Complete Characterization of the Ground Space Structure of Two-Body Frustration-Free Hamiltonians for Qubits

Physical Review A (Impact Factor: 2.81). 10/2010; 84(4). DOI: 10.1103/PhysRevA.84.042338
Source: arXiv


The problem of finding the ground state of a frustration-free Hamiltonian
carrying only two-body interactions between qubits is known to be solvable in
polynomial time. It is also shown recently that, for any such Hamiltonian,
there is always a ground state that is a product of single- or two-qubit
states. However, it remains unclear whether the whole ground space is of any
succinct structure. Here, we give a complete characterization of the ground
space of any two-body frustration-free Hamiltonian of qubits. Namely, it is a
span of tree tensor network states of the same tree structure. This
characterization allows us to show that the problem of determining the ground
state degeneracy is as hard as, but no harder than, its classical analog.

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    • "This value is positive if and only if H is frustration-free, and greater than one if H is also degenerate. The name #2-QSAT is chosen (see also Ref. [12]) in analogy to the problem #2-SAT of counting the satisfying assignments to an instance of 2-SAT . The dimension of the ground-state manifold of a frustration-free spin-1/2 Hamiltonian is simply the size of a basis for the solution space: if the projectors h u,v are all diagonal operators, this problem is #2-SAT. "
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    ABSTRACT: The problem 2-quantum-satisfiability (2-QSAT) is the generalisation of the 2-CNF-SAT problem to quantum bits, and is equivalent to determining whether or not a spin-1/2 Hamiltonian with two-body terms is frustration-free. Similarly to the classical problem #2-SAT, the counting problem #2-QSAT of determining the size (i.e. the dimension) of the set of satisfying states is #P-complete. However, if we consider random instances of 2-QSAT in which constraints are sampled from the Haar measure, intractible instances have measure zero. An apparent reason for this is that almost all two-qubit constraints are entangled, which more readily give rise to long-range constraints. We investigate under what conditions product constraints also give rise to families of #2-QSAT instances which are efficiently solvable, by considering #2-QSAT involving only discrete distributions over tensor product operators. This special case of #2-QSAT interpolates between classical #2-SAT, and #2-QSAT involving arbitrary product constraints. We find that such instances of #2-QSAT, defined on Erdos--Renyi graphs or bond-percolated lattices, are almost surely efficiently solvable except to the extent that they are biased to resemble monotone instances #2-SAT.
    Preview · Article · Mar 2014
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    • "As in the classical case, it is conjectured that a satisfiability threshold α c (k) exists, above which the probability that a random instance is satisfiable approaches zero as n → ∞ and below which this probability approaches one [17]. Some bounds on this threshold value have been proven using a quantum version of the Lovász local lemma [3] and by using graph-theoretic techniques [6] but only the case k = 2 is fully understood [17] [11]. Other previous work has focused on quantum satisfiability with qudit variables of dimension d > 2 [20] [22] [8] [5] or in restricted geometries [20] [7]. "
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    • "Chen et al. provided a no-go theorem showing that there does not exist a unique ground state for a two-body frustration-free Hamiltonian that can simultaneously be a resource state for MBQC [32] for a qubit system. It was further shown by Ji et al. that indeed for two-level systems , the structure of the ground-state space for any two-body frustration-free Hamiltonian can be fully characterized [33] and none of these states corresponds to a resource state for MBQC. These results herald bad news for the practical realization of MBQC based on VBS resource states since spin-1/2 systems appear to be the most prevalent systems in nature. "
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    ABSTRACT: Measurement-based quantum computing (MBQC) is a model of quantum computing that proceeds by sequential measurements of individual spins in an entangled resource state. However, it remains a challenge to produce efficiently such resource states. Would it be possible to generate these states by simply cooling a quantum many-body system to its ground state? Cluster states, the canonical resource states for MBQC, do not occur naturally as unique ground states of physical systems. This inherent hurdle has led to a significant effort to identify alternative resource states that appear as ground states in spin lattices. Recently, some interesting candidates have been identified with various valence-bond-solid (VBS) states. In this review, we provide a pedagogical introduction to recent progress regarding MBQC with VBS states as possible resource states. This study has led to an interesting interdisciplinary research area at the interface of quantum information science and condensed matter physics.
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