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arXiv:quant-ph/0603179 · Physical Review A

Quantum speed limit for perfect state transfer in one dimension

Man‐Hong Yung

Published Sep 14, 2006• 127 citations
arXivHTMLPDF

Abstract

The basic idea of spin-chain engineering for perfect quantum state transfer (QST) is to find a set of coupling constants in the Hamiltonian such that a particular state initially encoded on one site will evolve freely to the opposite site without any dynamical controls. The minimal possible evolution time represents a speed limit for QST. We prove that the optimal solution is the one simulating the precession of a spin in a static magnetic field. We also argue that, at least for solid-state systems where interactions are local, it is more realistic to characterize the computation power by the couplings than the initial energy.

DOI
10.1103/physreva.74.030303

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The basic idea of spin-chain engineering for perfect quantum state transfer (QST) is to find a set of coupling constants in the Hamiltonian such that a particular state initially encoded on one site will evolve freely to the opposite site without any dynamical controls. The minimal possible evolution time represents a speed limit for QST. We prove that the optimal solution is the one simulating the precession of a spin in a static magnetic field. We also argue that, at least for solid-state systems where interactions are local, it is more realistic to characterize the computation power by the couplings than the initial energy.
Christandl et al. have noted that the ddd-dimensional hypercube can be projected to a linear chain with d+1d+1d+1 sites so that, by considering fixed but different couplings between the qubits assigned to the sites, the perfect state transfer (PST) can be achieved over arbitrarily long distances in the chain [Phys. Rev. Lett. 92, 187902 (2004); Phys. Rev. A 71, 032312 (2005)]. In this work we consider distance-regular graphs as spin networks and note that any such network (not just the hypercube) can be projected to a linear chain and so can allow PST over long distances. We consider some particular spin Hamiltonians which are the extended version of those of Christandl et al. Then, by using techniques such as stratification of distance-regular graphs and spectral analysis methods, we give a procedure for finding a set of coupling constants in the Hamiltonians so that a particular state initially encoded on one site will evolve freely to the opposite site without any dynamical control, i.e., we show how to derive the parameters of the system so that PST can be achieved. It is seen that PST is only allowed in distance-regular spin networks for which, starting from an arbitrary vertex as reference vertex (prepared in the initial state which we wish to transfer), the last stratum of the networks with respect to the reference state contains only one vertex; i.e., stratification of these networks plays an important role which determines in which kinds of networks and between which vertices of them, PST can be allowed. As examples, the cycle network with even number of vertices and ddd-dimensional hypercube are considered in details and the method is applied for some important distance-regular networks.
One of the ambitious goals of artificial intelligence is to build a machine that outperforms human intelligence, even if limited knowledge and data are provided. Reinforcement learning (RL) provides one such possibility to reach this goal. In this work, we consider a specific task from quantum physics, i.e., quantum state transfer in a one-dimensional spin chain. The mission for the machine is to find transfer schemes with the fastest speeds while maintaining high transfer fidelities. The first scenario we consider is when the Hamiltonian is time independent. We update the coupling strength by minimizing a loss function dependent on both the fidelity and the speed. Compared with a scheme proven to be at the quantum speed limit for the perfect state transfer, the scheme provided by RL is faster while maintaining the infidelity below $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. In the second scenario where a time-dependent external field is introduced, we convert the state transfer process into a Markov decision process that can be understood by the machine. We solve it with the deep Q-learning algorithm. After training, the machine successfully finds transfer schemes with high fidelities and speeds, which are faster than previously known ones. These results show that reinforcement learning can be a powerful tool for quantum control problems.
Abstract Quantum state transfer (QST) is investigated beyond the nearest-neighbour coupling scheme in long spin- 1 2 linear chains. Exploiting the properties of the next-nearest neighbour Hamiltonian’s dispersion relation, it is shown that with minimal engineering, i.e. an on-site magnetic field on the two end sites and only a few symmetrically-modified end inter-site couplings, an average transfer fidelity above 99% can be achieved. To leading order, the required time scales linearly with the length of the chain. Such a fast, high-quality QST is based on the ballistic propagation of the wave packet centred in the linear region of the dispersion relation by means of the on-site magnetic field. At the same time, the wave packet width, modulated by the inter-site couplings at the chain ends, whose values are found via a carefully designed genetic algorithm, is constrained mostly in the linear region of the dispersion relation. Our coupling scheme is shown to hold for arbitrary values of the next-nearest inter-site coupling and can be straightforwardly applied to longer-range coupling schemes.
Abstract The speed limit of quantum state transfer (QST) in a system of interacting particles is not only important for quantum information processing, but also directly linked to Lieb–Robinson-type bounds that are crucial for understanding various aspects of quantum many-body physics. For strongly long-range interacting systems such as a fully-connected quantum computer, such a speed limit is still unknown. Here we develop a new quantum brachistochrone method that can incorporate inequality constraints on the Hamiltonian. This method allows us to prove an exactly tight bound on the speed of QST on a subclass of Hamiltonians experimentally realizable by a fully-connected quantum computer.
Quantum theory sets a bound on the speed of quantum evolution. The shortest time needed for a quantm system to evolve from an initial state to the target state is known as the quantum speed limit time. The study of this time in open and closed quantum systems has been the subject of much work in quantum information theory. Quantum speed limit time is inversely related to the evolution rate of a quantum process. This relation is such that the speed of evolution decreases with increasing quantum speed limit time and vice versa. In this work we study the QSL time for the case in which a qubit interacts with a quantum critical environment. We choose the environment to be an Ising spin chain in a transverse field. It is observed that for the quantum speed limit time has a direct relation with quantum coherence of the initial state of the system. We will also study the effect of perturbation coupling on quantum speed limit time. It is observed that the quantum speed limit time decreases with increasing the coupling parameter. It is also observed that the quantum speed limit time decreases with increasing the number of spin in spin chain.
In a one-dimensional Heisenberg chain, we show that there are no sets of coupling strengths such that the evolution perfectly transfers a quantum state between the two ends of the chain without the addition of magnetic fields. In lieu of perfect transfer, we consider a range of options for achieving high quality transfer, whether in finite time, or via "pretty good" transfer where one waits long times in the hope of getting arbitrarily close to perfect transfer. In attempting to engineer arbitrarily accurate transfer, we explore a new paradigm that facilitates time estimates for achieving any target accuracy εεε for the transfer.