The standard quantum search lacks a feature, enjoyed by many classical algorithms, of having a fixed point, i.e. monotonic convergence towards the solution. Recently a fixed point quantum search algorithm has been discovered, referred to as the Phase-$π/3$ search algorithm, which gets around this limitation. While searching a database for a target state, this algorithm reduces the error probability from $ε$ to $ε^{2q+1}$ using $q$ oracle queries, which has since been proved to be asymptotically optimal. A different algorithm is presented here, which has the same worst-case behavior as the Phase-$π/3$ search algorithm but much better average-case behavior. Furthermore the new algorithm gives $ε^{2q+1}$ convergence for all integral $q$, whereas the Phase-$π/3$ search algorithm requires $q$ to be $(3^{n}-1)/2$ with $n$ a positive integer. In the new algorithm, the operations are controlled by two ancilla qubits, and fixed point behavior is achieved by irreversible measurement operations applied to these ancillas. It is an example of how measurement can allow us to bypass some restrictions imposed by unitarity on quantum computing.
Grover recently presented the fixed-point search algorithm. In this letter, we study the fixed-point search algorithm obtained by replacing equal phase shifts of $π/3$ by different phase shifts.
Building quantum devices using fixed operators is a must to simplify the hardware construction. Quantum search engine is not an exception. In this paper, a fixed phase quantum search algorithm that searches for M matches in an unstructured search space of size N will be presented. Selecting phase shifts of 1.91684πin the standard amplitude amplification will make the technique perform better so as to get probability of success at least 99.58% in O(sqrt(N/M)) better than any know fixed operator quantum search algorithms. The algorithm will be able to handle either a single match or multiple mat…
Grover presented the fixed-point search by replacing the selective inversions by selective phase shifts of $π/3$. In this paper, we investigate the convergence behavior of the fixed-point search algorithm with general but equal phase shifts for any iterations.
We describe an implementation of Grover's fixed-point quantum search algorithm on a nuclear magnetic resonance (NMR) quantum computer, searching for either one or two matching items in an unsorted database of four items. In this new algorithm the target state (an equally weighted superposition of the matching states) is a fixed point of the recursive search operator, and so the algorithm always moves towards the desired state. The effects of systematic errors in the implementation are briefly explored.
Fixed-point quantum search algorithms succeed at finding one of $M$ target items among $N$ total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property --- the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang gave an optimal gate-model search algorithm with the fixed-point property. Meanwhile, it is known that an adiabatic quantum algorithm, operating by continuously varying a …
Distributed Quantum Computing Based on Fixed-point Quantum Search
School of Computer Science and Engineering, Sun Yat-sen University（中山大学计算机科学与工程学院）；The Guangdong Key Laboratory of Information Security Technology, Sun Yat-sen University（中山大学信息安全技术广东省重点实验室）
Grover's quantum search and its generalization, quantum amplitude amplification, provide quadratic advantage over classical algorithms for a diverse set of tasks, but are tricky to use without knowing beforehand what fraction $λ$ of the initial state is comprised of the target states. In contrast, fixed-point search algorithms need only a reliable lower bound on this fraction, but, as a consequence, lose the very quadratic advantage that makes Grover's algorithm so appealing. Here we provide the first version of amplitude amplification that achieves fixed-point behavior without sacrificing the…
A quantum algorithm is a set of instructions for a quantum computer, however, unlike algorithms in classical computer science their results cannot be guaranteed. Quantum search algorithm can be described as the rotation of state vectors in a Hilbert space. The state vectors uniformly rotate by iterative sequences until they hit the target position. To optimize the algorithm, it is necessary to have the precise knowledge about some parameters like the number of target positions and total number of states. Here, we demonstrate the implementation of optimal fixed-point quantum search algorithm in…
In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such Kronecker graphs can be searched by a continuous-time quantum walk in optimal $Θ(\sqrt{N})$ time. First, if the initiator is regular and its adjacency matrix has a dominant principal eigenvalue, meaning its unique largest eigenvalue asymptotically dominates the other eigenvalues in magnitude, then the Kronecker graphs generated by this initiator can be quantum…
The standard quantum search algorithm lacks a feature, enjoyed by many classical algorithms, of having a fixed-point, i.e. a monotonic convergence towards the solution. Here we present two variations of the quantum search algorithm, which get around this limitation. The first replaces selective inversions in the algorithm by selective phase shifts of $\fracπ{3}$. The second controls the selective inversion operations using two ancilla qubits, and irreversible measurement operations on the ancilla qubits drive the starting state towards the target state. Using $q$ oracle queries, these variatio…
For the unsorted database quantum search with the unknown fraction $λ$ of target items, there are mainly two kinds of methods, i.e., fixed-point or trail-and-error. (i) In terms of the fixed-point method, Yoder et al. [Phys. Rev. Lett. 113, 210501 (2014)] claimed that the quadratic speedup over classical algorithms has been achieved. However, in this paper, we point out that this is not the case, because the query complexity of Yoder's algorithm is actually in $O(1/\sqrt{λ_0})$ rather than $O(1/\sqrtλ)$, where $λ_0$ is a known lower bound of $λ$. (ii) In terms of the trail-and-error method, cu…
A robust control over quantum dynamics is of paramount importance for quantum technologies. Many of the existing control techniques are based on smooth Hamiltonian modulations involving repeated calculations of basic unitaries resulting in time complexities scaling rapidly with the length of the control sequence. On the other hand, the bang-bang controls need one-time calculation of basic unitaries and hence scale much more efficiently. By employing a global optimization routine such as the genetic algorithm, it is possible to synthesize not only highly intricate unitaries, but also certain no…
Quantum Algorithm for the Fixed-Radius Neighbor Search
Dipartimento di Fisica dell’Università di Trieste（的里雅斯特大学物理系）；Fondazione Istituto Italiano di Tecnologia, Center for Life Nano-Neuroscience at la Sapienza（意大利技术研究院基金会，萨皮恩扎大学生命纳米神经科学中心）；INAF - Osservatorio Astronomico di Trieste（意大利国家天体物理研究所的里雅斯特天文台）；MOX – Department of Mathematics Politecnico di Milano（米兰理工大学数学系MOX）；Center for Computational Astrophysics, Flatiron Institute（平顿研究所计算天体物理学中心）
Quantum search algorithm for similar subgraph identification under fixed edge removal
Institute of Climate and Energy Systems (ICE-1), Forschungszentrum Jülich（于利希研究中心气候与能源系统研究所）；Institute for Quantum Information, RWTH Aachen University（亚琛工业大学量子信息研究所）；Institute for Quantum Computing Analytics (PGI-12), Forschungszentrum Jülich（于利希研究中心量子计算分析研究所）；Computational Methods for Energy Systems Engineering, RWTH Aachen University（亚琛工业大学能源系统工程计算方法）
Generic quantum search algorithm searches for target entity in an unsorted database by repeatedly applying canonical Grover's quantum rotation transform to reach near the vicinity of the target entity represented by a basis state in the Hilbert space associated with the qubits. Thus, when qubits are measured, there is a high probability of finding the target entity. However, the number of times quantum rotation transform is to be applied for reaching near the vicinity of the target is a function of the number of target entities present in the unsorted database, which is generally unknown. A wr…
Generic quantum search algorithm searches for target entity in an unsorted database by repeatedly applying canonical Grover's quantum rotation transform to reach near the vicinity of the target entity. Thus, upon measurement, there is a high probability of finding the target entity. However, the number of times quantum rotation transform is to be applied for reaching near the vicinity of the target is a function of the number of target entities present in an unsorted database, which is generally unknown. A wrong estimate of the number of target entities can lead to overshooting or undershootin…
