12,343 research outputs found
Optimal quantum algorithm for polynomial interpolation
We consider the number of quantum queries required to determine the
coefficients of a degree-d polynomial over GF(q). A lower bound shown
independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2
quantum queries are needed to solve this problem with bounded error, whereas an
algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We
show that the lower bound is achievable: d/2+1/2 quantum queries suffice to
determine the polynomial with bounded error. Furthermore, we show that d/2+1
queries suffice to achieve probability approaching 1 for large q. These upper
bounds improve results of Boneh and Zhandry on the insecurity of cryptographic
protocols against quantum attacks. We also show that our algorithm's success
probability as a function of the number of queries is precisely optimal.
Furthermore, the algorithm can be implemented with gate complexity poly(log q)
with negligible decrease in the success probability. We end with a conjecture
about the quantum query complexity of multivariate polynomial interpolation.Comment: 17 pages, minor improvements, added conjecture about multivariate
interpolatio
General error estimate for adiabatic quantum computing
Most investigations devoted to the conditions for adiabatic quantum computing
are based on the first-order correction . However, it is
demonstrated that this first-order correction does not yield a good estimate
for the computational error. Therefore, a more general criterion is proposed,
which includes higher-order corrections as well and shows that the
computational error can be made exponentially small -- which facilitates
significantly shorter evolution times than the above first-order estimate in
certain situations. Based on this criterion and rather general arguments and
assumptions, it can be demonstrated that a run-time of order of the inverse
minimum energy gap is sufficient and necessary, i.e.,
T=\ord(\Delta E_{\rm min}^{-1}). For some examples, these analytical
investigations are confirmed by numerical simulations. PACS: 03.67.Lx,
03.67.-a.Comment: 8 pages, 6 figures, several modification
- …