Article
Complete Characterization of the Ground Space Structure of TwoBody FrustrationFree Hamiltonians for Qubits
Physical Review A (Impact Factor: 2.81). 10/2010; 84(4). DOI: 10.1103/PhysRevA.84.042338
Source: arXiv
ABSTRACT
The problem of finding the ground state of a frustrationfree Hamiltonian
carrying only twobody 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 twoqubit
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 twobody frustrationfree 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.
carrying only twobody 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 twoqubit
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 twobody frustrationfree 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.
Fulltext preview
ArXiv Available from: Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.

 "This value is positive if and only if H is frustrationfree, and greater than one if H is also degenerate. The name #2QSAT is chosen (see also Ref. [12]) in analogy to the problem #2SAT of counting the satisfying assignments to an instance of 2SAT . The dimension of the groundstate manifold of a frustrationfree spin1/2 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 #2SAT. "
[Show abstract] [Hide abstract]
ABSTRACT: The problem 2quantumsatisfiability (2QSAT) is the generalisation of the 2CNFSAT problem to quantum bits, and is equivalent to determining whether or not a spin1/2 Hamiltonian with twobody terms is frustrationfree. Similarly to the classical problem #2SAT, the counting problem #2QSAT of determining the size (i.e. the dimension) of the set of satisfying states is #Pcomplete. However, if we consider random instances of 2QSAT in which constraints are sampled from the Haar measure, intractible instances have measure zero. An apparent reason for this is that almost all twoqubit constraints are entangled, which more readily give rise to longrange constraints. We investigate under what conditions product constraints also give rise to families of #2QSAT instances which are efficiently solvable, by considering #2QSAT involving only discrete distributions over tensor product operators. This special case of #2QSAT interpolates between classical #2SAT, and #2QSAT involving arbitrary product constraints. We find that such instances of #2QSAT, defined on ErdosRenyi graphs or bondpercolated lattices, are almost surely efficiently solvable except to the extent that they are biased to resemble monotone instances #2SAT. 
 "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 graphtheoretic 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]. "
Article: Quantum 3SAT is QMA1complete
[Show abstract] [Hide abstract]
ABSTRACT: Quantum satisfiability is a constraint satisfaction problem that generalizes classical boolean satisfiability. In the quantum kSAT problem, each constraint is specified by a klocal projector and is satisfied by any state in its nullspace. Bravyi showed that quantum 2SAT can be solved efficiently on a classical computer and that quantum kSAT with k greater than or equal to 4 is QMA1complete. Quantum 3SAT was known to be contained in QMA1, but its computational hardness was unknown until now. We prove that quantum 3SAT is QMA1hard, and therefore complete for this complexity class. 
 "Chen et al. provided a nogo theorem showing that there does not exist a unique ground state for a twobody frustrationfree 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 twolevel systems , the structure of the groundstate space for any twobody frustrationfree 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 spin1/2 systems appear to be the most prevalent systems in nature. "
[Show abstract] [Hide abstract]
ABSTRACT: Measurementbased 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 manybody 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 valencebondsolid (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.