Article

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

Physical Review A (Impact Factor: 3.04). 10/2010; DOI: 10.1103/PhysRevA.84.042338

Source: arXiv

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**ABSTRACT:**We study the ground-state space properties for frustration-free Hamiltonians. We introduce a concept of `reduced spaces' to characterize local structures of ground-state spaces. For a many-body system, we characterize mathematical structures for the set $\Theta_k$ of all the $k$-particle reduced spaces, which with a binary operation called join forms a semilattice that can be interpreted as an abstract convex structure. The smallest nonzero elements in $\Theta_k$, called atoms, are analogs of extreme points. We study the properties of atoms in $\Theta_k$ and discuss its relationship with ground states of $k$-local frustration-free Hamiltonians. For spin-1/2 systems, we show that all the atoms in $\Theta_2$ are unique ground states of some 2-local frustration-free Hamiltonians. Moreover, we show that the elements in $\Theta_k$ may not be the join of atoms, indicating a richer structure for $\Theta_k$ beyond the convex structure. Our study of $\Theta_k$ deepens the understanding of ground-state space properties for frustration-free Hamiltonians, from a new angle of reduced spaces.Journal of Mathematical Physics 12/2011; 53(10). · 1.30 Impact Factor - [Show abstract] [Hide abstract]

**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.International Journal of Modern Physics B 11/2011; 26(02). · 0.46 Impact Factor -
##### Article: Quantum 3-SAT is QMA1-complete

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**ABSTRACT:**Quantum satisfiability is a constraint satisfaction problem that generalizes classical boolean satisfiability. In the quantum k-SAT problem, each constraint is specified by a k-local projector and is satisfied by any state in its nullspace. Bravyi showed that quantum 2-SAT can be solved efficiently on a classical computer and that quantum k-SAT with k greater than or equal to 4 is QMA1-complete. Quantum 3-SAT was known to be contained in QMA1, but its computational hardness was unknown until now. We prove that quantum 3-SAT is QMA1-hard, and therefore complete for this complexity class.Foundations of Computer Science, 1975., 16th Annual Symposium on 02/2013;

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