158 research outputs found
Optimal lower bounds for quantum automata and random access codes
Consider the finite regular language L_n = {w0 : w \in {0,1}^*, |w| \le n}.
It was shown by Ambainis, Nayak, Ta-Shma and Vazirani that while this language
is accepted by a deterministic finite automaton of size O(n), any one-way
quantum finite automaton (QFA) for it has size 2^{Omega(n/log n)}. This was
based on the fact that the evolution of a QFA is required to be reversible.
When arbitrary intermediate measurements are allowed, this intuition breaks
down. Nonetheless, we show a 2^{Omega(n)} lower bound for such QFA for L_n,
thus also improving the previous bound. The improved bound is obtained by
simple entropy arguments based on Holevo's theorem. This method also allows us
to obtain an asymptotically optimal (1-H(p))n bound for the dense quantum codes
(random access codes) introduced by Ambainis et al. We then turn to Holevo's
theorem, and show that in typical situations, it may be replaced by a tighter
and more transparent in-probability bound.Comment: 8 pages, 1 figure, Latex2e. Extensive modifications have been made to
increase clarity. To appear in FOCS'9
Inverting a permutation is as hard as unordered search
We show how an algorithm for the problem of inverting a permutation may be
used to design one for the problem of unordered search (with a unique
solution). Since there is a straightforward reduction in the reverse direction,
the problems are essentially equivalent.
The reduction we present helps us bypass the hybrid argument due to Bennett,
Bernstein, Brassard, and Vazirani (1997) and the quantum adversary method due
to Ambainis (2002) that were earlier used to derive lower bounds on the quantum
query complexity of the problem of inverting permutations. It directly implies
that the quantum query complexity of the problem is asymptotically the same as
that for unordered search, namely in Theta(sqrt(n)).Comment: 5 pages. Numerous changes to improve the presentatio
Approximate Randomization of Quantum States With Fewer Bits of Key
Randomization of quantum states is the quantum analogue of the classical
one-time pad. We present an improved, efficient construction of an
approximately randomizing map that uses O(d/epsilon^2) Pauli operators to map
any d-dimensional state to a state that is within trace distance epsilon of the
completely mixed state. Our bound is a log d factor smaller than that of
Hayden, Leung, Shor, and Winter (2004), and Ambainis and Smith (2004).
Then, we show that a random sequence of essentially the same number of
unitary operators, chosen from an appropriate set, with high probability form
an approximately randomizing map for d-dimensional states. Finally, we discuss
the optimality of these schemes via connections to different notions of
pseudorandomness, and give a new lower bound for small epsilon.Comment: 18 pages, Quantum Computing Back Action, IIT Kanpur, March 2006,
volume 864 of AIP Conference Proceedings, pages 18--36. Springer, New Yor
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