The task of finding an element in an unstructured database is known as spatial search and can be expressed as a quantum-walk evolution on a graph. In this article, we modify the usual search problem by adding an extra trapping vertex to the graph, which is only connected to the target element. We study the transfer efficiency of the walker to a trapping site, using the search problem as a case study. Thus, our model offers no computational advantage for the search problem, but focuses on information transport in an open environment with a search Hamiltonian. The walker evolution is a mix between classical and quantum-walk search dynamics. The balance between unitary and nonunitary dynamics is tuned with a parameter, and we numerically show that, depending on the graph topology and the connectivity of the target element, this hybrid approach can outperform a purely classical or quantum evolution for reaching the trapping site. We show that this behavior is only observed in the presence of an extra trapping site, and that, depending on the topology and a tunable parameter controlling the strength of the oracle, a hybrid regime composed of 90% coherent dynamics can lead to either the highest or worst transfer efficiency to the trapping site. We also relate the performance of a hybrid regime to the entropy's decay rate. As the introduction of nonunitary operations may be considered noise, we interpret this phenomenon as a noise-assisted quantum evolution.
The task of finding an element in an unstructured database is known as spatial search and can be expressed as a quantum-walk evolution on a graph. In this article, we modify the usual search problem by adding an extra trapping vertex to the graph, which is only connected to the target element. We study the transfer efficiency of the walker to a trapping site, using the search problem as a case study. Thus, our model offers no computational advantage for the search problem, but focuses on information transport in an open environment with a search Hamiltonian. The walker evolution is a mix between classical and quantum-walk search dynamics. The balance between unitary and nonunitary dynamics is tuned with a parameter, and we numerically show that, depending on the graph topology and the connectivity of the target element, this hybrid approach can outperform a purely classical or quantum evolution for reaching the trapping site. We show that this behavior is only observed in the presence of an extra trapping site, and that, depending on the topology and a tunable parameter controlling the strength of the oracle, a hybrid regime composed of 90% coherent dynamics can lead to either the highest or worst transfer efficiency to the trapping site. We also relate the performance of a hybrid regime to the entropy's decay rate. As the introduction of nonunitary operations may be considered noise, we interpret this phenomenon as a noise-assisted quantum evolution.
The task of finding an element in an unstructured database is known as spatial search and can be expressed as a quantum walk evolution on a graph. In this article, we modify the usual search problem by adding an extra trapping vertex to the graph, which is only connected to the target element. The walker evolution is a mix between classical and quantum walk search dynamics. The balance between unitary and non-unitary dynamics is tuned with a parameter, and we numerically show that depending on the graph topology and the connectivity of the target element, this hybrid approach can outperform a purely classical or quantum evolution for reaching the trapping site. We show that this behavior is only observed in the presence of an extra trapping site, and that depending on the topology, the increase of non-unitary operations can be compensated by increasing the strength of the quantum walk exploration. This compensation comes at the cost of reducing the searching feature of the evolution induced by the Hamiltonian. We also relate the optimal hybrid regime to the entropy's decay rate. As the introduction of non-unitary operations may be considered as noise, we interpret this phenomena as a noisy-assisted quantum evolution.
Coherent transport of an excitation through a network corresponds to continuous-time quantum walk on a graph, and the transport properties of the system may be radically different depending on the graph and on the initial state. The transport efficiency, i.e., the integrated probability of trapping at a certain vertex, is a measure of the success rate of the transfer process. Purely coherent quantum transport is known to be less efficient than the observed excitation transport, e.g., in biological systems, and there is evidence that environmental noise is indeed crucial for excitation transport. At variance with this picture, we here address purely coherent transport on highly symmetric graphs, and show analytically that it is possible to enhance the transport efficiency without environmental noise, i.e., using only a minimal perturbation of the graph. In particular, we show that adding an extra weight to one or two edges, depending on whether the initial state is localized or in a superposition of two vertex states, breaks the inherent symmetries of the graph and may be sufficient to achieve unit transport efficiency. We also briefly discuss the conditions to obtain a null transport efficiency, i.e., to avoid trapping.
The task of finding an entry in an unsorted list of N elements famously takes O(N) queries to an oracle for a classical computer and O(N) queries for a quantum computer using Grover's algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete- and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large N, the random walks converge to the same evolution, both taking Nln(1/ϵ) time to reach a success probability of 1−ϵ. In contrast, the discrete-time quantum walk asymptotically takes πN/22 timesteps to reach a success probability of 1/2, while the continuous-time quantum walk takes πN/2 time to reach a success probability of 1.
Abstract The quantum search of a target node on a cycle graph by means of a quantum walk assisted by continuous measurement and feedback are addressed. Unlike previous spatial search approaches, where the oracle is described as a projector on the target state, a dynamical oracle implemented through a feedback Hamiltonian is considered. The idea is based on continuously monitoring the position of the quantum walker on the graph and then applying a unitary feedback operation based on the information obtained from measurement. The feedback changes the couplings between the nodes and it is optimized at each time via a numerical procedure. The stochastic trajectories describing the evolution for graphs of dimensions up to are numerically simulated, and the performance of the protocol is quantified via the average fidelity between the state of the walker and the target node. Different constraints on the control strategy are discussed. For unbounded controls, the protocol is able to quickly localize the walker on the target node. How the performance is lowered by posing an upper bound on the control couplings is then discussed. Finally, it is shown how a digital feedback protocol seems in general as efficient as the continuous bounded one.
A quantum walk on a lattice is a paradigm of a quantum search in a database. The database qubit strings are the lattice sites, qubit rotations are tunneling events, and the target site is tagged by an energy shift. For quantum walks on a continuous time, the walker diffuses across the lattice and the search ends when it localizes at the target site. The search time T can exhibit Grover's optimal scaling with the lattice size N , namely, T ∼ N , on an all-connected, complete lattice. For finite-range tunneling between sites, instead, Grover's optimal scaling is warranted when the lattice is a hypercube of d > 4 dimensions. Here, we show that Grover's optimum can be reached in lower dimensions on lattices of long-range interacting particles, when the interaction strength scales algebraically with the distance r as 1 / r α and 0 < α < 3 d / 2 . For α < d the dynamics mimics the one of a globally connected graph. For d < α < d + 2 , the quantum search on the graph can be mapped to a short-range model on a hypercube with spatial dimension d s = 2 d / ( α − d ) , indicating that the search is optimal for d s > 4 . Our work identifies an exact relation between criticality of long-range and short-range systems, it provides a quantitative demonstration of the resources that long-range interactions provide for quantum technologies, and indicates when existing experimental platforms can implement efficient analog quantum search algorithms.
Quantum search algorithms provide a way to speed up combinatorial search, and have found several applications in modern quantum technology. In particular, spatial search on graphs, based on continuous-time quantum walks (CTQW), represents a promising platform for the implementation of quantum search in condensed matter systems. CTQW-based algorithms, however, work exactly on complete graphs, while they are known to perform poorly on realistic graphs with low connectivity. In this paper, we put forward an alternative search algorithm, based on structuring the oracle operator, which allows one to improve the localization properties of the walker by tuning only the on-site energies of the graph, i.e., without altering its topology. As such, the proposed algorithm is suitable for implementation in systems with low connectivity, e.g., rings of quantum dots or superconducting circuits. Oracle parameters are determined by Hamiltonian constraints, without the need for numerical optimization.