630 research outputs found
How fast do stabilizer Hamiltonians thermalize?
We present rigorous bounds on the thermalization time of the family of
quantum mechanical spin systems known as stabilizer Hamiltonians. The
thermalizing dynamics are modeled by a Davies master equation that arises from
a weak local coupling of the system to a large thermal bath. Two temperature
regimes are considered. First we clarify how in the low temperature regime, the
thermalization time is governed by a generalization of the energy barrier
between orthogonal ground states. When no energy barrier is present the
Hamiltonian thermalizes in a time that is at most quadratic in the system size.
Secondly, we show that above a universal critical temperature, every stabilizer
Hamiltonian relaxes to its unique thermal state in a time which scales at most
linearly in the size of the system. We provide an explicit lower bound on the
critical temperature. Finally, we discuss the implications of these result for
the problem of self-correcting quantum memories with stabilizer Hamiltonians
Divide and conquer method for proving gaps of frustration free Hamiltonians
Providing system-size independent lower bounds on the spectral gap of local
Hamiltonian is in general a hard problem. For the case of finite-range,
frustration free Hamiltonians on a spin lattice of arbitrary dimension, we show
that a property of the ground state space is sufficient to obtain such a bound.
We furthermore show that such a condition is necessary and equivalent to a
constant spectral gap. Thanks to this equivalence, we can prove that for
gapless models in any dimension, the spectral gap on regions of diameter is
at most for any positive
.Comment: This is an author-created, un-copyedited version of an article
accepted for publication/published in Journal of Statistical Mechanics:
Theory and Experiment. IOP Publishing Ltd is not responsible for any errors
or omissions in this version of the manuscript or any version derived from
it. The Version of Record is available online at
http://dx.doi.org/10.1088/1742-5468/aaa793, Journal of Statistical Mechanics:
Theory and Experiment, March 201
Quantum logarithmic Sobolev inequalities and rapid mixing
A family of logarithmic Sobolev inequalities on finite dimensional quantum
state spaces is introduced. The framework of non-commutative \bL_p-spaces is
reviewed and the relationship between quantum logarithmic Sobolev inequalities
and the hypercontractivity of quantum semigroups is discussed. This
relationship is central for the derivation of lower bounds for the logarithmic
Sobolev (LS) constants. Essential results for the family of inequalities are
proved, and we show an upper bound to the generalized LS constant in terms of
the spectral gap of the generator of the semigroup. These inequalities provide
a framework for the derivation of improved bounds on the convergence time of
quantum dynamical semigroups, when the LS constant and the spectral gap are of
the same order. Convergence bounds on finite dimensional state spaces are
particularly relevant for the field of quantum information theory. We provide a
number of examples, where improved bounds on the mixing time of several
semigroups are obtained; including the depolarizing semigroup and quantum
expanders.Comment: Updated manuscript, 30 pages, no figure
Non-commutative Nash inequalities
A set of functional inequalities - called Nash inequalities - are introduced
and analyzed in the context of quantum Markov process mixing. The basic theory
of Nash inequalities is extended to the setting of non-commutative Lp spaces,
where their relationship to Poincare and log-Sobolev inequalities are fleshed
out. We prove Nash inequalities for a number of unital reversible semigroups
Mutual information area laws for thermal free fermions
We provide a rigorous and asymptotically exact expression of the mutual
information of translationally invariant free fermionic lattice systems in a
Gibbs state. In order to arrive at this result, we introduce a novel
frameworkfor computing determinants of Toeplitz operators with smooth symbols,
and for treating Toeplitz matrices with system size dependent entries. The
asymptotically exact mutual information for a partition of the one-dimensional
lattice satisfies an area law, with a prefactor which we compute explicitly. As
examples, we discuss the fermionic XX model in one dimension and free fermionic
models on the torus in higher dimensions in detail. Special emphasis is put
onto the discussion of the temperature dependence of the mutual information,
scaling like the logarithm of the inverse temperature, hence confirming an
expression suggested by conformal field theory. We also comment on the
applicability of the formalism to treat open systems driven by quantum noise.
In the appendix, we derive useful bounds to the mutual information in terms of
purities. Finally, we provide a detailed error analysis for finite system
sizes. This analysis is valuable in its own right for the abstract theory of
Toeplitz determinants.Comment: 42 pages, 4 figures, replaced by final versio
Walkabout Activities
Postcard from Samantha O\u27Connor, during the Linfield College Semester Abroad Program at James Cook University in Cairns, Australi
Radicalization in Europe
The spectacular attacks on New York City on 11 September 2001 carried out by 19 suicide bombers belonging to the al-Qaeda network kicked off the century. Other, more recent attacks in different European cities, this time claimed by the Islamic State, have made terrorist acts daily news across the globe. Despite the differences in organization (a network such as al-Qaeda or grouped by territories such as Islamic State), these youth, engaged on the path of violence in the name of jihad, are guided by the force of the singular narrative of membership in the Ummah, the world-wide Muslim community, that lends all its strength to the appropriation of an ideology and the transition to violence..
Dynamic Evaluation of Job Search Assistance
This paper evaluates a job search assistance program for unemployment insurance recipients. The assignment to the program is dynamic. We provide a discussion on dynamic treatment effects and identification conditions. In the empirical analyses we use administrative data from a unique institutional environment. This allows us to compare different microeconometric evaluation estimators. All estimators find that the job search assistance program reduces the exit to work, in particular when provided early during the spell of unemployment. Furthermore, continuous-time (timing-of-events and regression discontinuity) methods are more robust than discrete-time (propensity score and regression discontinuity) methods.treatment evaluation, dynamic enrollment, empirical evaluation
Finite correlation length implies efficient preparation of quantum thermal states
Preparing quantum thermal states on a quantum computer is in general a
difficult task. We provide a procedure to prepare a thermal state on a quantum
computer with a logarithmic depth circuit of local quantum channels assuming
that the thermal state correlations satisfy the following two properties: (i)
the correlations between two regions are exponentially decaying in the distance
between the regions, and (ii) the thermal state is an approximate Markov state
for shielded regions. We require both properties to hold for the thermal state
of the Hamiltonian on any induced subgraph of the original lattice. Assumption
(ii) is satisfied for all commuting Gibbs states, while assumption (i) is
satisfied for every model above a critical temperature. Both assumptions are
satisfied in one spatial dimension. Moreover, both assumptions are expected to
hold above the thermal phase transition for models without any topological
order at finite temperature. As a building block, we show that exponential
decay of correlation (for thermal states of Hamiltonians on all induced
subgraph) is sufficient to efficiently estimate the expectation value of a
local observable. Our proof uses quantum belief propagation, a recent
strengthening of strong sub-additivity, and naturally breaks down for states
with topological order.Comment: 16 pages, 4 figure
- …