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Chinese Physics B

Spatial search by pseudo-Hermitian continuous-time quantum walks

Miaomiao Yu, Yan Wang, Pingyu Zhu, Kun Wang, Ping Xu

Published Oct 9, 2026• 0 citations
Publisher

Abstract

Abstract We apply pseudo-Hermitian continuous-time quantum walks to the spatial search problem, derive exact analytical solutions for the resulting dynamics, and determine the first hitting time at which the target probability is maximal. In an idealized setting with a fixed coupling strength g, we show that the formal first-maximum time t = π / g is independent of the database size N . However, we emphasize that this does not imply a general complexity improvement: making t = π / g arbitrarily small requires a corresponding increase in the Hamiltonian matrix elements (energy scale), a fundamental resource trade-off. A fair assessment must also account for the measurement and postselection overhead required to extract the target probability and for the cost of implementing the evolution. Our results provide an analytically tractable non-Hermitian model that supports state preparation and parameter-controlled quantum dynamics.

DOI
10.1088/1674-1056/aeb252

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Abstract We apply pseudo-Hermitian continuous-time quantum walks to the spatial search problem, derive exact analytical solutions for the resulting dynamics, and determine the first hitting time at which the target probability is maximal. In an idealized setting with a fixed coupling strength g, we show that the formal first-maximum time t = π / g is independent of the database size N . However, we emphasize that this does not imply a general complexity improvement: making t = π / g arbitrarily small requires a corresponding increase in the Hamiltonian matrix elements (energy scale), a fundamental resource trade-off. A fair assessment must also account for the measurement and postselection overhead required to extract the target probability and for the cost of implementing the evolution. Our results provide an analytically tractable non-Hermitian model that supports state preparation and parameter-controlled quantum dynamics.
Abstract The spatial search problem aims to find a marked vertex of a finite graph using a dynamic with two constraints: (1) the walker has no compass and (2) the walker can check whether a vertex is marked only after reaching it. This problem is a generalization of unsorted database search and has many applications to algorithms. Classical algorithms that solve the spatial search problem are based on random walks and the computational complexity is determined by the hitting time. On the other hand, quantum algorithms are based on quantum walks and the computational complexity is determined not only by the number of steps to reach a marked vertex, but also by the success probability, since we need to perform a measurement at the end of the algorithm to determine the walker’s position. In this work, we address the spatial search problem on Johnson graphs using the coined quantum walk model. Since Johnson graphs are vertex- and distance-transitive, we have found an invariant subspace of the Hilbert space, which aids in the calculation of the computational complexity. We have shown that, for every fixed diameter, the asymptotic success probability is 1/2 after taking π N / ( 2 2 ) steps, where N is the number of vertices of the Johnson graph.
Spatial search by continuous-time quantum walks on complex networks is focused on using a quantum walk in continuous time in order to find a single or multiple marked vertices within the complex network. The specific formalism used here is to consider a coupling constant that shifts the state of the quantum walker from the initial state to the target state, which is the marked vertex. The thesis begins with establishing the mathematical framework of network theory, quantum walks and numerical methods that will be used in the remainder of the thesis. Then spatial search by continuous-time quantum walk is studied on regular and semi-regular graphs, where most analytical results can be found. This will get us acquainted with spatial search by quantum walk. The complex networks studied are Barabasi-Albert graphs and the Internet network on the level of autonomous systems. Different renormalized and pruned versions of the Internet network are studied. The parameters of the quantum walk that are focused on are the optimal values for the coupling constant, success probability, time and search time.
In this paper we present a model exhibiting a new type of continuous-time quantum walk (as a quantum-mechanical transport process) on networks, which is described by a non-Hermitian Hamiltonian possessing a real spectrum. We call it pseudo-Hermitian continuous-time quantum walk. We introduce a method to obtain the probability distribution of walk on any vertex and then study a specific system. We observe that the probability distribution on certain vertices increases compared to that of the Hermitian case. This formalism makes the transport process faster and can be useful for search algorithms.
Continuous-time quantum walks provide a natural framework to tackle the fundamental problem of finding a node among a set of marked nodes in a graph, known as spatial search. Whether spatial search by continuous-time quantum walk provides a quadratic advantage over classical random walks has been an outstanding problem. Thus far, this advantage is obtained only for specific graphs or when a single node of the underlying graph is marked. In this Letter, we provide a new continuous-time quantum walk search algorithm that completely resolves this: our algorithm can find a marked node in any graph with any number of marked nodes, in a time that is quadratically faster than classical random walks. The overall algorithm is quite simple, requiring time evolution of the quantum walk Hamiltonian followed by a projective measurement. A key component of our algorithm is a purely analogue procedure to perform operations on a state of the form e^{-tH^{2}}|ψ⟩, which, for a given Hamiltonian H, only requires evolving H for time scaling as sqrt[t]. This allows us to quadratically fast-forward the dynamics of a continuous-time classical random walk. The applications of our Letter thus go beyond the realm of quantum walks and can lead to new analog quantum algorithms for preparing ground states of Hamiltonians or solving optimization problems.
Spatial search and walk processes. Spatial search by quantum walk is database search with the additional constraint that one is required to move through the search space that obeys some locality structure. For example, the data items may be stored at the vertices of a two-dimensional grid. The requirement of moves along the edges of the grid captures the cost of accessing different items starting from some fixed position in the database. One of possible ways of carrying out spatial search is by performing a random walk on the search space or its quantum analog, a quantum walk. The complexity of spatial search by quantum walk is strongly tied to the quantum hitting time [19] of the walk. Let S , with jS j D n, be a finite set of states. Assume that a subset M S of states are marked. We are given a procedure C that, on input x 2 S and an associated data structure d.x/, checks whether the state x is marked. The goal is either to find a marked state when promised that M ¤ ; (search version) or to determine whether M is nonempty (decision version). The algorithm progresses in stages. In the setup stage, we access some state of S (usually a random state). In the walk stage we move from state to state, performing a spatial walk as described below. The moves are called updates. In addition, in the walk stage we perform checks to see if the current state is marked at steps selected by the algorithm. In the classical setting, the transition probabilities of the spatial walk are described by a stochastic matrix P D .px;y/x;y2S . This makes the walk a Markov chain. In every move the algorithm must perform a random transition according to P . The possible x ! y moves, i.e., those with px;y ¤ 0, form the edges of a (directed) graph G, and we say that the Markov chain P has locality structure G. We define the search problem in the classical setting, which carries over to the quantum case with little modification:
One approach to the development of quantum search algorithms is the quantum walk. A spatial search can be effected by the continuous-time evolution of a single quantum particle on a lattice or graph containing a marked site. In most conceivable physical applications, however, one would rather expect to have multiple interacting particles. In bosonic systems at zero temperature, the dynamics would be well-described by a discrete non-linear Schrodinger equation. In this thesis we investigate the role of non-linearity in determining the efficiency of the spatial search algorithm within the quantum walk model, for the complete graph. Our analytical results indicate that the search time for this non-linear quantum search scales with size of the database N like square root of N, equivalent to linear spatial search time. The analytical results will be compared with numerical calculations of multiple interacting quantum walkers.