Quantum search algorithm by adiabatic evolution under a priori probability (original) (raw)

Quantum search by local adiabatic evolution

Physical Review A, 2002

The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state which tends to the solution [1]. We apply this time-dependent Hamiltonian approach to the Grover's problem, i. e., searching a marked item in an unstructured database. We find that, by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order √ N , where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N . This is in contrast with the constant-rate adiabatic approach developed in [1], where the requirement of adiabaticity is expressed only globally, resulting in a time of order N .

Complete Adiabatic Quantum Search in Unsorted Databases

2008

We propose a new adiabatic algorithm for the unsorted database search problem. This algorithm saves two thirds of qubits than Grover's algorithm in realizations. Meanwhile, we analyze the time complexity of the algorithm by both perturbative method and numerical simulation. The results show it provides a better speedup than the previous adiabatic search algorithm.

Adiabatic quantum search algorithm for structured problems

Physical Review A, 2003

The study of quantum computation has been motivated by the hope of finding efficient quantum algorithms for solving classically hard problems. In this context, quantum algorithms by local adiabatic evolution have been shown to solve an unstructured search problem with a quadratic speedup over a classical search, just as Grover's algorithm. In this paper, we study how the structure of the search problem may be exploited to further improve the efficiency of these quantum adiabatic algorithms. We show that by nesting a partial search over a reduced set of variables into a global search, it is possible to devise quantum adiabatic algorithms with a complexity that, although still exponential, grows with a reduced order in the problem size.

Non-Adiabatic Quantum Dynamics of Grover's Adiabatic Search Algorithm

We study quantum dynamics of Grover's adiabatic search algorithm with the equivalent two-level system. Its adiabatic and non-adiabatic evolutions are visualized as trajectories of Bloch vectors on a Bloch sphere. We find the change in the non-adiabatic transition probability from exponential decay for short running time to inverse-square decay for long running time. The size dependence of the critical running time is expressed in terms of Lambert W function. The transitionless driving Hamiltonian is obtained to make a quantum state follow the adiabatic path. We demonstrate that a constant Hamiltonian, approximate to the exact time-dependent driving Hamiltonian, can alter the non-adiabatic transition probability from the inverse square decay to the inverse fourth power decay with running time. This may open up a new way of reducing errors in adiabatic quantum computation. PACS numbers: 03.67.Ac, 03.65.-w, 03.67.-a

Quantum-circuit model of Hamiltonian search algorithms

Physical Review A, 2003

We analyze three different quantum search algorithms, the traditional Grover's algorithm, its continuous-time analogue by Hamiltonian evolution, and finally the quantum search by local adiabatic evolution. We show that they are closely related algorithms in the sense that they all perform a rotation, at a constant angular velocity, from a uniform superposition of all states to the solution state. This make it possible to implement the last two algorithms by Hamiltonian evolution on a conventional quantum circuit, while keeping the quadratic speedup of Grover's original algorithm.

A Modification of Grover's Quantum Search Algorithm

We propose a quantum search method, based on Grover’s algorithm. We show that to search for a single marked element from an unsorted search space of N elements the number of queries required using this algorithm varies as N**1.3 when compared to N**1/2 for the Grover’s algorithm.

Local adiabatic quantum search with different paths

2003

We report on a detailed analysis of generalization of the local adiabatic search algorithm. Instead of evolving directly from an initial ground state Hamiltonian to a solution Hamiltonian a different evolution path is introduced and is shown that the time required to find an item in a database of size NNN can be made to be independent of the size of the database by modifying the Hamiltonian used to evolve the system.

Grover’s quantum search algorithm

Proceedings of Symposia in Applied Mathematics, 2002

This paper ia a written version of a one hour lecture given on Lov Grover's quantum database search algorithm. It is based on [4], [5], and [9]. Contents 1. Problem definition 1 2. The quantum mechanical perspective 2 3. Properties of the inversion I |ψ 4 4. The method in Lov's "madness" 5 5. Summary of Grover's algorithm 8 6. An example of Grover's algorithm 10 References 12 1. Problem definition We consider the problem of searching an unstructured database of N = 2 n records for exactly one record which has been specifically marked. This can be rephrased in mathematical terms as an oracle problem as follows:

An Optimum Algorithm for Quantum Search

arXiv: Quantum Physics, 2020

This paper discusses an improvement to Grover's algorithm for searches where target states are Hamming weight eigenstates and search space is not ordered. It is shown that under these conditions search efficiency depends on the smaller number of 0's and 1's, not the total length, of binary string of target state, and that Grover's algorithm can be improved whenever number of 0's and number 1's are not equal. In particular, improvement can be exponential when number of 0's or number of 1's is very small relative to binary string length. One interesting application is that Dicke state preparation, which in Grover's algorithm is P on average, can be made poly-efficient in all cases. For decision making process, this improvement won't improve computation efficiency, but can make implementation much simpler.