173 research outputs found
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
A Rydberg Quantum Simulator
Following Feynman and as elaborated on by Lloyd, a universal quantum
simulator (QS) is a controlled quantum device which reproduces the dynamics of
any other many particle quantum system with short range interactions. This
dynamics can refer to both coherent Hamiltonian and dissipative open system
evolution. We investigate how laser excited Rydberg atoms in large spacing
optical or magnetic lattices can provide an efficient implementation of a
universal QS for spin models involving (high order) n-body interactions. This
includes the simulation of Hamiltonians of exotic spin models involving
n-particle constraints such as the Kitaev toric code, color code, and lattice
gauge theories with spin liquid phases. In addition, it provides the
ingredients for dissipative preparation of entangled states based on
engineering n-particle reservoir couplings. The key basic building blocks of
our architecture are efficient and high-fidelity n-qubit entangling gates via
auxiliary Rydberg atoms, including a possible dissipative time step via optical
pumping. This allows to mimic the time evolution of the system by a sequence of
fast, parallel and high-fidelity n-particle coherent and dissipative Rydberg
gates.Comment: 8 pages, 4 figure
Universal Quantum Computation with the Exchange Interaction
Experimental implementations of quantum computer architectures are now being
investigated in many different physical settings. The full set of requirements
that must be met to make quantum computing a reality in the laboratory [1] is
daunting, involving capabilities well beyond the present state of the art. In
this report we develop a significant simplification of these requirements that
can be applied in many recent solid-state approaches, using quantum dots [2],
and using donor-atom nuclear spins [3] or electron spins [4]. In these
approaches, the basic two-qubit quantum gate is generated by a tunable
Heisenberg interaction (the Hamiltonian is between spins and ), while the one-qubit gates require the control
of a local Zeeman field. Compared to the Heisenberg operation, the one-qubit
operations are significantly slower and require substantially greater materials
and device complexity, which may also contribute to increasing the decoherence
rate. Here we introduce an explicit scheme in which the Heisenberg interaction
alone suffices to exactly implement any quantum computer circuit, at a price of
a factor of three in additional qubits and about a factor of ten in additional
two-qubit operations. Even at this cost, the ability to eliminate the
complexity of one-qubit operations should accelerate progress towards these
solid-state implementations of quantum computation.Comment: revtex, 2 figures, this version appeared in Natur
Quantum Computing with Very Noisy Devices
In theory, quantum computers can efficiently simulate quantum physics, factor
large numbers and estimate integrals, thus solving otherwise intractable
computational problems. In practice, quantum computers must operate with noisy
devices called ``gates'' that tend to destroy the fragile quantum states needed
for computation. The goal of fault-tolerant quantum computing is to compute
accurately even when gates have a high probability of error each time they are
used. Here we give evidence that accurate quantum computing is possible with
error probabilities above 3% per gate, which is significantly higher than what
was previously thought possible. However, the resources required for computing
at such high error probabilities are excessive. Fortunately, they decrease
rapidly with decreasing error probabilities. If we had quantum resources
comparable to the considerable resources available in today's digital
computers, we could implement non-trivial quantum computations at error
probabilities as high as 1% per gate.Comment: 47 page
Non-Abelian statistics and topological quantum information processing in 1D wire networks
Topological quantum computation provides an elegant way around decoherence,
as one encodes quantum information in a non-local fashion that the environment
finds difficult to corrupt. Here we establish that one of the key
operations---braiding of non-Abelian anyons---can be implemented in
one-dimensional semiconductor wire networks. Previous work [Lutchyn et al.,
arXiv:1002.4033 and Oreg et al., arXiv:1003.1145] provided a recipe for driving
semiconducting wires into a topological phase supporting long-sought particles
known as Majorana fermions that can store topologically protected quantum
information. Majorana fermions in this setting can be transported, created, and
fused by applying locally tunable gates to the wire. More importantly, we show
that networks of such wires allow braiding of Majorana fermions and that they
exhibit non-Abelian statistics like vortices in a p+ip superconductor. We
propose experimental setups that enable the Majorana fusion rules to be probed,
along with networks that allow for efficient exchange of arbitrary numbers of
Majorana fermions. This work paves a new path forward in topological quantum
computation that benefits from physical transparency and experimental realism.Comment: 6 pages + 17 pages of Supp. Mat.; 10 figures. Supp. Mat. has doubled
in size to establish results more rigorously; many other improvements as wel
Identifying Topological Order by Entanglement Entropy
Topological phases are unique states of matter incorporating long-range
quantum entanglement, hosting exotic excitations with fractional quantum
statistics. We report a practical method to identify topological phases in
arbitrary realistic models by accurately calculating the Topological
Entanglement Entropy (TEE) using the Density Matrix Renormalization Group
(DMRG). We argue that the DMRG algorithm naturally produces a minimally
entangled state, from amongst the quasi-degenerate ground states in a
topological phase. This proposal both explains the success of this method, and
the absence of ground state degeneracy found in prior DMRG sightings of
topological phases. We demonstrate the effectiveness of the calculational
procedure by obtaining the TEE for several microscopic models, with an accuracy
of order when the circumference of the cylinder is around ten times
the correlation length. As an example, we definitively show the ground state of
the quantum antiferromagnet on the kagom\'e lattice is a topological
spin liquid, and strongly constrain the full identification of this phase of
matter.Comment: 20 pages, 6 figure
On Quantum Effects in a Theory of Biological Evolution
We construct a descriptive toy model that considers quantum effects on biological evolution starting from Chaitin's classical framework. There are smart evolution scenarios in which a quantum world is as favorable as classical worlds for evolution to take place. However, in more natural scenarios, the rate of evolution depends on the degree of entanglement present in quantum organisms with respect to classical organisms. If the entanglement is maximal, classical evolution turns out to be more favorable
Evidence of Majorana fermions in an Al - InAs nanowire topological superconductor
Majorana fermions are the only fermionic particles that are expected to be
their own antiparticles. While elementary particles of the Majorana type were
not identified yet, quasi-particles with Majorana like properties, born from
interacting electrons in the solid, were predicted to exist. Here, we present
thorough experimental studies, backed by numerical simulations, of a system
composed of an aluminum superconductor in proximity to an indium arsenide
nanowire, with the latter possessing strong spin-orbit coupling. An induced 1d
topological superconductor - supporting Majorana fermions at both ends - is
expected to form. We concentrate on the characteristics of a distinct zero bias
conductance peak (ZBP), and its splitting in energy, both appearing only with a
small magnetic field applied along the wire. The ZBP was found to be robustly
tied to the Fermi energy over a wide range of system parameters. While not
providing a definite proof of a Majorana state, the presented data and the
simulations support strongly its existence
How to realize a robust practical Majorana chain in a quantum dot-superconductor linear array
Semiconducting nanowires in proximity to superconductors are promising
experimental systems for Majorana fermions, which may ultimately be used as
building blocks for topological quantum computers. A serious challenge in the
experimental realization of the Majorana fermions is the supression of
topological superconductivity by disorder. We show that Majorana fermions
protected by a robust topological gap can occur at the ends of a chain of
quantum dots connected by s-wave superconductors. In the appropriate parameter
regime, we establish that the quantum dot/superconductor system is equivalent
to a 1D Kitaev chain, which can be tuned to be in a robust topological phase
with Majorana end modes even in the case where the quantum dots and
superconductors are both strongly disordered. Such a spin-orbit coupled quantum
dot - s-wave superconductor array provides an ideal experimental platform for
the observation of non-Abelian Majorana modes.Comment: 8 pages; 3 figures; version 2: Supplementary material updated to
include more general proof for localized Majorana fermion
Anyonic interferometry and protected memories in atomic spin lattices
Strongly correlated quantum systems can exhibit exotic behavior called
topological order which is characterized by non-local correlations that depend
on the system topology. Such systems can exhibit remarkable phenomena such as
quasi-particles with anyonic statistics and have been proposed as candidates
for naturally fault-tolerant quantum computation. Despite these remarkable
properties, anyons have never been observed in nature directly. Here we
describe how to unambiguously detect and characterize such states in recently
proposed spin lattice realizations using ultra-cold atoms or molecules trapped
in an optical lattice. We propose an experimentally feasible technique to
access non-local degrees of freedom by performing global operations on trapped
spins mediated by an optical cavity mode. We show how to reliably read and
write topologically protected quantum memory using an atomic or photonic qubit.
Furthermore, our technique can be used to probe statistics and dynamics of
anyonic excitations.Comment: 14 pages, 6 figure
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