1,802 research outputs found
Sculpting Quantum Speedups
Given a problem which is intractable for both quantum and classical
algorithms, can we find a sub-problem for which quantum algorithms provide an
exponential advantage? We refer to this problem as the "sculpting problem." In
this work, we give a full characterization of sculptable functions in the query
complexity setting. We show that a total function f can be restricted to a
promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only
if f has a large number of inputs with large certificate complexity. The proof
uses some interesting techniques: for one direction, we introduce new
relationships between randomized and quantum query complexity in various
settings, and for the other direction, we use a recent result from
communication complexity due to Klartag and Regev. We also characterize
sculpting for other query complexity measures, such as R(f) vs. R_0(f) and
R_0(f) vs. D(f).
Along the way, we prove some new relationships for quantum query complexity:
for example, a nearly quadratic relationship between Q(f) and D(f) whenever the
promise of f is small. This contrasts with the recent super-quadratic query
complexity separations, showing that the maximum gap between classical and
quantum query complexities is indeed quadratic in various settings - just not
for total functions!
Lastly, we investigate sculpting in the Turing machine model. We show that if
there is any BPP-bi-immune language in BQP, then every language outside BPP can
be restricted to a promise which places it in PromiseBQP but not in PromiseBPP.
Under a weaker assumption, that some problem in BQP is hard on average for
P/poly, we show that every paddable language outside BPP is sculptable in this
way.Comment: 30 page
The zero-error randomized query complexity of the pointer function
The pointer function of G{\"{o}}{\"{o}}s, Pitassi and Watson
\cite{DBLP:journals/eccc/GoosP015a} and its variants have recently been used to
prove separation results among various measures of complexity such as
deterministic, randomized and quantum query complexities, exact and approximate
polynomial degrees, etc. In particular, the widest possible (quadratic)
separations between deterministic and zero-error randomized query complexity,
as well as between bounded-error and zero-error randomized query complexity,
have been obtained by considering {\em
variants}~\cite{DBLP:journals/corr/AmbainisBBL15} of this pointer function.
However, as was pointed out in \cite{DBLP:journals/corr/AmbainisBBL15}, the
precise zero-error complexity of the original pointer function was not known.
We show a lower bound of on the zero-error
randomized query complexity of the pointer function on bits;
since an upper bound is already known
\cite{DBLP:conf/fsttcs/MukhopadhyayS15}, our lower bound is optimal up to a
factor of \polylog\, n
Quantum complexities of ordered searching, sorting, and element distinctness
We consider the quantum complexities of the following three problems:
searching an ordered list, sorting an un-ordered list, and deciding whether the
numbers in a list are all distinct. Letting N be the number of elements in the
input list, we prove a lower bound of \frac{1}{\pi}(\ln(N)-1) accesses to the
list elements for ordered searching, a lower bound of \Omega(N\log{N}) binary
comparisons for sorting, and a lower bound of \Omega(\sqrt{N}\log{N}) binary
comparisons for element distinctness. The previously best known lower bounds
are {1/12}\log_2(N) - O(1) due to Ambainis, \Omega(N), and \Omega(\sqrt{N}),
respectively. Our proofs are based on a weighted all-pairs inner product
argument.
In addition to our lower bound results, we give a quantum algorithm for
ordered searching using roughly 0.631 \log_2(N) oracle accesses. Our algorithm
uses a quantum routine for traversing through a binary search tree faster than
classically, and it is of a nature very different from a faster algorithm due
to Farhi, Goldstone, Gutmann, and Sipser.Comment: This new version contains new results. To appear at ICALP '01. Some
of the results have previously been presented at QIP '01. This paper subsumes
the papers quant-ph/0009091 and quant-ph/000903
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