26 research outputs found
Topological Order and Reflection Positivity
The interplay between the two fundamental concepts of topological order and
reflection positivity allows one to characterize the ground states of certain
many-body Hamiltonians. We define topological order in an appropriate fashion
and show that certain operators have positive expectation value in all ground
states. We apply our method to vortex loops in a model relevant to topological
quantum memories.Comment: 3 pages, 1 figur
Majorana braiding with thermal noise
We investigate the self-correcting properties of a network of Majorana wires,
in the form of a trijunction, in contact with a parity-preserving thermal
environment. As opposed to the case where Majorana bound states (MBSs) are
immobile, braiding MBSs within a trijunction introduces dangerous error
processes that we identify. Such errors prevent the lifetime of the memory from
increasing with the size of the system. We confirm our predictions with Monte
Carlo simulations. Our findings put a restriction on the degree of
self-correction of this specific quantum computing architecture.Comment: 6 pages, 3 figures, long version: arXiv: 1507.0089
Long-Range Interaction of Spin-Qubits via Ferromagnets
We propose a mechanism of coherent coupling between distant spin qubits
interacting dipolarly with a ferromagnet. We derive an effective two-spin
interaction Hamiltonian and estimate the coupling strength. We discuss the
mechanisms of decoherence induced solely by the coupling to the ferromagnet and
show that there is a regime where it is negligible. Finally, we present a
sequence for the implementation of the entangling CNOT gate and estimate the
corresponding operation time to be a few tens of nanoseconds. A particularly
promising application of our proposal is to atomistic spin-qubits such as
silicon-based qubits and NV-centers in diamond to which existing coupling
schemes do not apply.Comment: 6 pages, 7 pages of appendi
Physical solutions of the Kitaev honeycomb model
We investigate the exact solution of the honeycomb model proposed by Kitaev
and derive an explicit formula for the projector onto the physical subspace.
The physical states are simply characterized by the parity of the total
occupation of the fermionic eigenmodes. We consider a general lattice on a
torus and show that the physical fermion parity depends in a nontrivial way on
the vortex configuration and the choice of boundary conditions. In the
vortex-free case with a constant gauge field we are able to obtain an
analytical expression of the parity. For a general configuration of the gauge
field the parity can be easily evaluated numerically, which allows the exact
diagonalization of large spin models. We consider physically relevant
quantities, as in particular the vortex energies, and show that their true
value and associated states can be substantially different from the one
calculated in the unprojected space, even in the thermodynamic limit
Quantum memory coupled to cavity modes
Inspired by spin-electric couplings in molecular magnets, we introduce in the
Kitaev honeycomb model a linear modification of the Ising interactions due to
the presence of quantized cavity fields. This allows to control the properties
of the low-energy toric code Hamiltonian, which can serve as a quantum memory,
by tuning the physical parameters of the cavity modes, like frequencies, photon
occupations, and coupling strengths. We study the properties of the model
perturbatively by making use of the Schrieffer-Wolff transformation and show
that, depending on the specific setup, the cavity modes can be useful in
several ways. They allow to detect the presence of anyons through frequency
shifts and to prolong the lifetime of the memory by enhancing the anyon
excitation energy or mediating long-range anyon-anyon interactions with tunable
sign. We consider both resonant and largely detuned cavity modes.Comment: 16 pages, 6 figure
Methodology for bus layout for topological quantum error correcting codes
Most quantum computing architectures can be realized as two-dimensional
lattices of qubits that interact with each other. We take transmon qubits and
transmission line resonators as promising candidates for qubits and couplers;
we use them as basic building elements of a quantum code. We then propose a
simple framework to determine the optimal experimental layout to realize
quantum codes. We show that this engineering optimization problem can be
reduced to the solution of standard binary linear programs. While solving such
programs is a NP-hard problem, we propose a way to find scalable optimal
architectures that require solving the linear program for a restricted number
of qubits and couplers. We apply our methods to two celebrated quantum codes,
namely the surface code and the Fibonacci code.Comment: 11 pages, 12 figure
Monte Carlo studies of the properties of the Majorana quantum error correction code: is self-correction possible during braiding?
The Majorana code is an example of a stabilizer code where the quantum
information is stored in a system supporting well-separated Majorana Bound
States (MBSs). We focus on one-dimensional realizations of the Majorana code,
as well as networks of such structures, and investigate their lifetime when
coupled to a parity-preserving thermal environment. We apply the Davies
prescription, a standard method that describes the basic aspects of a thermal
environment, and derive a master equation in the Born-Markov limit. We first
focus on a single wire with immobile MBSs and perform error correction to
annihilate thermal excitations. In the high-temperature limit, we show both
analytically and numerically that the lifetime of the Majorana qubit grows
logarithmically with the size of the wire. We then study a trijunction with
four MBSs when braiding is executed. We study the occurrence of dangerous error
processes that prevent the lifetime of the Majorana code from growing with the
size of the trijunction. The origin of the dangerous processes is the braiding
itself, which separates pairs of excitations and renders the noise nonlocal;
these processes arise from the basic constraints of moving MBSs in 1D
structures. We confirm our predictions with Monte Carlo simulations in the
low-temperature regime, i.e. the regime of practical relevance. Our results put
a restriction on the degree of self-correction of this particular 1D
topological quantum computing architecture.Comment: Main text: 20 pages, Supplementary Material: 66 pages. Short version:
arXiv:1505.0371
Majorana states in inhomogeneous spin ladders
We propose an inhomogeneous open spin ladder, related to the Kitaev honeycomb
model, which can be tuned between topological and nontopological phases. In
extension of Lieb's theorem, we show numerically that the ground state of the
spin ladder is either vortex free or vortex full. We study the robustness of
Majorana end states (MES) which emerge at the boundary between sections in
different topological phases and show that while the MES in the homogeneous
ladder are destroyed by single-body perturbations, in the presence of
inhomogeneities at least two-body perturbations are required to destabilize
MES. Furthermore, we prove that x, y, or z inhomogeneous magnetic fields are
not able to destroy the topological degeneracy. Finally, we present a
trijunction setup where MES can be braided. A network of such spin ladders
provides thus a promising platform for realization and manipulation of MES