77 research outputs found
Universal Quantum Computation with the nu=5/2 Fractional Quantum Hall State
We consider topological quantum computation (TQC) with a particular class of
anyons that are believed to exist in the Fractional Quantum Hall Effect state
at Landau level filling fraction nu=5/2. Since the braid group representation
describing statistics of these anyons is not computationally universal, one
cannot directly apply the standard TQC technique. We propose to use very noisy
non-topological operations such as direct short-range interaction between
anyons to simulate a universal set of gates. Assuming that all TQC operations
are implemented perfectly, we prove that the threshold error rate for
non-topological operations is above 14%. The total number of non-topological
computational elements that one needs to simulate a quantum circuit with
gates scales as .Comment: 17 pages, 12 eps figure
On measurement-based quantum computation with the toric code states
We study measurement-based quantum computation (MQC) using as quantum
resource the planar code state on a two-dimensional square lattice (planar
analogue of the toric code). It is shown that MQC with the planar code state
can be efficiently simulated on a classical computer if at each step of MQC the
sets of measured and unmeasured qubits correspond to connected subsets of the
lattice.Comment: 9 pages, 5 figure
A short proof of stability of topological order under local perturbations
Recently, the stability of certain topological phases of matter under weak
perturbations was proven. Here, we present a short, alternate proof of the same
result. We consider models of topological quantum order for which the
unperturbed Hamiltonian can be written as a sum of local pairwise
commuting projectors on a -dimensional lattice. We consider a perturbed
Hamiltonian involving a generic perturbation that can be written
as a sum of short-range bounded-norm interactions. We prove that if the
strength of is below a constant threshold value then has well-defined
spectral bands originating from the low-lying eigenvalues of . These bands
are separated from the rest of the spectrum and from each other by a constant
gap. The width of the band originating from the smallest eigenvalue of
decays faster than any power of the lattice size.Comment: 15 page
Matrix Product State and mean field solutions for one-dimensional systems can be found efficiently
We consider the problem of approximating ground states of one-dimensional
quantum systems within the two most common variational ansatzes, namely the
mean field ansatz and Matrix Product States. We show that both for mean field
and for Matrix Product States of fixed bond dimension, the optimal solutions
can be found in a way which is provably efficient (i.e., scales polynomially).
This implies that the corresponding variational methods can be in principle
recast in a way which scales provably polynomially. Moreover, our findings
imply that ground states of one-dimensional commuting Hamiltonians can be found
efficiently.Comment: 5 pages; v2: accepted version, Journal-ref adde
The Fragility of Quantum Information?
We address the question whether there is a fundamental reason why quantum
information is more fragile than classical information. We show that some
answers can be found by considering the existence of quantum memories and their
dimensional dependence.Comment: Essay on quantum information: no new results. Ten pages, published in
Lec. Notes in Comp. Science, Vol. 7505, pp. 47-56 (2012. One reference adde
Quantum picturalism for topological cluster-state computing
Topological quantum computing is a way of allowing precise quantum
computations to run on noisy and imperfect hardware. One implementation uses
surface codes created by forming defects in a highly-entangled cluster state.
Such a method of computing is a leading candidate for large-scale quantum
computing. However, there has been a lack of sufficiently powerful high-level
languages to describe computing in this form without resorting to single-qubit
operations, which quickly become prohibitively complex as the system size
increases. In this paper we apply the category-theoretic work of Abramsky and
Coecke to the topological cluster-state model of quantum computing to give a
high-level graphical language that enables direct translation between quantum
processes and physical patterns of measurement in a computer - a "compiler
language". We give the equivalence between the graphical and topological
information flows, and show the applicable rewrite algebra for this computing
model. We show that this gives us a native graphical language for the design
and analysis of topological quantum algorithms, and finish by discussing the
possibilities for automating this process on a large scale.Comment: 18 pages, 21 figures. Published in New J. Phys. special issue on
topological quantum computin
Computational Complexity of interacting electrons and fundamental limitations of Density Functional Theory
One of the central problems in quantum mechanics is to determine the ground
state properties of a system of electrons interacting via the Coulomb
potential. Since its introduction by Hohenberg, Kohn, and Sham, Density
Functional Theory (DFT) has become the most widely used and successful method
for simulating systems of interacting electrons, making their original work one
of the most cited in physics. In this letter, we show that the field of
computational complexity imposes fundamental limitations on DFT, as an
efficient description of the associated universal functional would allow to
solve any problem in the class QMA (the quantum version of NP) and thus
particularly any problem in NP in polynomial time. This follows from the fact
that finding the ground state energy of the Hubbard model in an external
magnetic field is a hard problem even for a quantum computer, while given the
universal functional it can be computed efficiently using DFT. This provides a
clear illustration how the field of quantum computing is useful even if quantum
computers would never be built.Comment: 8 pages, 3 figures. v2: Version accepted at Nature Physics; differs
significantly from v1 (including new title). Includes an extra appendix (not
contained in the journal version) on the NP-completeness of Hartree-Fock,
which is taken from v
Bipartite entanglement and localization of one-particle states
We study bipartite entanglement in a general one-particle state, and find
that the linear entropy, quantifying the bipartite entanglement, is directly
connected to the paricitpation ratio, charaterizing the state localization. The
more extended the state is, the more entangled the state. We apply the general
formalism to investigate ground-state and dynamical properties of entanglement
in the one-dimensional Harper model.Comment: 4 pages and 3 figures. Version
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