3,748 research outputs found
Inequalities and bounds for quasi-symmetric 3-designs
AbstractQuasi-symmetric 3-designs with block intersection numbers x and y(0⩽x<y<k) are studied, several inequalities satisfied by the parameters of a quasi-symmetric 3-designs are obtained. Let D be a quasi-symmetric 3-design with the block size k and intersection numbers x, y; y>x⩾1 and suppose D′ denote the complement of D with the block size k′ and intersection numbers x′ and y′. If k −1 ⩽x + y then it is proved that x′ + y′ ⩽ k′. Using this it is shown that the quasi-symmetric 3-designs corresponding to y = x + 1, x + 2 are either extensions of symmetric designs or designs corresponding to the Witt-design (or trivial design, i.e., v = k + 2) or the complement of above designs
On the number of perfect lattices
We show that the number of non-similar perfect -dimensional
lattices satisfies eventually the
inequalities for arbitrary
smallstrictly positive
Extrema of graph eigenvalues
In 1993 Hong asked what are the best bounds on the 'th largest eigenvalue
of a graph of order . This challenging question has
never been tackled for any . In the present paper tight bounds are
obtained for all and even tighter bounds are obtained for the 'th
largest singular value
Some of these bounds are based on Taylor's strongly regular graphs, and other
on a method of Kharaghani for constructing Hadamard matrices. The same kind of
constructions are applied to other open problems, like Nordhaus-Gaddum problems
of the kind: How large can be
These constructions are successful also in another open question: How large
can the Ky Fan norm be
Ky Fan norms of graphs generalize the concept of graph energy, so this question
generalizes the problem for maximum energy graphs.
In the final section, several results and problems are restated for
-matrices, which seem to provide a more natural ground for such
research than graphs.
Many of the results in the paper are paired with open questions and problems
for further study.Comment: 32 page
Approximate unitary -designs by short random quantum circuits using nearest-neighbor and long-range gates
We prove that -depth local random quantum circuits
with two qudit nearest-neighbor gates on a -dimensional lattice with n
qudits are approximate -designs in various measures. These include the
"monomial" measure, meaning that the monomials of a random circuit from this
family have expectation close to the value that would result from the Haar
measure. Previously, the best bound was due to
Brandao-Harrow-Horodecki (BHH) for . We also improve the "scrambling" and
"decoupling" bounds for spatially local random circuits due to Brown and Fawzi.
One consequence of our result is that assuming the polynomial hierarchy (PH)
is infinite and that certain counting problems are -hard on average,
sampling within total variation distance from these circuits is hard for
classical computers. Previously, exact sampling from the outputs of even
constant-depth quantum circuits was known to be hard for classical computers
under the assumption that PH is infinite. However, to show the hardness of
approximate sampling using this strategy requires that the quantum circuits
have a property called "anti-concentration", meaning roughly that the output
has near-maximal entropy. Unitary 2-designs have the desired anti-concentration
property. Thus our result improves the required depth for this level of
anti-concentration from linear depth to a sub-linear value, depending on the
geometry of the interactions. This is relevant to a recent proposal by the
Google Quantum AI group to perform such a sampling task with 49 qubits on a
two-dimensional lattice and confirms their conjecture that depth
suffices for anti-concentration. We also prove that anti-concentration is
possible in depth O(log(n) loglog(n)) using a different model
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