Continuous-time quantum walks can be used to solve the spatial search problem, which is an essential component for many quantum algorithms that run quadratically faster than their classical counterpart, in O(sqrt[n]) time for n entries. However, the capability of models found in nature is largely unexplored-e.g., in one dimension only nearest-neighbor Hamiltonians have been considered so far, for which the quadratic speedup does not exist. Here, we prove that optimal spatial search, namely with O(sqrt[n]) run time and high fidelity, is possible in one-dimensional spin chains with long-range interactions that decay as 1/r^{α} with distance r. In particular, near unit fidelity is achieved for α≈1 and, in the limit n→∞, we find a continuous transition from a region where optimal spatial search does exist (α<1.5) to where it does not (α>1.5). Numerically, we show that spatial search is robust to dephasing noise and that, for reasonable chain lengths, α≲1.2 should be sufficient to demonstrate optimal spatial search experimentally with near unit fidelity.
Continuous-time quantum walks can be used to solve the spatial search problem, which is an essential component for many quantum algorithms that run quadratically faster than their classical counterpart, in O(sqrt[n]) time for n entries. However, the capability of models found in nature is largely unexplored-e.g., in one dimension only nearest-neighbor Hamiltonians have been considered so far, for which the quadratic speedup does not exist. Here, we prove that optimal spatial search, namely with O(sqrt[n]) run time and high fidelity, is possible in one-dimensional spin chains with long-range interactions that decay as 1/r^{α} with distance r. In particular, near unit fidelity is achieved for α≈1 and, in the limit n→∞, we find a continuous transition from a region where optimal spatial search does exist (α<1.5) to where it does not (α>1.5). Numerically, we show that spatial search is robust to dephasing noise and that, for reasonable chain lengths, α≲1.2 should be sufficient to demonstrate optimal spatial search experimentally with near unit fidelity.
Continuous-time quantum walks can be used to solve the spatial search problem, which is an essential component for many quantum algorithms that run quadratically faster than their classical counterpart, in O(n) time for n entries. However, the capability of models found in nature is largely unexplored - e.g., in one dimension only nearest-neighbor Hamiltonians have been considered so far, for which the quadratic speedup does not exist. Here, we prove that optimal spatial search, namely with O(n) run time and high fidelity, is possible in one-dimensional spin chains with long-range interactions that decay as 1/rα with distance r. In particular, near unit fidelity is achieved for α≈1 and, in the limit n→∞, we find a continuous transition from a region where optimal spatial search does exist (α<1.5) to where it does not (α>1.5). Numerically, we show that spatial search is robust to dephasing noise and that, for reasonable chain lengths, α≲1.2 should be sufficient to demonstrate optimal spatial search experimentally with near unit fidelity.
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.
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.
Continuous-time quantum walks provide an alternative method for quantum search problems. Most earlier studies confirmed that quadratic speedup exists in some synthetic Hamiltonians, but whether there is quadratic speedup in real physical systems remains elusive. Here, we investigate three physical systems with long-range atom-atom interaction which are possibly good candidates for realizing the quantum search, including one-dimensional atom arrays either trapped in an optical lattice or coupled to a waveguide near the band edge or dispersively coupled to a good cavity. We find that all three systems can provide a near-optimal quantum search if there is no dissipation. However, if the dissipation is considered, only the latter two systems (i.e., waveguide-QED and cavity-QED systems) can still have high success probabilities because they can significantly enhance the atom-atom interaction even if they are far apart and the spectra gap can be much larger, which can reduce the search time and the effects of dissipation significantly. Our studies here can provide helpful instructions for realizing quantum search in real physical systems in the noisy intermediate-scale quantum era.
Quantum-walk-based algorithms that search a marked location among N locations on a d-dimensional lattice succeeds in time O(N) for d>2, while this is not found to be possible when d=2. In this paper, we consider a spatial search algorithm using continuous-time quantum walk on a two-dimensional square lattice with the existence of additional long-range edges. We examined such a search on a probabilistic graph model where an edge connecting non-nearest-neighbor lattice points i and j apart by a distance $|i\ensuremath{-}j|$ is added by probability ${p}_{ij}={|i\ensuremath{-}j|}^{\ensuremath{-}\ensuremath{\alpha}}\phantom{\rule{1em}{0ex}}(\ensuremath{\alpha}\ensuremath{\ge}0)$. Through numerical analysis, we found that the search succeeds in time O(N) when $\ensuremath{\alpha}\ensuremath{\le}{\ensuremath{\alpha}}_{c}=2.4\ifmmode\pm\else\textpm\fi{}0.1$. For $\ensuremath{\alpha}>2$, the expectation value of the additional long-range edges on each node scales as a constant when $N\ensuremath{\rightarrow}\ensuremath{\infty}$, which means that search time of O(N) is achieved on a graph with average degree scaling as a constant.
The problem of finding a marked node in a graph can be solved by the spatial search algorithm based on continuous-time quantum walks (CTQW). However, this algorithm is known to run in optimal time only for a handful of graphs. In this work, we prove that for Erdös-Renyi random graphs, i.e., graphs of n vertices where each edge exists with probability p, search by CTQW is almost surely optimal as long as p≥log^{3/2}(n)/n. Consequently, we show that quantum spatial search is in fact optimal for almost all graphs, meaning that the fraction of graphs of n vertices for which this optimality holds tends to one in the asymptotic limit. We obtain this result by proving that search is optimal on graphs where the ratio between the second largest and the largest eigenvalue is bounded by a constant smaller than 1. Finally, we show that we can extend our results on search to establish high fidelity quantum communication between two arbitrary nodes of a random network of interacting qubits, namely, to perform quantum state transfer, as well as entanglement generation. Our work shows that quantum information tasks typically designed for structured systems retain performance in very disordered structures.