220 research outputs found
Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range
We give a general method for proving quantum lower bounds for problems with
small range. Namely, we show that, for any symmetric problem defined on
functions , its polynomial degree is the same
for all . Therefore, if we have a quantum lower bound for some
(possibly, quite large) range which is shown using polynomials method, we
immediately get the same lower bound for all ranges . In particular,
we get and quantum lower bounds for
collision and element distinctness with small range.Comment: 9 pages, LaTeX, v2 new result on degree lower bound for AND-OR added,
v3 many small change
Quantum search with variable times
Since Grover's seminal work, quantum search has been studied in great detail.
In the usual search problem, we have a collection of n items and we would like
to find a marked item. We consider a new variant of this problem in which
evaluating the i-th item may take a different number of time steps for
different i.
Let t_i be the number of time steps required to evaluate the i-th item. If
the numbers t_i are known in advance, we give an algorithm that solves the
problem in O(\sqrt{t_1^2+t_2^2+...+t_n^2}) steps. This is optimal, as we also
show a matching lower bound. The case, when t_i are not known in advance, can
be solved with a polylogarithmic overhead. We also give an application of our
new search algorithm to computing read-once functions.Comment: 19 page
Variable time amplitude amplification and a faster quantum algorithm for solving systems of linear equations
We present two new quantum algorithms. Our first algorithm is a
generalization of amplitude amplification to the case when parts of the quantum
algorithm that is being amplified stop at different times.
Our second algorithm uses the first algorithm to improve the running time of
Harrow et al. algorithm for solving systems of linear equations from O(kappa^2
log N) to O(kappa log^3 kappa log N) where \kappa is the condition number of
the system of equations.Comment: 17 pages, no figures, v2: various small correction
Quantum algorithms for formula evaluation
We survey the recent sequence of algorithms for evaluating Boolean formulas
consisting of NAND gates.Comment: 11 pages, survey for NATO ARW "Quantum Cryptography and Computing",
Gdansk, September 200
A New Protocol and Lower Bounds for Quantum Coin Flipping
We present a new protocol and two lower bounds for quantum coin flipping. In
our protocol, no dishonest party can achieve one outcome with probability more
than 0.75. Then, we show that our protocol is optimal for a certain type of
quantum protocols.
For arbitrary quantum protocols, we show that if a protocol achieves a bias
of at most , it must use at least rounds of communication. This implies that the parallel
repetition fails for quantum coin flipping. (The bias of a protocol cannot be
arbitrarily decreased by running several copies of it in parallel.)Comment: 20 pages, submitted to Journal of Computer and System Sciences,
earlier version in proceedings of STOC'0
Understanding Quantum Algorithms via Query Complexity
Query complexity is a model of computation in which we have to compute a
function of variables which can be accessed via
queries. The complexity of an algorithm is measured by the number of queries
that it makes. Query complexity is widely used for studying quantum algorithms,
for two reasons. First, it includes many of the known quantum algorithms
(including Grover's quantum search and a key subroutine of Shor's factoring
algorithm). Second, one can prove lower bounds on the query complexity,
bounding the possible quantum advantage. In the last few years, there have been
major advances on several longstanding problems in the query complexity. In
this talk, we survey these results and related work, including:
- the biggest quantum-vs-classical gap for partial functions (a problem
solvable with 1 query quantumly but requiring queries
classically);
- the biggest quantum-vs-determistic and quantum-vs-probabilistic gaps for
total functions (for example, a problem solvable with queries quantumly but
requiring queries probabilistically);
- the biggest probabilistic-vs-deterministic gap for total functions (a
problem solvable with queries probabilistically but requiring
queries deterministically);
- the bounds on the gap that can be achieved for subclasses of functions (for
example, symmetric functions);
- the connections between query algorithms and approximations by low-degree
polynomials.Comment: 20 page survey of recent results, for Proceedings of International
Congress of Mathematicians'201
Polynomial degree vs. quantum query complexity
The degree of a polynomial representing (or approximating) a function f is a
lower bound for the number of quantum queries needed to compute f. This
observation has been a source of many lower bounds on quantum algorithms. It
has been an open problem whether this lower bound is tight.
We exhibit a function with polynomial degree M and quantum query complexity
\Omega(M^{1.321...}). This is the first superlinear separation between
polynomial degree and quantum query complexity. The lower bound is shown by a
new, more general version of quantum adversary method.Comment: 23 pages, 1 figure, v4 "proof by old method" corrected, moderate
changes to presentation elsewher
Quantum walk algorithm for element distinctness
We use quantum walks to construct a new quantum algorithm for element
distinctness and its generalization. For element distinctness (the problem of
finding two equal items among N given items), we get an O(N^{2/3}) query
quantum algorithm. This improves the previous O(N^{3/4}) query quantum
algorithm of Buhrman et.al. (quant-ph/0007016) and matches the lower bound by
Shi (quant-ph/0112086). The algorithm also solves the generalization of element
distinctness in which we have to find k equal items among N items. For this
problem, we get an O(N^{k/(k+1)}) query quantum algorithm.Comment: 33 pages, 1 figure, v9 typos with signs corrected on pages 11-1
A note on quantum black-box complexity of almost all Boolean functions
We show that, for almost all N-variable Boolean functions f, at least
N/4-O(\sqrt{N} log N) queries are required to compute f in quantum black-box
model with bounded error.Comment: 4 pages, LaTe
A nearly optimal discrete query quantum algorithm for evaluating NAND formulas
We present an O(\sqrt{N}) discrete query quantum algorithm for evaluating
balanced binary NAND formulas and an O(N^{{1/2}+O(\frac{1}{\sqrt{\log N}})})
discrete query quantum algorithm for evaluating arbitrary binary NAND formulas.Comment: 21 pages, 2 figure
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