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arXiv:0912.4071

Experimental NMR implementation of a robust quantum search algorithm

Avik Mitra, Avatar Tulsi, Anil Kumar

Published Dec 21, 2009• 1 citation
arXivHTMLPDF

Abstract

Grover's quantum search algorithm, involving a large number of qubits, is highly sensitive to errors in the physical implementation of the unitary operators. This poses an intrinsic limitation to the size of the database that can be practically searched. The lack of robustness of Grover's algorithm for a large number of qubits is due to quite stringent "phase-matching" condition. To overcome this limitation, Tulsi suggested a modified search algorithm [PRA 78, 022332] which succeeds as long as the errors are reproducible and reversible while Grover's algorithm fails. Such systematic errors arise often from imperfections in apparatus setup e.g. the errors arising from imperfect pulse calibration and offset effect in NMR systems. In this paper, we report the experimental NMR implementation of the modified search algorithm and its comparison with the original Grover's algorithm. We experimentally validate the theoretical predictions made by Tulsi.

DOI
10.48550/arxiv.0912.4071

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Grover's quantum search algorithm, involving a large number of qubits, is highly sensitive to errors in the physical implementation of the unitary operators. This poses an intrinsic limitation to the size of the database that can be practically searched. The lack of robustness of Grover's algorithm for a large number of qubits is due to quite stringent "phase-matching" condition. To overcome this limitation, Tulsi suggested a modified search algorithm [PRA 78, 022332] which succeeds as long as the errors are reproducible and reversible while Grover's algorithm fails. Such systematic errors arise often from imperfections in apparatus setup e.g. the errors arising from imperfect pulse calibration and offset effect in NMR systems. In this paper, we report the experimental NMR implementation of the modified search algorithm and its comparison with the original Grover's algorithm. We experimentally validate the theoretical predictions made by Tulsi.
We briefly introduced some of our recent work related to the phase matching condition in quantum searching algorithms and the improved Grover algorithm. When one replaces the two phase inversions in the Grover algorithm with arbitrary phase rotations, the modified algorithm usually fails in searching the marked state unless a phase matching condition is satisfied between the two phases. The Grover algorithm is not 100% in success rate, an improved Grover algorithm with zero-failure rate is given by replacing the phase inversions with angles that depends on the size of the database. Other aspects of the Grover algorithm such as the SO(3) picture of quantum searching, the dominant gate imperfections in the Grover algorithm are also mentioned.
We describe an implementation of Grover's fixed-point quantum search algorithm on a nuclear magnetic resonance quantum computer, searching for either one or two matching items in an unsorted database of four items. In this algorithm the target state (an equally weighted superposition of the matching states) is a fixed point of the recursive search operator, so that the algorithm always moves towards the desired state. The effects of systematic errors in the implementation are briefly explored.
The search problem is to find a state satisfying certain properties out of a given set. Grover's algorithm drives a quantum computer from a prepared initial state to the target state and solves the problem quadratically faster than a classical computer. The algorithm uses selective transformations to distinguish the initial state and target state from other states. It does not succeed unless the selective transformations are very close to phase inversions. Here we show a way to go beyond this limitation. An important application lies in quantum error correction, where the errors can cause the selective transformations to deviate from phase inversions. The algorithms presented here are robust to errors as long as the errors are reproducible and reversible. This particular class of systematic errors arises often from imperfections in the apparatus setup. Hence our algorithms offer a significant flexibility in the physical implementation of quantum search.
Currently,the NMR technology is one of the effective physical systems to realize quantum computation.As the multiple-quantum operator algebra theory mentions,any unitary transformation can be decomposed into a sequence of a limited number of one-qubit quantum gates and two-qubit diagonal gates.Based on these backgrounds,we propose a method to form NMR pulse sequences to realize the multi-qubit Grover quantum search algorithm.Experiments finish the algorithm with three-qubit,on a Quantum Computer Emulator.
Grover’s quantum search algorithm delivers quadratic speedup over classical counterparts for finding an item in unstructured database. But the accuracy of the algorithm degrades as the number of qubits are increased. Noises of various types affect the accuracy of the Grover’s algorithm. The imperfect measurement on the qubit results in measurement errors appears to be one of the factors which limit the scalability of near-term quantum devices/algorithms. This paper is focused on the implementation of measurement error mitigation technique on Grover’s algorithm up to 4-qubits. The measurement error mitigation model for Grover’s algorithm is developed and implemented on the real-time quantum computer. The accuracy of Grover’s algorithm up to 4-qubits with and without measurement errors is evaluated and compared. The results indicate that measurement error mitigation technique mitigates the measurement errors of the Grover’s algorithm and improves its accuracy.
Abstract Quantum search algorithms have important applications in quantum information, which have been experimentally demonstrated in several quantum systems with database sizes up to 16. We first propose an efficient quantum search scheme based on an optimized Grover’s algorithm and a novel encoding method. With this scheme, a target element can be deterministically searched in a large database using only a single qubit. Then we implement this search scheme in linear optics with database sizes up to 40. Experimental results fit well with the theoretical results, with all fidelities greater than 99.7%. The limitation of the number of elements encoded in a single qubit is also investigated in the experiment. The experimental results indicate that a target state can be successfully searched from a database of size 36 using a single qubit in our laboratory. Since only one qubit and one-qubit gates are required, this search scheme is quite efficient and general, and can be adopted to implement the same task in most quantum systems.