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arXiv:0805.3975 · Physical Review A

Local controllability of quantum networks

Daniel Klaus Burgarth, Sougato Bose, Christoph Bruder, Vittorio Giovannetti

Published Jun 15, 2009• 97 citations
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Abstract

We give a sufficient criterion that guarantees that a many-body quantum system can be controlled by properly manipulating the (local) Hamiltonian of one of its subsystems. The method can be applied to a wide range of systems: it does not depend on the details of the couplings but only on their associated topology. As a special case, we prove that Heisenberg and Affleck-Kennedy-Lieb-Tasaki chains can be controlled by operating on one of the spins at their ends. In principle, arbitrary quantum algorithms can be performed on such chains by acting on a single qubit.

DOI
10.1103/physreva.79.060305

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We give a sufficient criterion that guarantees that a many-body quantum system can be controlled by properly manipulating the (local) Hamiltonian of one of its subsystems. The method can be applied to a wide range of systems: it does not depend on the details of the couplings but only on their associated topology. As a special case, we prove that Heisenberg and Affleck-Kennedy-Lieb-Tasaki chains can be controlled by operating on one of the spins at their ends. In principle, arbitrary quantum algorithms can be performed on such chains by acting on a single qubit.
Key properties of a physical system depend on whether it is gapped, i.e., whether its spectral gap has a positive lower bound that is independent of system size. Here, we provide a prescription for renormalizing a spin chain Hamiltonian in such a way that the renormalized Hamiltonian is gapped if and only if the original Hamiltonian is gapped. Then, we articulate a set of conditions that guarantees the renormalized Hamiltonian is gapped. These conditions are built on a certain strong notion of decaying correlations involving an operator norm of non-commuting terms in the renormalized Hamiltonian. We apply the method to show that two interesting models, with forms motivated by quantum circuits, are gapped. We also confirm the generality of the method by successfully applying it to a somewhat different case, the well-known Affleck-Kennedy-Lieb-Tasaki (AKLT) model.
We propose and analyze a new approach to the coherent control and manipulation of quantum degrees of freedom in disordered, interacting systems in the many-body localized phase. Our approach leverages a number of unique features of many-body localization: a lack of thermalization, a locally gapped spectrum, and slow dephasing. Using the technique of quantum phase estimation, we demonstrate a protocol that enables the local preparation of a many-body system into an effective eigenstate. This leads to the ability to encode information and control interactions without full microscopic knowledge of the underlying Hamiltonian. Finally, we analyze the effects of weak coupling to an external bath and provide an estimate for the fidelity of our protocol.
We provide a complete classification of the integrability and nonintegrability of the spin-1 bilinear-biquadratic model with a uniaxial anisotropic field, which includes the Heisenberg model and the Affleck-Kennedy-Lieb-Tasaki model. It is rigorously shown that, within this class, the only integrable systems are those that have been solved by the Bethe ansatz method, and that all other systems are nonintegrable, in the sense that they do not have nontrivial local conserved quantities. Here, “nontrivial” excludes quantities like the Hamiltonian or the total magnetization, and “local” refers to sums of operators that act only on sites within a finite distance. This result establishes the nonintegrability of the Affleck-Kennedy-Lieb-Tasaki model and, consequently, demonstrates that the quantum many-body scars observed in this model emerge independently of any conservation laws of local quantities. Furthermore, we extend the proof of nonintegrability to more general spin-1 models that encompass anisotropic extensions of the bilinear-biquadratic Hamiltonian and completely classified the integrability of generic Hamiltonians that possess translational symmetry, U ( 1 ) symmetry, time-reversal symmetry, and spin-flip symmetry. Our result has accomplished a breakthrough in nonintegrability proofs by expanding their scope to spin-1 systems.
If the interaction between qubits is not diagonal in the computational basis (e.g., the Heisenberg interaction), then one must be able to "switch it off" in order to prevent uncontrolled propagation of states. Therefore, schemes for quantum computing typically demand local control of the interaction strength between each pair of neighboring qubits. Here we demonstrate that this degree of control is not necessary: it suffices to switch the interaction collectively, something that can in principle be achieved by global fields rather than with local manipulations. This observation may offer a significant simplification for various implementations, including solid state and optical lattice schemes.
Affleck-Kennedy-Lieb-Tasaki (AKLT) states are an important class of many-body quantum states that are useful in quantum information processing, including measurement-based quantum computation in particular. Here we propose a general approach for constructing efficient verification protocols for AKLT states on arbitrary graphs with local spin measurements. Our verification protocols build on bond verification protocols and matching covers (including edge coloring) of the underlying graphs, which have a simple geometric and graphic picture. We also provide rigorous performance guarantee that is required for practical applications. With our approach, most AKLT states of wide interest, including those defined on one- and two-dimensional lattices, can be verified with a constant sample cost, which is independent of the system size and is dramatically more efficient than all previous approaches. As an illustration, we construct concrete verification protocols for AKLT states on various lattices and on arbitrary graphs up to five vertices.