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Physical Review A

Multiqubit maximally entangled states in the NMR model

Bin Zhou, Ruibao Tao, Shun-Qing Shen

Published Aug 19, 2004• 6 citations
Publisher

Abstract

A single-step operation is proposed to produce multiqubit maximally entangled states in the NMR model. In the scheme, all qubits are initially in the ground state, and one single pulse of a multifrequency coherent magnetic radiation is applied to manipulate simultaneously the ``active states'' that satisfy the resonant conditions while all other ``inactive states'' remain unchanged. An effective Hamiltonian is derived in a generalized rotating frame, which allows us to predict the time evolution of ``active states'' generated by the magnetic pulse. The magnetic pulse parameters, such as frequencies, phases, amplitudes, and duration time, are obtained analytically to implement a Bell state of two qubits and a Greenberger-Horne-Zeilinger state of three qubits. The scheme has been generalized to create an $N$-qubit entangled state. Two rules are found to calculate the magnetic pulse parameters numerically, which are required to realize entangled states for even and odd qubits, respectively. The rules are successfully checked in the cases of $4\ensuremath{\leqslant}N\ensuremath{\leqslant}10$. The relation of $\mathrm{ln}\phantom{\rule{0.3em}{0ex}}{t}_{0}$ and $\mathrm{ln}\phantom{\rule{0.3em}{0ex}}N$ is found to be linear, and the duration time ${t}_{0\phantom{\rule{0.3em}{0ex}}}$is approximately proportional to $\sqrt{N}$.

DOI
10.1103/physreva.70.022311

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A single-step operation is proposed to produce multiqubit maximally entangled states in the NMR model. In the scheme, all qubits are initially in the ground state, and one single pulse of a multifrequency coherent magnetic radiation is applied to manipulate simultaneously the active states'' that satisfy the resonant conditions while all other inactive states'' remain unchanged. An effective Hamiltonian is derived in a generalized rotating frame, which allows us to predict the time evolution of ``active states'' generated by the magnetic pulse. The magnetic pulse parameters, such as frequencies, phases, amplitudes, and duration time, are obtained analytically to implement a Bell state of two qubits and a Greenberger-Horne-Zeilinger state of three qubits. The scheme has been generalized to create an NNN-qubit entangled state. Two rules are found to calculate the magnetic pulse parameters numerically, which are required to realize entangled states for even and odd qubits, respectively. The rules are successfully checked in the cases of 4\ensuremath⩽N\ensuremath⩽104\ensuremath{\leqslant}N\ensuremath{\leqslant}104\ensuremath⩽N\ensuremath⩽10. The relation of lnt0\mathrm{ln}\phantom{\rule{0.3em}{0ex}}{t}_{0}lnt0​ and lnN\mathrm{ln}\phantom{\rule{0.3em}{0ex}}NlnN is found to be linear, and the duration time t0{t}_{0\phantom{\rule{0.3em}{0ex}}}t0​is approximately proportional to N\sqrt{N}N​.
We theoretically study the macroscopic quantum entanglement in superconducting flux qubits. A phase-coupling scheme is proposed to offer enough strength of interactions between qubits. It is shown that due to the two-qubit tunneling processes both the ground state and excited states of coupled two flux qubits can be Bell type states, maximally entangled, in experimentally accessible regimes. The parameter regimes for the Bell states are discussed in terms of external magnetic flux and Josephson coupling energies. We also investigate two types of genuine three-qubit entanglement, known as the Greenberger-Horne- Zeilinger(GHZ) and W states. While an excited state can be the W state, the GHZ state is formed at the ground state of the coupled three flux qubits. The GHZ and W states are shown to be robust against external flux fluctuations for feasible experimental realizations.
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We propose a way for generating nnn-qubit Greenberger-Horne-Zeilinger (GHZ) entangled states with a three-level qubit system and $(n\ensuremath{-}1)$ four-level qubit systems in a cavity. This proposal does not require identical qubit-cavity coupling constants and thus is tolerant to qubit-system parameter nonuniformity and nonexact placement of qubits in a cavity. The proposal does not require adjustment of the qubit-system level spacings during the entire operation. Moreover, it is shown that entanglement can be deterministically generated using this method and the operation time is independent of the number of qubits. The present proposal is quite general, which can be applied to physical systems such as various types of superconducting devices coupled to a resonator or atoms trapped in a cavity.
Quantum simulations, where controllable quantum systems are used to simulate other quantum systems, originally proposed by Richard Feynman, are one of the most remarkable applications of quantum information science. Compared to computation, quantum simulations require much less number of qubits for the m to be practical. In the work described in this thesis, we have performed a few quantum simulations of small quantum systems using Nuclear Magnetic Resonance(NMR) techniques. These simulations have been used to experimentally demonstrate the underlying interesting quantum protocols. All the experiments presented have been carried out using liquid-state or liquid crystal NMR. Numerical pulse optimization techniques have been utilized in some of the experiments, to achieve better control over the spin systems. The first chapter contains “Introduction” to quantum information processing, NMR, and numerical pulse optimization techniques. In chapter 2, we describe quantum simulation of a 3-spin Heisenberg-XY spin chain having only nearest neighbour interactions. Recently, spin chains having pre-engineered short-range interactions have been proposed to efficiently transfer quantum information between different parts of a quantum information processor. Other important proposals involving these spin chains include generating entangled states and universal quantum computation. However, such engineered interactions do not occur naturally in any system. In such a scenario, the experimental viability of these proposals can be tested by simulating the spin chains in other controllable quantum systems. In this work, we first theoretically study the time evolution of bipartite and tripartite entanglement measures for a 3-spin open ended XY spin chain. Then, by simulating the XY interactions in a 3-spin nuclear spin system, we experimentally generate, (i)a bipartite maximally(pseudo-)entangled state(Bell state) between end qubits, and(ii) multipartite(pseudo-)entangled states(Wand GHZ states),starting from separable pseudo-pure states. Bell state has been generated by using only the natural unitary evolution of the XY spin chain. W-state and GHZ-state have been generated by applying a single-qubit rotation to the second qubit, and a global rotation of all the three qubits respectively after the unitary evolution of the spin chain. In chapter 3, we simulate a 3-spin quantum transverse Ising spin system in a triangular configuration, and show that multipartite quantum correlations can be used to distinguish between the frustrated and non-frustrated regimes in the ground state of this spin system. The ground state of the spin system has been prepared by using adiabatic state preparation method. Gradient ascent pulse engineering technique has been utilized to efficiently realize the adiabatic evolution of the spin system. To analyse the experimental ground state of the system, we employ two different multipartite quantum correlation measures, generated from monogamy studies of bipartite quantum correlations. Chapter 4 contains a digital…
Implementing high-fidelity multiqubit gates is critical for reducing circuit depth in near-term quantum processors and fault-tolerant architectures. However, realizing multiqubit interactions and entanglement remains a critical challenge that limits the implementation of multiqubit gates. Here we propose an all-microwave scheme to realize single-step multiqubit gates incorporating n control and m target qubits, tailored for frequency-tunable superconducting transmon networks. By leveraging cross-resonance (CR) drives, this approach induces effective two-body ZX interactions that are significantly stronger than those achieved in conventional resonant regimes, circumventing the need for tunable couplers. The system's effective Hamiltonian is analytically derived using both quad frame rotation and rigorous block diagonalization. These combined theoretical methods facilitate the identification of optimal parameter regimes that simultaneously enhance desired target couplings and suppress parasitic higher-order terms, such as ZXX interactions. Through numerical simulations and gradient-based pulse-shape optimization, we demonstrate three-qubit gates achieving fidelities exceeding 99.9% within a 60 ns gate duration. Furthermore, we evaluate the fundamental scalability of this framework by extending the optimization to a five-qubit CXXXX architecture, confirming the physical viability of the multi-target driving scheme. Because these multiqubit operations naturally emulate stabilizer measurements, this architecture provides a hardware-efficient strategy for reducing the circuit depth of stabilizer-type operations by replacing sequential two-qubit gate decompositions with single-step multiqubit gates.