We propose a new method for designing quantum search algorithms for finding a element in the state space of a classical Markov chain. The algorithm is based on a quantum walk \'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.
We propose a new method for designing quantum search algorithms for finding a element in the state space of a classical Markov chain. The algorithm is based on a quantum walk 'a la Szegedy (2004) that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis (2004) and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.
We propose a new method for designing quantum search algorithms for finding a “marked” element in the state space of a classical Markov chain. The algorithm is based on a quantum walk à la Szegedy [Quantum speed-up of Markov chain based algorithms, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 2004, pp. 32–41] that is defined in terms of the Markov chain. The main new idea is to apply quantum phase estimation to the quantum walk in order to implement an approximate reflection operator. This operator is then used in an amplitude amplification scheme. As a result we considerably expand the scope of the previous approaches of Ambainis [Quantum walk algorithm for Element Distinctness, in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 2004, pp. 22–31] and Szegedy (2004). Our algorithm combines the benefits of these approaches in terms of being able to find marked elements, incurring the smaller cost of the two, and being applicable to a larger class of Markov chains. In addition, it is conceptually simple and avoids some technical difficulties in the previous analyses of several algorithms based on quantum walk.
This thesis provides an introduction to the quantum walk based search method which is the quantum analogue of the classical Markov chain based search. We develop the basic theory of the Markov chain based search together with its quantisation due to Mario Szegedy[Sz2]. We deeply analyse their relationship, and sketch some of the possible algorithmic applications. The thesis also addresses the question of the implementation, which is largely hidden by the high level description. We show how to implement the quantum walk based search algorithm in the case of the element distinctness problem. We implement the necessary operations using only very basic data structures, mainly arrays. Also while we use a weak and basic computational model we could still significantly reduce the exponent of the polylog terms arising from counting non oracle calls. Another interesting achievement is that we could parallelise the Setup operation reducing its circuit depth form O(N ) to O(polylog(N)). Probably the most important tool we introduced for the implementation is the use of reversible sorting networks, but we introduce some other tricks as well. The optimal quantum query algorithm for the element distinctness problem was first described by Andris Ambainis [Amb3] and provided the basis of the generalised method. In this seminal paper Ambainis also addressed the question of implementation. However that implementation method does not fully cast into the general framework which was introduced later. The operations described by Ambainis use some involved data structures, such as skip lists and hash table. Also it uses a strong computational model for example, a qRAM query has cost only 1. In contrast with Ambainis’s single node implementation method ours uses bipartite style approach fitting the general framework of Szegedy. At first sight it may seem unnecessary since it results in a duplication of a large amount of data stored during the walk steps, but in fact it simplifies the implementation process. On the structure of the thesis: Section 1 covers some background on quantum walks and basic techniques related to the quantum walk based search method. Section 2-5 gives a comprehensive description of the general scheme introduced by Szegedy and derives the main theorems showing the quadratic speed-up compared to the classical case. These Sections roughly follow the structure of Szegedy’s original paper [Sz2], but the proofs are restructured, so that I could state the main theorem in a slightly stronger form with a much improved constant. Section 6 addresses the implementation issues already mentioned. Finally, Section 7 gives a brief overview of the further generalised scheme introduced by Frederic Magniez, Ashwin Nayak, Jeremie Roland and Miklos Santha. Quantum walk based search methods and algorithmic applications 2
We will give an introduction into the quantum search algorithms on the Markov chains introduced by Szegedy and recent modifications based on partially absorbing Markov chains due to Krovi et al.Algorithmica 74, 851 (2016) It has been shown that a quantum search can find a set of marked vertices quadratically faster (in units of the hitting time) for any reversible Markov chain.The proofs are based on certain properties of the stationary state of the partially absorbing walk and rely on quantum phase estimation techniques.We will offer an alternative view of the underlying mechanism of the quantum search based on spectral properties of the quantum walk operator.By considering the complete graph as an example, we identify the relevant two-level quantum subspace leading to a Grover-like rotation in the underlying vector space.
The hitting time of a classical random walk (Markov chain) is the time required to detect the presence of—or equivalently, to find—a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the detection and finding problems have the same time complexity. In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Then, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem (Tulsi A.: Phys. Rev. A 78:012310 2008) for the 2D grid. Extending his result, we show that we can find a unique marked element with constant probability and with the same complexity as detection for a large class of quantum walks—the quantum analogue of state-transitive reversible ergodic Markov chains. Further, we prove that for any reversible ergodic Markov chain P, the quantum hitting time of the quantum analogue of P has the same order as the square root of the classical hitting time of P. We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. In doing so, we define a notion of reversibility for a broad class of quantum walks and show how to derive from any such quantum walk a classical analogue. For the special case of quantum walks built on reflections, we show that the hitting time of the classical analogue is exactly the square of the quantum walk.
The hitting time of a classical random walk (Markov chain) is the time required to detect the presence of—or equivalently, to find —a marked state. The hitting time of a quantum walk is subtler to define; in particular, it is unknown whether the detection and finding problems have the same time complexity. In this paper we define new Monte Carlo type classical and quantum hitting times, and we prove several relationships among these and the already existing Las Vegas type definitions. In particular, we show that for some marked state the two types of hitting time are of the same order in both the classical and the quantum case. Then, we present new quantum algorithms for the detection and finding problems. The complexities of both algorithms are related to the new, potentially smaller, quantum hitting times. The detection algorithm is based on phase estimation and is particularly simple. The finding algorithm combines a similar phase estimation based procedure with ideas of Tulsi from his recent theorem (Tulsi A.: Phys. Rev. A 78 :012310 2008 ) for the 2D grid. Extending his result, we show that we can find a unique marked element with constant probability and with the same complexity as detection for a large class of quantum walks—the quantum analogue of state-transitive reversible ergodic Markov chains. Further, we prove that for any reversible ergodic Markov chain P , the quantum hitting time of the quantum analogue of P has the same order as the square root of the classical hitting time of P . We also investigate the (im)possibility of achieving a gap greater than quadratic using an alternative quantum walk. In doing so, we define a notion of reversibility for a broad class of quantum walks and show how to derive from any such quantum walk a classical analogue. For the special case of quantum walks built on reflections, we show that the hitting time of the classical analogue is exactly the square of the quantum walk.
Quantum walks provide a versatile framework for quantum algorithms across a wide range of applications. We develop efficient classical simulation methods for Szegedy quantum walks that avoid explicit construction of the full unitary evolution operator. Unlike previous approaches restricted to a particular walk formulation, our framework is built from fundamental update and reflection operators, enabling the simulation of a broader class of Szegedy walk formulations. We further extend these methods to phase-estimation-based algorithms coupled to the walk, including implementations suitable for large sparse graphs. The resulting methods achieve optimal O(N2) complexity for dense graphs with N nodes. For sparse graphs, the computational cost scales linearly with the number of edges, which is O(N) in many cases. We implement the framework in the Python package SQWLib and illustrate its capabilities through simulations of representative algorithms, including quantum simulated annealing and quantum search on graphs. These results provide a practical tool for studying Szegedy-walk-based algorithms numerically beyond purely analytical treatments.