6,191 research outputs found
Interpolation in Valiant's theory
We investigate the following question: if a polynomial can be evaluated at
rational points by a polynomial-time boolean algorithm, does it have a
polynomial-size arithmetic circuit? We argue that this question is certainly
difficult. Answering it negatively would indeed imply that the constant-free
versions of the algebraic complexity classes VP and VNP defined by Valiant are
different. Answering this question positively would imply a transfer theorem
from boolean to algebraic complexity. Our proof method relies on Lagrange
interpolation and on recent results connecting the (boolean) counting hierarchy
to algebraic complexity classes. As a byproduct we obtain two additional
results: (i) The constant-free, degree-unbounded version of Valiant's
hypothesis that VP and VNP differ implies the degree-bounded version. This
result was previously known to hold for fields of positive characteristic only.
(ii) If exponential sums of easy to compute polynomials can be computed
efficiently, then the same is true of exponential products. We point out an
application of this result to the P=NP problem in the Blum-Shub-Smale model of
computation over the field of complex numbers.Comment: 13 page
Bounds on the Power of Constant-Depth Quantum Circuits
We show that if a language is recognized within certain error bounds by
constant-depth quantum circuits over a finite family of gates, then it is
computable in (classical) polynomial time. In particular, our results imply
EQNC^0 is contained in P, where EQNC^0 is the constant-depth analog of the
class EQP. On the other hand, we adapt and extend ideas of Terhal and
DiVincenzo (quant-ph/0205133) to show that, for any family F of quantum gates
including Hadamard and CNOT gates, computing the acceptance probabilities of
depth-five circuits over F is just as hard as computing these probabilities for
circuits over F. In particular, this implies that NQNC^0 = NQACC = NQP = coC=P
where NQNC^0 is the constant-depth analog of the class NQP. This essentially
refutes a conjecture of Green et al. that NQACC is contained in TC^0
(quant-ph/0106017)
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