Implementation of Grover’s quantum search algorithm in a scalable system

Abstract

We report the implementation of Grover's quantum search algorithm in the scalable system of trapped atomic ion quantum bits. Any one of four possible states of a two-qubit memory is marked, and following a single query of the search space, the marked element is successfully recovered with an average probability of 60(2)%. This exceeds the performance of any possible classical search algorithm, which can only succeed with a maximum average probability of 50%. PACS numbers:Quantum computers promise dramatic speedup over conventional computers in some applications owing to the power of entangled superpositions [1]. Among the best-known quantum applications is Grover's search algorithm, which can search an unsorted database quadratically faster than any known classical search [2]. A common analogy for this searching algorithm is the problem of finding a person's name in a phone book given only their phone number [3]: for N entries in the phonebook, this requires of order N queries. However, if the correlation between name and phone number is encoded with quantum bits, the name can be found after only about √ N queries. While Grover's algorithm does not attain the exponential speedup of Shor's quantum factoring algorithm [4], it may be more versatile, by providing quadratic gains for almost any quantum algorithm [5] or accelerating NP-complete problems through exhaustive searches over possible solutions [6].We implement the Grover search algorithm over a space of N=4 elements using two trapped atomic ion qubits [7,8]. Grover's algorithm has been implemented with ensembles of molecules using nuclear magnetic resonance [9,10,11], with states of light using linear optical techniques [12,13], and with Rydberg states within individual atoms [14]. None of these systems are scalable however, as they require exponential resources as the number of qubits grows. The implementation of Grover's algorithm reported here complements the repertoire of multi-qubit quantum algorithms recently demonstrated in the scalable system of trapped atomic ions [15,16,17,18]. Unlike these earlier ion trap demonstrations, we use magnetically-insensitive "clock state" qubits and particular entangling gates that are uniquely suited to such qubits while remaining insensitive to external phase drifts between gates [19,20,21].At the heart of Grover's algorithm is the "oracle query," which quickly checks if a proposed input "x" is a solution to the search problem. The oracle marks a particular component of a quantum superposition by flipping the sign of its amplitude. Following the oracle, a number of quantum operations amplify the weighting of the marked state independent of which state is

Quantum computing with trapped ions

“…Furthermore, Haljan et al (2005b) created all four Bell states and implemented Grover's seach algorithm with 111 Cd + ions using this gate operation (see also Refs. Lee et al (2005), Brickman et al (2005) and Brickman (2007)). …”

Quantum Computing and High Performance Computing

“…To minimize the numerical error in (12), instead of calculating the density matrix ρ once after the full time step ∆t, QX divides the time step by some value n, and repeatedly calculates the next value of ρ. Figure 5 shows this function.…”

“…• Deutsch-Jozsa [11] • Grover [12] • Semi-classical quantum Fourier transform (QFT) [13] After reviewing the published data, we identified three main project risks and developed a plan to address them. The first risk was that we needed to obtain complete and consistent experimental data.…”

“…The purpose of the Grover algorithm as experimentally implemented is to find a target state in a search space of four states [12]. Figure 36 is a schematic illustration of the general algorithm.…”

Technologies for trapped-ion quantum information systems

“…Images courtesy of University of Innsbruck using laser or microwave pulses to carry out quantum gates [5][6][7][8][9], which were then read out via the fluorescence emitted by the ions. With this hardware, quantum protocols such as teleportation [10][11][12], state mapping from an ion to a photon [13], and quantum simulation [14][15][16][17][18][19][20] have been realized, and quantum algorithms including the quantum Fourier transform [21,22], the Deutsch-Josza algorithm [23], Grover search [24], and quantum error correction [25] have been demonstrated.…”