53 research outputs found
A Constructive Quantum Lov\'asz Local Lemma for Commuting Projectors
The Quantum Satisfiability problem generalizes the Boolean satisfiability
problem to the quantum setting by replacing classical clauses with local
projectors. The Quantum Lov\'asz Local Lemma gives a sufficient condition for a
Quantum Satisfiability problem to be satisfiable [AKS12], by generalizing the
classical Lov\'asz Local Lemma.
The next natural question that arises is: can a satisfying quantum state be
efficiently found, when these conditions hold? In this work we present such an
algorithm, with the additional requirement that all the projectors commute. The
proof follows the information theoretic proof given by Moser's breakthrough
result in the classical setting [Mos09].
Similar results were independently published in [CS11,CSV13]
The Quantum PCP Conjecture
The classical PCP theorem is arguably the most important achievement of
classical complexity theory in the past quarter century. In recent years,
researchers in quantum computational complexity have tried to identify
approaches and develop tools that address the question: does a quantum version
of the PCP theorem hold? The story of this study starts with classical
complexity and takes unexpected turns providing fascinating vistas on the
foundations of quantum mechanics, the global nature of entanglement and its
topological properties, quantum error correction, information theory, and much
more; it raises questions that touch upon some of the most fundamental issues
at the heart of our understanding of quantum mechanics. At this point, the jury
is still out as to whether or not such a theorem holds. This survey aims to
provide a snapshot of the status in this ongoing story, tailored to a general
theory-of-CS audience.Comment: 45 pages, 4 figures, an enhanced version of the SIGACT guest column
from Volume 44 Issue 2, June 201
An improved 1D area law for frustration-free systems
We present a new proof for the 1D area law for frustration-free systems with
a constant gap, which exponentially improves the entropy bound in Hastings' 1D
area law, and which is tight to within a polynomial factor. For particles of
dimension , spectral gap and interaction strength of at most
, our entropy bound is S_{1D}\le \orderof{1}X^3\log^8 X where
X\EqDef(J\log d)/\epsilon. Our proof is completely combinatorial, combining
the detectability lemma with basic tools from approximation theory.
Incorporating locality into the proof when applied to the 2D case gives an
entanglement bound that is at the cusp of being non-trivial in the sense that
any further improvement would yield a sub-volume law.Comment: 15 pages, 6 figures. Some small style corrections and updated ref
How local is the information in MPS/PEPS tensor networks?
Two dimensional tensor networks such as projected entangled pairs states
(PEPS) are generally hard to contract. This is arguably the main reason why
variational tensor network methods in 2D are still not as successful as in 1D.
However, this is not necessarily the case if the tensor network represents a
gapped ground state of a local Hamiltonian; such states are subject to many
constraints and contain much more structure. In this paper we introduce a new
approach for approximating the expectation value of a local observable in
ground states of local Hamiltonians that are represented as PEPS
tensor-networks. Instead of contracting the full tensor-network, we try to
estimate the expectation value using only a local patch of the tensor-network
around the observable. Surprisingly, we demonstrate that this is often easier
to do when the system is frustrated. In such case, the spanning vectors of the
local patch are subject to non-trivial constraints that can be utilized via a
semi-definite program to calculate rigorous lower- and upper-bounds on the
expectation value. We test our approach in 1D systems, where we show how the
expectation value can be calculated up to at least 3 or 4 digits of precision,
even when the patch radius is smaller than the correlation length.Comment: 11 pages, 5 figures, RevTeX4.1. Comments are welcome. (v2) Minor
corrections and slightly modified intro. Matches the published versio
Connecting global and local energy distributions in quantum spin models on a lattice
Generally, the local interactions in a many-body quantum spin system on a
lattice do not commute with each other. Consequently, the Hamiltonian of a
local region will generally not commute with that of the entire system, and so
the two cannot be measured simultaneously. The connection between the
probability distributions of measurement outcomes of the local and global
Hamiltonians will depend on the angles between the diagonalizing bases of these
two Hamiltonians. In this paper we characterize the relation between these two
distributions. On one hand, we upperbound the probability of measuring an
energy in a local region, if the global system is in a superposition of
eigenstates with energies . On the other hand, we bound the
probability of measuring a global energy in a bipartite system that
is in a tensor product of eigenstates of its two subsystems. Very roughly, we
show that due to the local nature of the governing interactions, these
distributions are identical to what one encounters in the commuting case, up to
some exponentially small corrections. Finally, we use these bounds to study the
spectrum of a locally truncated Hamiltonian, in which the energies of a
contiguous region have been truncated above some threshold energy . We
show that the lower part of the spectrum of this Hamiltonian is exponentially
close to that of the original Hamiltonian. A restricted version of this result
in 1D was a central building block in a recent improvement of the 1D area-law.Comment: 23 pages, 2 figures. A new version with tigheter bounds and a
re-written introductio
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