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arXiv:1012.3695 · IEEE Transactions on Automatic Control

Symmetry and Subspace Controllability for Spin Networks With a Single-Node Control

Xiaoting Wang, Peter J. Pemberton-Ross, S. G. Schirmer

Published May 31, 2012• 41 citations

Abstract

We consider the relation of symmetries and subspace controllability for spin-1/2 networks with XXZ couplings, subject to perturbation of a single node by a local potential (Z-control). The Hamiltonians for such networks decompose into excitation subspaces. Focusing on the single excitation subspace, it is shown for single-node Z-controls that external symmetries are characterized by eigenstates of the system Hamiltonian which have zero overlap with the control node, and there are no internal symmetries. It is further shown that there are symmetries which persist even in the presence of random perturbations. For XXZ chains with uniform coupling strengths, a characterization of all possible symmetries is given which shows a strong dependence on the position of the node we control. We then show for Heisenberg and XX chains with uniform coupling strength subject to single-node Z-control that the lack of symmetry is both necessary and sufficient for subspace controllability. Finally, the latter approach is generalized to establish controllability results for simple branched networks.

DOI: 10.1109/tac.2012.2202057

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We consider the relation of symmetries and subspace controllability for spin-1/2 networks with XXZ couplings, subject to perturbation of a single node by a local potential (Z-control). The Hamiltonians for such networks decompose into excitation subspaces. Focusing on the single excitation subspace, it is shown for single-node Z-controls that external symmetries are characterized by eigenstates of the system Hamiltonian which have zero overlap with the control node, and there are no internal symmetries. It is further shown that there are symmetries which persist even in the presence of random perturbations. For XXZ chains with uniform coupling strengths, a characterization of all possible symmetries is given which shows a strong dependence on the position of the node we control. We then show for Heisenberg and XX chains with uniform coupling strength subject to single-node Z-control that the lack of symmetry is both necessary and sufficient for subspace controllability. Finally, the latter approach is generalized to establish controllability results for simple branched networks.

We develop a technique to prove simultaneous subspace controllability on multiple invariant subspaces, which specifically enables us study the controllability properties of spin systems that are not amenable to standard controllability arguments based on energy level connectivity graphs or simple induction arguments on the length of the chain. The technique is applied to establish simultaneous subspace controllability for Heisenberg spin chains subject to limited local controls. This model is theoretically important and the controllability result shows that a single control can be sufficient for complete controllability of an exponentially large subspace and universal quantum computation in the exponentially large subspace. The controllability results are extended to prove subspace controllability in the presence of control field leakage and discuss minimal control resources required to achieve controllability over the entire spin chain space.

We consider a class of spin networks where each spin in a certain set interacts via Ising coupling with a single central spin. This is a common situation for instance in NV centers in diamonds. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invariant subspaces and the dynamical Lie algebra of this class of systems and prove subspace controllability.

We consider a class of spin networks where each spin in a certain set interacts, via Ising coupling, with a set of central spins, and the control acts simultaneously on all the spins. This is a common situation for instance in NV centers in diamonds, and we focus on the physical case of up to two central spins. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invariant subspaces and the dynamical Lie algebra of this class of systems and prove subspace controllability in every case.

We investigate the universality of multi-spin systems in architectures of various symmetries of coupling type and topology. Explicit reachability sets under symmetry constraints are provided. Thus for a given (possibly symmetric) experimental coupling architecture several decision problems can be solved in a unified way: (i) can a target Hamiltonian be simulated? (ii) can a target gate be synthesised? (iii) to which extent is the system observable by a given set of detection operators? and, as a special case of the latter, (iv) can an underlying system Hamiltonian be identified with a given set of detection operators? Finally, in turn, lack of symmetry provides a convenient necessary condition for full controllability. Though often easier to assess than the well-established Lie-algebra rank condition, this is not sufficient unless the candidate dynamic simple Lie algebra can be pre-identified uniquely, which is fortunately less complicated than expected.

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.

We show that we can achieve global density-operator controllability for most

N

-dimensional bilinear Hamiltonian control systems with general fixed couplings using a single, locally acting actuator that modulates one energy-level transition. Controllability depends upon the position of the actuator and relies on the absence of either decompositions into noninteracting subgroups or symmetries restricting the dynamics to a subgroup of

\mathrm{SU}(N)

. These results are applied to spin-chain systems and used to explicitly construct control sequences for a single binary-valued switch actuator.