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arXiv:1110.6583 · Physical Review A

From ground states to local Hamiltonians

Jianxin Chen, Zhengfeng Ji, Bei Zeng, Dong Zhou

Published Aug 30, 2012• 31 citations
arXivHTMLPDF

Abstract

Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interested in local Hamiltonians with certain interaction patterns, such as nearest-neighbor interactions on some types of lattices. A necessary condition for a space $V$ to be the ground-state space of some local Hamiltonian with a given interaction pattern is that the maximally mixed state supported on $V$ is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have $V$ satisfying the necessary condition mentioned above as its ground-state space by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles.

DOI
10.1103/physreva.86.022339

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Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interested in local Hamiltonians with certain interaction patterns, such as nearest-neighbor interactions on some types of lattices. A necessary condition for a space VVV to be the ground-state space of some local Hamiltonian with a given interaction pattern is that the maximally mixed state supported on VVV is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have VVV satisfying the necessary condition mentioned above as its ground-state space by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles.
The structure of the ground spaces of quantum systems consisting of local interactions is of fundamental importance to different areas of physics. In this Letter, we present a necessary and sufficient condition for a subspace to be the ground space of a k-local Hamiltonian. Our analysis are motivated by the concept of irreducible correlations studied by [Linden et al., PRL 89, 277906] and [Zhou, PRL 101, 180505], which is in turn based on the principle of maximum entropy. It establishes a better understanding of the ground spaces of local Hamiltonians and builds an intimate link of ground spaces to the correlations of quantum states.
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 Θk 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 Θk, called atoms, are analogs of extreme points. We study the properties of atoms in Θk 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 Θ2 are unique ground states of some 2-local frustration-free Hamiltonians. Moreover, we show that the elements in Θk may not be the join of atoms, indicating a richer structure for Θk beyond the convex structure. Our study of Θk deepens the understanding of ground-state space properties for frustration-free Hamiltonians, from the new perspective of reduced spaces.
A broad range of quantum optimization problems can be phrased as the question of whether a specific system has a ground state at zero energy, i.e., whether its Hamiltonian is frustration-free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms to, at least, partially answer this question. Here we prove a general criterion-a sufficient condition-under which a local Hamiltonian is guaranteed to be frustration-free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hardcore lattice gas at negative fugacity on the Hamiltonian's interaction graph, which, as a statistical mechanics problem, is of interest in its own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass physics that permit us to obtain new bounds on the satisfiable to unsatisfiable transition in random quantum satisfiability. We are then led to natural conjectures for when such bounds will be tight, as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this work underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.
A central problem in many-body quantum physics is the determination of the ground state of a thermodynamically large physical system. We construct a cluster expansion for ground states of local Hamiltonians, which naturally incorporates physical requirements inherited by locality as conditions on its cluster amplitudes. Applying a diagrammatic technique we derive the relation of these amplitudes to thermodynamic quantities and local observables. Moreover we derive a set of functional equations that determine the cluster amplitudes for a general Hamiltonian, verify the consistency with perturbation theory and discuss non-perturbative approaches. Lastly we verify the persistence of locality features of the cluster expansion under unitary evolution with a local Hamiltonian and provide applications to out-of-equilibrium problems: a simplified proof of equilibration to the GGE and a cumulant expansion for the statistics of work, for an interacting-to-free quantum quench.
A broad range of quantum optimisation problems can be phrased as the question whether a specific system has a ground state at zero energy, i.e. whether its Hamiltonian is frustration free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms in order to, at least, partially answer this question. Here we prove a general criterion -- a sufficient condition -- under which a local Hamiltonian is guaranteed to be frustration free by lifting Shearer's theorem from classical probability theory to the quantum world. Remarkably, evaluating this condition proceeds via a fully classical analysis of a hard-core lattice gas at negative fugacity on the Hamiltonian's interaction graph which as a statistical mechanics problem is of interest in their own right. We concretely apply this criterion to local Hamiltonians on various regular lattices, while bringing to bear the tools of spin glass theory which permit us to obtain new bounds on the SAT/UNSAT transition in random quantum satisfiability. These also lead us to natural conjectures for when such bounds will be exact (tight), as well as to a novel notion of universality for these computer science problems. Besides providing concrete algorithms leading to detailed and quantitative insights, this underscores the power of marrying classical statistical mechanics with quantum computation and complexity theory.
We consider the problem of reconstructing global quantum states from local data. Because the reconstruction problem has many solutions in general, we consider the reconstructed state of maximum global entropy consistent with the local data. We show that unique ground states of local Hamiltonians are exactly reconstructed as the maximal entropy state. More generally, we show that if the state in question is a ground state of a local Hamiltonian with a degenerate space of locally indistinguishable ground states, then the maximal entropy state is close to the ground state projector. We also show that local reconstruction is possible for thermal states of local Hamiltonians. Finally, we discuss a procedure to certify that the reconstructed state is close to the true global state. We call the entropy of our reconstructed maximum entropy state the "reconstruction entropy," and we discuss its relation to emergent geometry in the context of holographic duality.