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arXiv:1211.2549 · Journal of Physics A Mathematical and Theoretical

The quantum brachistochrone problem for two spins-$\frac{1}{2}$ with anisotropic Heisenberg interaction

Andrij Romanovych Kuzmak, V M Tkachuk

Published Apr 3, 2013• 17 citations
arXivHTMLPDF

Abstract

We study the quantum brachistochrone evolution for a system of two spins-$\frac{1}{2}$ described by an anisotropic Heisenberg Hamiltonian without $zx$, $zy$ interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by $\|\uparrow\uparrow\rangle$, $\|\downarrow\downarrow\rangle$ and $\|\uparrow\downarrow\rangle$, $\|\downarrow\uparrow\rangle$ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an entangler gate, SWAP gate, iSWAP gate et al.

DOI
10.1088/1751-8113/46/15/155305

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The quantum brachistochrone problem for two spins-12\frac{1}{2}21​ with anisotropic Heisenberg interaction
We study the quantum brachistochrone evolution for a system of two spins-12\frac{1}{2}21​ described by an anisotropic Heisenberg Hamiltonian without zxzxzx, zyzyzy interacting couplings in magnetic field directed along the z-axis. This Hamiltonian realizes quantum evolution in two subspaces spanned by ∥↑↑⟩\|\uparrow\uparrow\rangle∥↑↑⟩, ∥↓↓⟩\|\downarrow\downarrow\rangle∥↓↓⟩ and ∥↑↓⟩\|\uparrow\downarrow\rangle∥↑↓⟩, ∥↓↑⟩\|\downarrow\uparrow\rangle∥↓↑⟩ separately and allows to consider the brachistochrone problem on each subspace separately. Using the evolution operator for this Hamiltonian we generate quantum gates, namely an entangler gate, SWAP gate, iSWAP gate et al.
We consider a physical system consisting of two interacting spins governed by the [Formula: see text]-type Heisenberg Hamiltonian in an external magnetic field. We investigate the quantum evolution and the Riemannian geometry of the two-spin state space by means of the relevant Fubini–Study metric. The geometrical phase accumulated by the two-spin state is also examined under arbitrary and cyclic evolutions. By computing the evolution speed and the corresponding geodesic distance, we solve the quantum brachistochrone problem. The entanglement between the two spins is also studied via two approaches: the first one deals with the entanglement effect on the Fubini–Study metric and the geometrical phase, while the second one treats the entanglement effect on the evolution speed and the corresponding geodesic distance. Finally, we solve the quantum brachistochrone problem using the entanglement degree.
We consider the two-step method [A. R. Kuzmak and V. M. Tkachuk, Phys. Lett. A 378 (2014) 1469] for preparation of an arbitrary quantum gate on two spins with anisotropic Heisenberg interaction. At the first step, the system evolves during some period of time. At the second step, we apply pulsed magnetic field individually to each spin. We obtain the conditions for realization of SWAP, iSWAP, [Formula: see text] and entangled gates. Finally, we consider the implementation of this method on the physical system of ultracold atoms in optical lattice.
In the present paper an influence of the anisotropic antisymmetric exchange interaction, the Dzialoshinskii-Moriya (DM) interaction, on entanglement of two qubits in various magnetic spin models, including the pure DM model and the most general XYZ model are studied. We find that the time evolution generated by DM interaction can implement the SWAP gate and discuss realistic quasi-one-dimensional magnets where it can be realized. It is shown that inclusion of the DM interaction to any Heisenberg model creates, when it does not exist, or strengthens, when it exists, the entanglement. We give physical explanation of these results by studying the ground state of the systems at T=0. Nonanalytic dependence of the concurrence on the DM interaction and its relation with quantum phase transition is indicated. Our results show that spin models with the DM coupling have some potential applications in quantum computations and the DM interaction could be an efficient control parameter of entanglement.
In the present paper, an influence of the anisotropic antisymmetric exchange interaction, the Dzialoshinskii–Moriya (DM) interaction, on entanglement of two qubits in various magnetic spin models, including the pure DM model and the most general XYZ model, are studied. We find that the time evolution generated by DM interaction can implement the SWAP gate and discuss realistic quasi-one-dimensional magnets where it can be realized. It is shown that inclusion of the DM interaction to any Heisenberg model creates, when it does not exist, or strengthens, when it exists, the entanglement. We give physical explanation of these results by studying the ground state of the systems at T = 0. Nonanalytic dependence of the concurrence on the DM interaction and its relation with quantum phase transition is indicated. Our results show that spin models with the DM coupling have some potential applications in quantum computations and the DM interaction could be an efficient control parameter of entanglement.
Spin model plays an important role in the realization of quantum information processing. Quantum entanglement can be constructed, also （swap）n quantum logic gate is available if anisotropy is ignored, when Heisenberg interaction between spins is used; thus a complete quantum computing gate will be realized by combining single-qubit rotation gate. To study the influence of anisotropy on logic gate has high academic value since anisotropy is inevitable in any solid material. In this paper, swap operation in a two-qubit anisotropy XYZ model in the presence of an inhomogeneous magnetic field is discussed. The swap errors caused by the anisotropy parameter ν of the exchange interaction between spins and the inhomogeneous field δ are evaluated. We can see clearly when the swap operation can be approximately used as an ideal swap gate.
Abstract We analytically investigate the role of entanglement in time-optimal state evolution as an application of the quantum brachistochrone, a general method for obtaining the optimal time-dependent Hamiltonian for reaching a target quantum state. As a model, we treat two qubits indirectly coupled through an intermediate qubit that is directly controllable, which represents a typical situation in quantum information processing. We find the time-optimal unitary evolution law and quantify residual entanglement by the two-tangle between the indirectly coupled qubits, for all possible sets of initial pure quantum states of a tripartite system. The integrals of the motion of the brachistochrone are determined by fixing the minimal time at which the residual entanglement is maximized. Entanglement plays a role for W and Greenberger–Horne–Zeilinger ( GHz ) initial quantum states, and for the bi-separable initial state in which the indirectly coupled qubits have a nonzero value of the 2-tangle.