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Physical Review A

Deterministic conversions between Greenberger-Horne-Zeilinger states and W states of spin qubits via Lie-transform-based inverse Hamiltonian engineering

Yi‐Hao Kang, Zhi‐Cheng Shi, Bi‐Hua Huang, Jie Song, Yan Xia

Published Jul 22, 2019• 28 citations
Publisher

Abstract

In this paper, we propose a protocol to realize the conversions between Greenberger-Horne-Zeilinger (GHZ) states and $W$ states of spin qubits. By analyzing and simplifying the dynamics of the system, the control fields are designed via the inverse Hamiltonian engineering based on the Lie transforms. Moreover, the states' conversions between multitype GHZ states and $W$ states can be executed deterministically and reversibly, which makes the protocol resource saving and flexible. We show in numerical simulations that the state conversions are robust against the systematic errors, random noise, and frequency mismatching of the control fields. Therefore, the protocol may provide some useful prospectives to the research of quantum mechanics and applications of quantum information processing based on GHZ states and $W$ states.

DOI
10.1103/physreva.100.012332

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In this paper, we propose a protocol to realize the conversions between Greenberger-Horne-Zeilinger (GHZ) states and WWW states of spin qubits. By analyzing and simplifying the dynamics of the system, the control fields are designed via the inverse Hamiltonian engineering based on the Lie transforms. Moreover, the states' conversions between multitype GHZ states and WWW states can be executed deterministically and reversibly, which makes the protocol resource saving and flexible. We show in numerical simulations that the state conversions are robust against the systematic errors, random noise, and frequency mismatching of the control fields. Therefore, the protocol may provide some useful prospectives to the research of quantum mechanics and applications of quantum information processing based on GHZ states and WWW states.
In the paper, we propose a protocol for the one-step interconversions between the Greenberger-Horne-Zeilinger (GHZ) states and the WWW states with deterministic success probabilities. The Rydberg interactions among three Rydberg atoms are utilized here to structure nonlocal operations for the interconversions. Additionally, we construct a Lewis-Riesenfeld invariant for a four-level system and afterwards use it to reversely design the Rabi frequencies of the effective Hamiltonian. The numerical simulations indicate that the protocol is robust against the spontaneous emission, the dephasing, the thermal noise, the systematic errors on Rabi frequencies, and the frequency mismatching. Therefore, the protocol may contribute to the research of the interconversions between entanglements.
Abstract Conversion between different types of entangled states is an interesting problem in quantum mechanics. But research on the conversion between the Greenberger–Horne–Zeilinger (GHZ) state and Knill–Laflamme–Milburn (KLM) state in an atomic system has not been reported. In this paper, we propose a scheme to realize the interconversion (one-step) between the GHZ state and KLM state with Rydberg atoms. By utilizing Rydberg-mediated interactions, we simplify the system. By combining a Lie-transform-based pulse design, the evolution path is built up to realize interconversion of the GHZ state and KLM state. The numerical simulation result shows that the present scheme is robust against decoherence and operational imperfection.
Conversion between different types of entangled states is an interesting problem in quantum mechanics. But research on the conversion between Greenberger-Horne-Zeilinger (GHZ) state and Knill-Laflamme-Milburn (KLM) state in atomic system is absent. In this paper, we propose a scheme to realize the interconversion (one-step) between GHZ state and KLM state with Rydberg atoms. By utilizing Rydberg-mediated interactions, we simplify the system. By combining Lie-transform-based pulse design, the evolution path is built up to realize interconversion of GHZ state and KLM state. The numerical simulation result shows that the present scheme is robust against decoherence and operational imperfection, the analysis shows that the scheme is feasible with current experimental technology.
We investigate the possibilities for a deterministic conversion between two important types of maximally entangled multiqubit states, namely, WWW and Greenberger-Horne-Zeilinger (GHZ) states, in the Rydberg-blockade regime of a neutral-atom system where each atom is subject to four external laser pulses. Such interconversions between WWW states and their GHZ counterparts have quite recently been addressed using the method of shortcuts to adiabaticity, more precisely techniques based on Lewis-Riesenfeld invariants [R.-H. Zheng et al., Phys. Rev. A 101, 012345 (2020)]. Motivated in part by this recent work, we revisit the WWW to GHZ state-conversion problem using a fundamentally different approach, which is based on the dynamical symmetries of the system and a Lie-algebraic parametrization of its permissible evolutions. In contrast to the previously used invariant-based approach, which leads to a state-conversion protocol characterized by strongly time-dependent Rabi frequencies of external lasers, ours can also yield one with time-independent Rabi frequencies. This feature makes our protocol more easily applicable experimentally, with the added advantage that it allows the desired state conversion to be carried out in a significantly shorter time with the same total laser pulse energy used.
We investigate the possibilities for a deterministic conversion between two important types of maximally-entangled multiqubit states -- namely, W and Greenberger-Horne-Zeilinger (GHZ) states -- in the Rydberg-blockade regime of a neutral-atom system where each atom is subject to four external laser pulses. Such interconversions between W states and their GHZ counterparts have quite recently been addressed using the method of shortcuts to adiabaticity, more precisely techniques based on Lewis-Riesenfeld invariants [R.-H. Zheng {\em et al.}, Phys. Rev. A {\bf 101}, 012345 (2020)]. Motivated in part by this recent work, we revisit the W-to-GHZ state-conversion problem using a fundamentally different approach, which is based on the dynamical symmetries of the system and a Lie-algebraic parametrization of its permissible evolutions. In contrast to the previously used invariant-based approach, which leads to a state-conversion protocol characterized by strongly time-dependent Rabi frequencies of external lasers, ours can also yield one with time-independent Rabi frequencies. This feature makes our protocol more easily applicable experimentally, with the added advantage that it allows the desired state conversion to be carried out in a significantly shorter time.
The N-qubit Greenberger-Horne-Zeilinger (GHZ) state is an important resource for quantum technologies. We consider the task of GHZ encoding using all-to-all interactions, which prepares the GHZ state in a special case, and is furthermore useful for quantum error correction, interaction-rate enhancement, and transmitting information using power-law interactions. The naive protocol based on parallelizing cnot gates takes O(1)-time of Hamiltonian evolution. In this work, we propose a fast protocol that achieves GHZ encoding with high accuracy. The evolution time O(log^{2}N/N) almost saturates the theoretical limit Ω(logN/N). Moreover, the final state is close to the ideal encoded one with high fidelity >1-10^{-3}, up to large system sizes N≲2000. The protocol only requires a few stages of time-independent Hamiltonian evolution; the key idea is to use the data qubit as control, and to use fast spin-squeezing dynamics generated by e.g., two-axis twisting.