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arXiv:1202.1033 · Physical Review A

Subspace controllability of spin- 12 chains with symmetries

Xiaoting Wang, Daniel Klaus Burgarth, S. G. Schirmer

Published Nov 14, 2016• 39 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.94.052319

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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 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 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 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.
Abstract Recent work has characterized rigorously what it means for one quantum system to simulate another and demonstrated the existence of universal Hamiltonians—simple spin lattice Hamiltonians that can replicate the entire physics of any other quantum many-body system. Previous universality results have required proofs involving complicated ‘chains’ of perturbative ‘gadgets.’ In this paper, we derive a significantly simpler and more powerful method of proving universality of Hamiltonians, directly leveraging the ability to encode quantum computation into ground states. This provides new insight into the origins of universal models and suggests a deep connection between universality and complexity. We apply this new approach to show that there are universal models even in translationally invariant spin chains in 1D. This gives as a corollary a new Hamiltonian complexity result that the local Hamiltonian problem for translationally invariant spin chains in one dimension with an exponentially small promise gap is PSPACE-complete. Finally, we use these new universal models to construct the first known toy model of 2D–1D holographic duality between local Hamiltonians.
Sufficient conditions for complete controllability of N-level quantum systems subject to a single control pulse that addresses multiple allowed transitions concurrently are established. The results are applied in particular to Morse and harmonic-oscillator systems, as well as some systems with degenerate energy levels. Controllability of these model systems is of special interest since they have many applications in physics, e.g., Morse and harmonic oscillators serve as models for molecular bonds, and the standard control approach of using a sequence of frequency-selective pulses to address a single transition at a time is either not applicable or only of limited utility for such systems.
For many quantum information protocols such as state transfer, entanglement transfer, and entanglement generation, standard notions of controllability for quantum systems are too strong. We introduce the weaker notion of accessible pairs and prove an upper bound on the achievable fidelity of a transformation between a pair of states based on the symmetries of the system. A large class of spin networks is presented for which this bound can be saturated. In this context, we show how the inaccessible dark states for a given excitation-preserving evolution can be calculated and illustrate how some of these can be accessed using extra catalytic excitations. This emphasizes that it is not sufficient for analyses of state transfer in spin networks to restrict to the single excitation subspace. One class of symmetries in these spin networks is exactly characterized in terms of the underlying graph properties.