -- Grover's diffusion transformation -- has a wide class of exceptional configurations of marked locations, for which the probability of finding any of the marked locations does not grow over time. This extends the class of known exceptional configurations; until now the only known such configuration was the "diagonal construction" by [AR08].
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.
Quantum walks on two-dimensional grids with multiple marked locations
The running time of a quantum walk search algorithm depends on both the structure of the search space (graph) and the configuration (the placement and the number) of marked locations. While the first dependence has been studied in a number of papers, the second dependence remains mostly unstudied.We study search by quantum walks on the two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [3]. The original paper analyses one and two marked locations only. We move beyond two marked locations and study the behaviour of the algorithm for several configurations of multiple marked locations.In this paper, we prove two results showing the importance of how the marked locations ar…
Grover’s Search with Faults on Some Marked Elements
Grover's algorithm is a quantum query algorithm solving the unstructured search problem of size N using $$O\sqrt{N}$$ queries. It provides a significant speed-up over any classical algorithm [2]. The running time of the algorithm, however, is very sensitive to errors in queries. Multiple authors have analysed the algorithm using different models of query errors and showed the loss of quantum speed-up [1, 4]. We study the behavior of Grover's algorithm in the model where the search space contains both faulty and non-faulty marked elements. We show that in this setting it is indeed possible to find one of marked elements in $$O\sqrt{N}$$ queries.
Lackadaisical Quantum Walks with Multiple Marked Vertices
The concept of lackadaisical quantum walk – quantum walk with self loops – was first introduced for discrete-time quantum walk on one-dimensional line [8]. Later it was successfully applied to improve the running time of the spacial search on two-dimensional grid [16].
Exceptional configurations of quantum walks with Grover's coin
We study search by quantum walk on a two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. We show what the most natural coin transformation - Grover's diffusion transformation - has a wide class of exceptional configurations of marked locations, for which the probability of finding any of the marked locations does not grow over time. This extends the class of known exceptional configurations; until now the only known such configuration was the "diagonal construction" by Ambainis and Rivosh [AR08]
Quantum Walks on Two-Dimensional Grids with Multiple Marked Locations
The running time of a quantum walk search algorithm depends on both the structure of the search space graph and the configuration of marked locations. While the first dependence has been studied in a number of papers, the second dependence remains mostly unstudied. We study search by quantum walks on the two-dimensional grid using the algorithm of Ambainis, Kempe and Rivosh [AKR05]. The original paper analyses one and two marked locations only. We move beyond two marked locations and study the behaviour of the algorithm for an arbitrary configuration of marked locations. In this paper, we prove two results showing the importance of how the marked locations are arranged. First, we present tw…
Adjacent Vertices Can Be Hard to Find by Quantum Walks
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 …
On the probability of finding marked connected components using quantum walks
Finding a marked vertex in a graph can be a complicated task when using quantum walks. Recent results show that for two or more adjacent marked vertices search by quantum walk with Grover's coin may have no speed-up over classical exhaustive search. In this paper, we analyze the probability of finding a marked vertex for a set of connected components of marked vertices. We prove two upper bounds on the probability of finding a marked vertex and sketch further research directions.
Adjacent vertices can be hard to find by quantum walks
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 $\Omega(N)$ steps, that is, it has no speed-up over the classical exhaustive search. The demonstrated configurations occur for certain placements of two or…
Grover’s Algorithm with Errors
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].
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 (quant-ph/0402107) takes O(\sqrt{N log N}) steps and finds a marked location with probability O(1/log N) for grid of size \sqrt{N} * \sqrt{N}. This probability is small, thus amplitude amplification is needed to achieve \Theta(1) success probability. The amplitude amplification adds an additional O(\sqrt{log N}) factor to the number of steps, making it O(\sqrt{N} log N). In this paper, we show that despite a small probability to find a marked location, the probability to be within an O(\sqrt{N}) neighbourhood (at an O(\sqrt[4]{N}) distance) of the marked location is \Theta(1). This all…
Grover’s Search with Faults on Some Marked Elements
Grover’s algorithm is a quantum query algorithm solving the unstructured search problem of size [Formula: see text] using [Formula: see text] queries. It provides a significant speed-up over any classical algorithm [3]. The running time of the algorithm, however, is very sensitive to errors in queries. Multiple authors have analysed the algorithm using different models of query errors and showed the loss of quantum speed-up [2, 6]. We study the behavior of Grover’s algorithm in the model where the search space contains both faulty and non-faulty marked elements. We show that in this setting it is indeed possible to find one of marked elements in [Formula: see text] queries. We also analyze…
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 $\frac{4}{\epsilon} \log 2p + O(1)$ 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.
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 \(O(\sqrt{N} \log{N})\).
Upperbounds on the probability of finding marked connected components using quantum walks
Quantum walk search may exhibit phenomena beyond the intuition from a conventional random walk theory. One of such examples is exceptional configuration phenomenon -- it appears that it may be much harder to find any of two or more marked vertices, that if only one of them is marked. In this paper, we analyze the probability of finding any of marked vertices in such scenarios and prove upper bounds for various sets of marked vertices. We apply the upper bounds to large collection of graphs and show that the quantum search may be slow even when taking real-world networks.
Kvantu automātu un meklēšanas algoritmu iespējas un ierobežojumi
ANOTĀCIJA Kvantu skaitļošana ir nozare, kas pēta uz kvantu mehānikas likumiem balstīto skaitļošanas modeļu īpašības. Disertācija ir veltīta kvantu skaitļošanas algoritmiskiem aspektiem. Piedāvāti rezultāti trijos virzienos: Kvantu galīgi automāti Analizēta stāvokļu efektivitāte kvantu vienvirziena galīgam automātam. Uzlabota labāka zināmā eksponenciālā atšķirība [AF98] starp kvantu un klasiskajiem galīgajiem automātiem. Grovera algoritma analīze Pētīta Grovera algoritma noturība pret kļūdām. Vispārināts [RS08] loģisko kļūdu modelis un piedāvāti vairāki jauni rezultāti. Kvantu klejošana Pētīta meklēšana 2D režģī izmantojot kvantu klejošanu. Paātrināts [AKR05] kvantu klejošanas meklēšanas alg…
The Power and the Limits of Quantum Automata and Search Algorithms
Kvantu skaitļošana ir nozare, kas pēta uz kvantu mehānikas likumiem balstīto skaitļošanas modeļu īpašības. Disertācija ir veltīta kvantu skaitļošanas algoritmiskiem aspektiem. Piedāvāti rezultāti trijos virzienos: Kvantu galīgi automāti Analizēta stāvokļu efektivitāte kvantu vienvirziena galīgam automātam. Uzlabota labāka zināmā eksponenciālā atšķirība [AF98] starp kvantu un klasiskajiem galīgajiem automātiem. Grovera algoritma analīze Pētīta Grovera algoritma noturība pret kļūdām. Vispārināts [RS08] loģisko kļūdu modelis un piedāvāti vairāki jauni rezultāti. Kvantu klejošana Pētīta meklēšana 2D režģī izmantojot kvantu klejošanu. Paātrināts [AKR05] kvantu klejošanas meklēšanas algoritms. At…
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 \frac{4}{\epsilon} \log 2p + O(1) 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 \log p than the previously known construction of Ambainis and Freivalds (quant-ph/9802062). Similarly to Ambainis and Freivalds, our construction is by a probabilistic argument. We consider the possibility to derandomize it and present some results in this direction.
Adjacent vertices can be hard to find by quantum walks
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. 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 Ω(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 adjace…