15,753 research outputs found
Quantum Information and the PCP Theorem
We show how to encode (classical) bits by a single
quantum state of size O(n) qubits, such that: for any constant and
any , the values of the bits
can be retrieved from by a one-round
Arthur-Merlin interactive protocol of size polynomial in . This shows how to
go around Holevo-Nayak's Theorem, using Arthur-Merlin proofs.
We use the new representation to prove the following results:
1) Interactive proofs with quantum advice: We show that the class
contains ALL languages. That is, for any language (even non-recursive), the
membership (for of length ) can be proved by a polynomial-size
quantum interactive proof, where the verifier is a polynomial-size quantum
circuit with working space initiated with some quantum state
(depending only on and ). Moreover, the interactive proof that we give
is of only one round, and the messages communicated are classical.
2) PCP with only one query: We show that the membership (for
of length ) can be proved by a logarithmic-size quantum state ,
together with a polynomial-size classical proof consisting of blocks of length
bits each, such that after measuring the state the
verifier only needs to read {\bf one} block of the classical proof.
While the first result is a straight forward consequence of the new
representation, the second requires an additional machinery of quantum
low-degree-test that may be interesting in its own right.Comment: 30 page
Arithmetic Circuit Lower Bounds via MaxRank
We introduce the polynomial coefficient matrix and identify maximum rank of
this matrix under variable substitution as a complexity measure for
multivariate polynomials. We use our techniques to prove super-polynomial lower
bounds against several classes of non-multilinear arithmetic circuits. In
particular, we obtain the following results :
As our main result, we prove that any homogeneous depth-3 circuit for
computing the product of matrices of dimension requires
size. This improves the lower bounds by Nisan and
Wigderson(1995) when .
There is an explicit polynomial on variables and degree at most
for which any depth-3 circuit of product dimension at most
(dimension of the space of affine forms feeding into each
product gate) requires size . This generalizes the lower bounds
against diagonal circuits proved by Saxena(2007). Diagonal circuits are of
product dimension 1.
We prove a lower bound on the size of product-sparse
formulas. By definition, any multilinear formula is a product-sparse formula.
Thus, our result extends the known super-polynomial lower bounds on the size of
multilinear formulas by Raz(2006).
We prove a lower bound on the size of partitioned arithmetic
branching programs. This result extends the known exponential lower bound on
the size of ordered arithmetic branching programs given by Jansen(2008).Comment: 22 page
On the zone of the boundary of a convex body
We consider an arrangement \A of hyperplanes in and the zone
in \A of the boundary of an arbitrary convex set in in such an
arrangement. We show that, whereas the combinatorial complexity of is
known only to be \cite{APS}, the outer part of the zone has
complexity (without the logarithmic factor). Whether this bound
also holds for the complexity of the inner part of the zone is still an open
question (even for )
Expert-oriented abilities vs. novice-oriented abilities: An alternative account of epistemic authority
According to a recent account of epistemic authority proposed by Linda Zagzebski (2012), it is rational for laypersons to believe on authority when they conscientiously judge that the authority is more likely to form true beliefs and avoid false ones than they are in some domain. Christoph Jäger (2016) has recently raised several objections to her view. By contrast, I argue that both theories fail to adequately capture what epistemic authority is, and I offer an alternative account grounded in the abilities that different kinds of authorities are required to possess
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