7,845 research outputs found
Stability estimates for the regularized inversion of the truncated Hilbert transform
In limited data computerized tomography, the 2D or 3D problem can be reduced
to a family of 1D problems using the differentiated backprojection (DBP)
method. Each 1D problem consists of recovering a compactly supported function
, where is a finite interval, from its
partial Hilbert transform data. When the Hilbert transform is measured on a
finite interval that only overlaps but does not cover
this inversion problem is known to be severely ill-posed [1].
In this paper, we study the reconstruction of restricted to the overlap
region . We show that with this restriction and by
assuming prior knowledge on the norm or on the variation of , better
stability with H\"older continuity (typical for mildly ill-posed problems) can
be obtained.Comment: added one remark, larger fonts for axis labels in figure
Hamiltonian simulation with nearly optimal dependence on all parameters
We present an algorithm for sparse Hamiltonian simulation whose complexity is
optimal (up to log factors) as a function of all parameters of interest.
Previous algorithms had optimal or near-optimal scaling in some parameters at
the cost of poor scaling in others. Hamiltonian simulation via a quantum walk
has optimal dependence on the sparsity at the expense of poor scaling in the
allowed error. In contrast, an approach based on fractional-query simulation
provides optimal scaling in the error at the expense of poor scaling in the
sparsity. Here we combine the two approaches, achieving the best features of
both. By implementing a linear combination of quantum walk steps with
coefficients given by Bessel functions, our algorithm's complexity (as measured
by the number of queries and 2-qubit gates) is logarithmic in the inverse
error, and nearly linear in the product of the evolution time, the
sparsity, and the magnitude of the largest entry of the Hamiltonian. Our
dependence on the error is optimal, and we prove a new lower bound showing that
no algorithm can have sublinear dependence on .Comment: 21 pages, corrects minor error in Lemma 7 in FOCS versio
Undecidability of the Spectral Gap in One Dimension
The spectral gap problem - determining whether the energy spectrum of a
system has an energy gap above ground state, or if there is a continuous range
of low-energy excitations - pervades quantum many-body physics. Recently, this
important problem was shown to be undecidable for quantum spin systems in two
(or more) spatial dimensions: there exists no algorithm that determines in
general whether a system is gapped or gapless, a result which has many
unexpected consequences for the physics of such systems. However, there are
many indications that one dimensional spin systems are simpler than their
higher-dimensional counterparts: for example, they cannot have thermal phase
transitions or topological order, and there exist highly-effective numerical
algorithms such as DMRG - and even provably polynomial-time ones - for gapped
1D systems, exploiting the fact that such systems obey an entropy area-law.
Furthermore, the spectral gap undecidability construction crucially relied on
aperiodic tilings, which are not possible in 1D.
So does the spectral gap problem become decidable in 1D? In this paper we
prove this is not the case, by constructing a family of 1D spin chains with
translationally-invariant nearest neighbour interactions for which no algorithm
can determine the presence of a spectral gap. This not only proves that the
spectral gap of 1D systems is just as intractable as in higher dimensions, but
also predicts the existence of qualitatively new types of complex physics in 1D
spin chains. In particular, it implies there are 1D systems with constant
spectral gap and non-degenerate classical ground state for all systems sizes up
to an uncomputably large size, whereupon they switch to a gapless behaviour
with dense spectrum.Comment: 7 figure
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