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Proceedings of the Samahang Pisika ng Pilipinas

Simulation of Grover's algorithm with position-variation of the ancilla qubit

Neris Ilano, Ronald S. Banzon, Cristine Dlr. Villagonzalo

Published Jun 3, 2015• 0 citations
OpenAlex

Abstract

We simulate Grover’s quantum search algorithm in an Ising spin chain quantum computer with nearest and second-nearest neighbour interactions. We determine the optimum Rabi frequency that satisfies the 2πk method by selecting a high fidelity value in the frequency scan for each database size. Also, we vary the position of the ancilla qubit and investigate the effect on the fidelity for different sizes of the database. The behavior of the fidelity as a function of the number of Grover iterations during the algorithm is considered. For a fixed location of the ancilla qubit, the overall fidelity decreases as we increase the size of the database.

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We simulate Grover’s quantum search algorithm in an Ising spin chain quantum computer with nearest and second-nearest neighbour interactions. We determine the optimum Rabi frequency that satisfies the 2πk method by selecting a high fidelity value in the frequency scan for each database size. Also, we vary the position of the ancilla qubit and investigate the effect on the fidelity for different sizes of the database. The behavior of the fidelity as a function of the number of Grover iterations during the algorithm is considered. For a fixed location of the ancilla qubit, the overall fidelity decreases as we increase the size of the database.
We implement Grover's quantum search algorithm on a nuclear spin-chain quantum computer, taking Ising-type interactions between nearest and second-nearest neighbours into account. The performance of this implementation is studied by numerical simulations with four spins. We determine the temporal behaviour of the fidelity during the algorithm, and we compute the final fidelity as a function of the Rabi frequency. For the latter, we obtain pronounced maxima at frequencies which fulfil the condition of the 2π k -method with respect to the second-nearest- neighbour interactions.
We consider Grover's search algorithm on a model quantum computer implemented on a chain of four or five nuclear spins with first- and second-neighbour Ising interactions. Noise is introduced into the system in terms of random fluctuations of the external fields. By averaging over many repetitions of the algorithm, the output state becomes effectively a mixed state. We study its overlap with the nominal output state of the algorithm, which is called the fidelity. We analyse the behaviour of the fidelity as a function of the noise intensity for static and random noise on either the Larmor frequencies or the Rabi frequency, and we compare our results to theoretical predictions and numerical simulations which are based on more abstract quantum computer models.
We study the implementation of Grover’s algorithm in an Ising spin chain quantum computer. In our scheme, we removed the application of time delay whenever a qubit operation is performed. Instead, we introduced a set of modified pulses that utilizes the interaction of the neighboring qubit. In principle, the absence of time delay pulses allows the overall rotation pulses to be applied simultaneously to execute a gate. We show that the fidelity remains high after continuous application of π/2 pulses. The error arising from the unitary evolution is minimized by using soft pulses. Finally, the probability of finding the target state remain bounded through the application of fixed-point quantum search scheme.
For searching an item in unstructured databases, Grover's quantum search algorithm offers quadratic speedup overclassical search algorithms. This paper reports 2 to 5 quantum-bit (Qubit) implementations of Grover's search algorithmusing the phase-flip method for oracle function realization without any extra ancilla qubit. A comprehensive estimation andanalysis of the theoretical and physical accuracies of the algorithm have been presented. The impact of increasing qubits onaccuracy has been computed and analyzed. The metrics delineated for comparison are the number of qubits and gates, depthof the circuit, execution time, and theoretical/physical accuracy. The results revealed a greater disparity between theoreticaland physical accuracy using a higher number of qubits perceived to be caused by noisy qubits utilized in computations. Thenovelty of the work is the investigation of variations caused by the noise in the accuracy and execution time of Grover'ssearch algorithm. The results indicate that because of noise, the accuracy of 2- and 3- qubit implementations declined by14.49% and 33.86%, whereas the execution time increased by 50% and 80%; respectively.
The oscillatory nature of the unitary Grover operator makes quantum searching difficult if one has no prior knowledge of the number of target states in the database. In this work, we coupled the database to an external qubit to transfer the target state in a space where the quantum search operation is not applied. This is followed by a measurement of the external and the ancilla qubit which result to a mixed state with a higher probability of finding the target state. The algorithm is simulated in a spin chain having first and second order neighbor interactions. The result shows that the oscillation is suppressed and the probability of success generally increased with iteration number.
When the Grover’s algorithm is applied to search an unordered database, the probability of success usually decreases with the increase of marked items. To address this phenomenon, a fixed-phase quantum search algorithm with more flexible behavior is proposed. In proposed algorithm, the phase shifts can be fixed at the different values to meet the needs of different practical problems. If research requires a relatively rapid speed, the value of the phase shifts should be appropriately increased, if search requires a higher success probability, the value of the phase shifts should be appropriately decreased. When the phase shifts are fixed at , the success probability of at least 99.38% can be obtained in iterations.