35 research outputs found
On the Probability of Generating a Lattice
We study the problem of determining the probability that m vectors selected
uniformly at random from the intersection of the full-rank lattice L in R^n and
the window [0,B)^n generate when B is chosen to be appropriately
large. This problem plays an important role in the analysis of the success
probability of quantum algorithms for solving the Discrete Logarithm Problem in
infrastructures obtained from number fields and also for computing fundamental
units of number fields.
We provide the first complete and rigorous proof that 2n+1 vectors suffice to
generate L with constant probability (provided that B is chosen to be
sufficiently large in terms of n and the covering radius of L and the last n+1
vectors are sampled from a slightly larger window). Based on extensive computer
simulations, we conjecture that only n+1 vectors sampled from one window
suffice to generate L with constant success probability. If this conjecture is
true, then a significantly better success probability of the above quantum
algorithms can be guaranteed.Comment: 18 page
Efficient Quantum Algorithm for Identifying Hidden Polynomials
We consider a natural generalization of an abelian Hidden Subgroup Problem
where the subgroups and their cosets correspond to graphs of linear functions
over a finite field F with d elements. The hidden functions of the generalized
problem are not restricted to be linear but can also be m-variate polynomial
functions of total degree n>=2.
The problem of identifying hidden m-variate polynomials of degree less or
equal to n for fixed n and m is hard on a classical computer since
Omega(sqrt{d}) black-box queries are required to guarantee a constant success
probability. In contrast, we present a quantum algorithm that correctly
identifies such hidden polynomials for all but a finite number of values of d
with constant probability and that has a running time that is only
polylogarithmic in d.Comment: 17 page
Weak Fourier-Schur sampling, the hidden subgroup problem, and the quantum collision problem
Schur duality decomposes many copies of a quantum state into subspaces
labeled by partitions, a decomposition with applications throughout quantum
information theory. Here we consider applying Schur duality to the problem of
distinguishing coset states in the standard approach to the hidden subgroup
problem. We observe that simply measuring the partition (a procedure we call
weak Schur sampling) provides very little information about the hidden
subgroup. Furthermore, we show that under quite general assumptions, even a
combination of weak Fourier sampling and weak Schur sampling fails to identify
the hidden subgroup. We also prove tight bounds on how many coset states are
required to solve the hidden subgroup problem by weak Schur sampling, and we
relate this question to a quantum version of the collision problem.Comment: 21 page