388 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
Universal topological phase of 2D stabilizer codes
Two topological phases are equivalent if they are connected by a local
unitary transformation. In this sense, classifying topological phases amounts
to classifying long-range entanglement patterns. We show that all 2D
topological stabilizer codes are equivalent to several copies of one universal
phase: Kitaev's topological code. Error correction benefits from the
corresponding local mappings.Comment: 4 pages, 3 figure
The Bose-Hubbard model is QMA-complete
The Bose-Hubbard model is a system of interacting bosons that live on the
vertices of a graph. The particles can move between adjacent vertices and
experience a repulsive on-site interaction. The Hamiltonian is determined by a
choice of graph that specifies the geometry in which the particles move and
interact. We prove that approximating the ground energy of the Bose-Hubbard
model on a graph at fixed particle number is QMA-complete. In our QMA-hardness
proof, we encode the history of an n-qubit computation in the subspace with at
most one particle per site (i.e., hard-core bosons). This feature, along with
the well-known mapping between hard-core bosons and spin systems, lets us prove
a related result for a class of 2-local Hamiltonians defined by graphs that
generalizes the XY model. By avoiding the use of perturbation theory in our
analysis, we circumvent the need to multiply terms in the Hamiltonian by large
coefficients
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
Lieb-Robinson Bounds and the Generation of Correlations and Topological Quantum Order
The Lieb-Robinson bound states that local Hamiltonian evolution in nonrelativistic quantum mechanical theories gives rise to the notion of an effective light cone with exponentially decaying tails. We discuss several consequences of this result in the context of quantum information theory. First, we show that the information that leaks out to spacelike separated regions is negligible and that there is a finite speed at which correlations and entanglement can be distributed. Second, we discuss how these ideas can be used to prove lower bounds on the time it takes to convert states without topological quantum order to states with that property. Finally, we show that the rate at which entropy can be created in a block of spins scales like the boundary of that block
Entanglement entropy of multipartite pure states
Consider a system consisting of -dimensional quantum particles and
arbitrary pure state of the whole system. Suppose we simultaneously
perform complete von Neumann measurements on each particle. One can ask: what
is the minimal possible value of the entropy of outcomes joint
probability distribution? We show that coincides with entanglement
entropy for bipartite states. We compute for two sample multipartite
states: the hexacode state () and determinant states (). The
generalization of determinant states to the case is considered.Comment: 7 pages, REVTeX, corrected some typo
A time-dependent variational principle for dissipative dynamics
We extend the time-dependent variational principle to the setting of
dissipative dynamics. This provides a locally optimal (in time) approximation
to the dynamics of any Lindblad equation within a given variational manifold of
mixed states. In contrast to the pure-state setting there is no canonical
information geometry for mixed states and this leads to a family of possible
trajectories --- one for each information metric. We focus on the case of the
operationally motivated family of monotone riemannian metrics and show further,
that in the particular case where the variational manifold is given by the set
of fermionic gaussian states all of these possible trajectories coincide. We
illustrate our results in the case of the Hubbard model subject to spin
decoherence.Comment: Published versio
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
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