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AIP conference proceedings

Simulation of quantum algorithm using modified qubit rotation

Neris I. Sombillo

Published Jan 1, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.1063/5.0163121

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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.
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.
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 present efficient methods to implement the quantum computing Grover search algorithm using the Rydberg blockade interaction. We show that simple π-pulse excitation sequences between ground and Rydberg excited states readily produce the key conditional phase shift and inversion-about-the-mean unitary operations for the Grover search. Multi-qubit implementation schemes suitable for different properties of the atomic interactions are identified and the error scaling of the protocols with system size is found to be promising for experimental investigation.
Models of Grover’s search algorithm is reviewed to build the foundation for the other algorithms. Thereafter, some preliminary modifications of the original algorithms by others are stated, that increases the applicability of the search procedure. A general quantum computation on an isolated system can be represented by a unitary matrix. In order to execute such a computation on a quantum computer, it is common to decompose the unitary into a quantum circuit, i.e., a sequence of quantum gates that can be physically implemented on a given architecture. There are different universal gate sets for quantum computation. Here we choose the universal gate set consisting of CNOT and single-qubit gates. We measure the cost of a circuit by the number of CNOT gates as they are usually more difficult to implement than single qubit gates and since the number of single-qubit gates is bounded by about twice the number of CNOT’s.
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.
Abstract This work concerns a three-qubit processor implemented as three transmon-type superconducting qubits capacitively coupled to a superconducting resonator driven by a microwave field. Based on an effective Hamiltonian of this system, we propose a scheme to realize an X -rotation gate for three qubits requiring only a single-step operation. During the gate operation, the cavity photon-number is canceled due to the strong microwave field. It does not require the cavity to be initially in the vacuum state and the scheme is insensitive to cavity decay. We also present an effective scheme to implement Grover’s search algorithm based on this quantum gate and a phase oracle. Using numerical simulation and analysis, we show that the proposed schemes allow for efficiently finding the target state with a high fidelity. The potential application of these proposals with current QED circuit technology is also discussed.