In recent years, quantum walks have been widely researched and have shown exciting properties. One such is a quadratic speed-up in hitting time compared to its classical counterpart. In this paper, we design a quantum circuit for the MNRS algorithm, which finds a marked node in a graph with a quantum walk, and use it to find a hitting time for the marked nodes in the walk. We do this by implementing the circuit on IBM quantum simulators and show that the execution on a noise-free simulator results in hitting times that agree with the theoretical expectations. We also run the algorithm on a mock backend that simulates the noise of the IBM Melbourne computer. As expected, the noise has an extensive impact on the output, resulting in outcomes far from the noise-free simulation.
In recent years, quantum walks have been widely researched and have shown exciting properties. One such is a quadratic speed-up in hitting time compared to its classical counterpart. In this paper, we design a quantum circuit for the MNRS algorithm, which finds a marked node in a graph with a quantum walk, and use it to find a hitting time for the marked nodes in the walk. We do this by implementing the circuit on IBM quantum simulators and show that the execution on a noise-free simulator results in hitting times that agree with the theoretical expectations. We also run the algorithm on a mock backend that simulates the noise of the IBM Melbourne computer. As expected, the noise has an extensive impact on the output, resulting in outcomes far from the noise-free simulation.
Quantum walks play an important role in the area of quantum algorithms. Many interesting problems can be reduced to searching marked states in a quantum Markov chain. In this context, the notion of quantum hitting time is very important, because it quantifies the running time of the algorithms. Markov chain-based algorithms are probabilistic, therefore the calculation of the success probability is also required in the analysis of the computational complexity. Using Szegedy's definition of quantum hitting time, which is a natural extension of the definition of the classical hitting time, we present analytical expressions for the hitting time and success probability of the quantum walk on the complete graph.
We demonstrate an implementation of the hitting time of a discrete time quantum random walk on cubelike graphs using IBM's Qiskit platform. Our implementation is based on efficient circuits for the Grover and Shift operators. We verify the results about the one-shot hitting time of quantum walks on a hypercube as proved in [https://link.springer.com/article/10.1007/s00440-004-0423-2]. We extend the study to another family of cubelike graphs called the augmented cubes [https://onlinelibrary.wiley.com/doi/abs/10.1002/net.10033]. Based on our numerical study, we conjecture that for all families of cubelike graphs there is a linear relationship between the degree of a cubelike graph and its hitting time which holds asymptotically. That is, for any cubelike graph of degree Δ, the probability of finding the quantum random walk at the target node at time 2πΔ approaches 1 as the degree Δ of the cubelike graph approaches infinity.
Quantum walks on graphs play an important role in the field of quantum algorithms. Fast hitting is one of the properties that quantum walk algorithms can utilize to outperform classical random walk algorithms. Fast hitting refers to a particle starting from the entrance node on a graph and trying to hit the exit node quickly. Especially, continuous-time quantum walks on random glued binary trees have been investigated in theories extensively for their exponentially faster hitting speed over classical random walks. Here, using heralded single photons to represent quantum walkers and laser-written waveguide arrays to simulate the theoretical graph, we are able to demonstrate the hitting efficiency of quantum walks with tree depth as high as 16 layers for the first time. Furthermore, we expand the graph’s branching rate from 2 to 5, revealing that quantum walks can exhibit more superiority over classical random walks as the branching rate increases. Our results may shed light on the physical implementation of quantum walk algorithms as well as quantum computation and quantum simulation.
Many quantum algorithms critically rely on quantum walk search, or the use of quantum walks to speed up search problems on graphs. However, the main results on quantum walk search are scattered over different, incomparable frameworks, such as the hitting time framework, the MNRS framework, and the electric network framework. As a consequence, a number of pieces are currently missing. For example, recent work by Ambainis et al. (STOC'20) shows how quantum walks starting from the stationary distribution can always find elements quadratically faster. In contrast, the electric network framework allows quantum walks to start from an arbitrary initial state, but it only detects marked elements. We present a new quantum walk search framework that unifies and strengthens these frameworks, leading to a number of new results. For example, the new framework effectively finds marked elements in the electric network setting. The new framework also allows to interpolate between the hitting time framework, minimizing the number of walk steps, and the MNRS framework, minimizing the number of times elements are checked for being marked. This allows for a more natural tradeoff between resources. In addition to quantum walks and phase estimation, our new algorithm makes use of quantum fast-forwarding, similar to the recent results by Ambainis et al. This perspective also enables us to derive more general complexity bounds on the quantum walk algorithms, e.g., based on Monte Carlo type bounds of the corresponding classical walk. As a final result, we show how in certain cases we can avoid the use of phase estimation and quantum fast-forwarding, answering an open question of Ambainis et al.
Let χ be a function that maps the vertices of a graph to zero or one. We want to find a vertex m, called a marked vertex, such that χ(m) = 1. Random walk is an algorithm for finding such a vertex. The measure of complexity for random walk is hitting time. There is a quantum algorithm that finds a marked vertex, and runs in the square root of the extended hitting time. For some configurations of marked vertices on certain graphs, the extended hitting time is asymptotically larger than the hitting time. This might cause the quantum algorithm for finding to be less efficient than a random walk. Two-dimensional torus and grid are canonical examples of such graphs. This thesis is about efficient quantum algorithms for finding a marked vertex on a two-dimensional torus. We prove the convexity of the extended hitting time, and provide a new upper bound on it. We show that, with a probability that is more than a constant, random walk on a torus is localized, and it is restricted to a sub-graph of the torus. From these observations, we introduce two algorithms that find a marked vertex on a torus. The complexity of these algorithms is in the order of the square root of the hitting time, up to logarithmic factors. They both start by finding an estimate to the hitting time. Using that, they estimate the size of a sub-graph that contains a marked vertex with more than a constant probability, and sample such a sub-graph. Our first algorithm finds a marked vertex on this sub-graph by using a quantum algorithm that runs in extended hitting time, and finds a marked vertex with probability at least a constant. Our second algorithm recursively runs the detection algorithm on cuts of the sub-graph, until reaching a constant sized sub-graph that contains a marked vertex. These algorithms are the first quantum algorithms that achieve close-to-quadratic speed-up in quantum walk on a two-dimensional torus, for all configurations of marked vertices.
Abstract A random walk is one of the widely adopted random processes for simulation and approximation in multiple areas of science and engineering. Quantum random walk is an analog version of the classical random walk. It was first introduced in 1993 by Y. Aharonov et al. They presented that, with the nature of quantum characteristic, the average length of the walking path on a line is possible to larger than a classical random walk can produce. Therefore, the quantum random walk can be used as a tool to construct many other quantum algorithms. Moreover, it can solve graph problems that many real-world problems can be formulated. However, quantum algorithms are only useful in practice if we can implement them efficiently on a quantum computer. In this study, we are then interested in designing and implementing a quantum circuit that can be run on a real quantum computing device. We firstly focus on the most straightforward, which is a one-dimension, quantum random walk algorithm. Then, the quantum circuits are developed and implemented on a real quantum computer and a quantum computing simulator using software development kits provided by IBM. Lastly, the performance and results of the quantum circuits tested on both computing platforms are presented.