10,374 research outputs found
Improved Quantum Communication Complexity Bounds for Disjointness and Equality
We prove new bounds on the quantum communication complexity of the
disjointness and equality problems. For the case of exact and non-deterministic
protocols we show that these complexities are all equal to n+1, the previous
best lower bound being n/2. We show this by improving a general bound for
non-deterministic protocols of de Wolf. We also give an O(sqrt{n}c^{log^*
n})-qubit bounded-error protocol for disjointness, modifying and improving the
earlier O(sqrt{n}log n) protocol of Buhrman, Cleve, and Wigderson, and prove an
Omega(sqrt{n}) lower bound for a large class of protocols that includes the
BCW-protocol as well as our new protocol.Comment: 11 pages LaTe
The Quantum Complexity of Set Membership
We study the quantum complexity of the static set membership problem: given a
subset S (|S| \leq n) of a universe of size m (m \gg n), store it as a table of
bits so that queries of the form `Is x \in S?' can be answered. The goal is to
use a small table and yet answer queries using few bitprobes. This problem was
considered recently by Buhrman, Miltersen, Radhakrishnan and Venkatesh, where
lower and upper bounds were shown for this problem in the classical
deterministic and randomized models. In this paper, we formulate this problem
in the "quantum bitprobe model" and show tradeoff results between space and
time.In this model, the storage scheme is classical but the query scheme is
quantum.We show, roughly speaking, that similar lower bounds hold in the
quantum model as in the classical model, which imply that the classical upper
bounds are more or less tight even in the quantum case. Our lower bounds are
proved using linear algebraic techniques.Comment: 19 pages, a preliminary version appeared in FOCS 2000. This is the
journal version, which will appear in Algorithmica (Special issue on Quantum
Computation and Quantum Cryptography). This version corrects some bugs in the
parameters of some theorem
Separations in Query Complexity Based on Pointer Functions
In 1986, Saks and Wigderson conjectured that the largest separation between
deterministic and zero-error randomized query complexity for a total boolean
function is given by the function on bits defined by a complete
binary tree of NAND gates of depth , which achieves . We show this is false by giving an example of a total
boolean function on bits whose deterministic query complexity is
while its zero-error randomized query complexity is . We further show that the quantum query complexity of the same
function is , giving the first example of a total function
with a super-quadratic gap between its quantum and deterministic query
complexities.
We also construct a total boolean function on variables that has
zero-error randomized query complexity and bounded-error
randomized query complexity . This is the first
super-linear separation between these two complexity measures. The exact
quantum query complexity of the same function is .
These two functions show that the relations and are optimal, up to poly-logarithmic factors. Further
variations of these functions give additional separations between other query
complexity measures: a cubic separation between and , a -power
separation between and , and a 4th power separation between
approximate degree and bounded-error randomized query complexity.
All of these examples are variants of a function recently introduced by
\goos, Pitassi, and Watson which they used to separate the unambiguous
1-certificate complexity from deterministic query complexity and to resolve the
famous Clique versus Independent Set problem in communication complexity.Comment: 25 pages, 6 figures. Version 3 improves separation between Q_E and
R_0 and updates reference
The quantum query complexity of the hidden subgroup problem is polynomial
We present a quantum algorithm which identifies with certainty a hidden
subgroup of an arbitrary finite group G in only a polynomial (in log |G|)
number of calls to the oracle. This is exponentially better than the best
classical algorithm. However our quantum algorithm requires exponential time,
as in the classical case. Our algorithm utilizes a new technique for
constructing error-free algorithms for non-decision problems on quantum
computers.Comment: To appear in Information Processing Letters (IPL
- …