4,498 research outputs found
Exponential Lower Bound for 2-Query Locally Decodable Codes via a Quantum Argument
A locally decodable code encodes n-bit strings x in m-bit codewords C(x), in
such a way that one can recover any bit x_i from a corrupted codeword by
querying only a few bits of that word. We use a quantum argument to prove that
LDCs with 2 classical queries need exponential length: m=2^{Omega(n)}.
Previously this was known only for linear codes (Goldreich et al. 02). Our
proof shows that a 2-query LDC can be decoded with only 1 quantum query, and
then proves an exponential lower bound for such 1-query locally
quantum-decodable codes. We also show that q quantum queries allow more
succinct LDCs than the best known LDCs with q classical queries. Finally, we
give new classical lower bounds and quantum upper bounds for the setting of
private information retrieval. In particular, we exhibit a quantum 2-server PIR
scheme with O(n^{3/10}) qubits of communication, improving upon the O(n^{1/3})
bits of communication of the best known classical 2-server PIR.Comment: 16 pages Latex. 2nd version: title changed, large parts rewritten,
some results added or improve
Weak Parity
We study the query complexity of Weak Parity: the problem of computing the
parity of an n-bit input string, where one only has to succeed on a 1/2+eps
fraction of input strings, but must do so with high probability on those inputs
where one does succeed. It is well-known that n randomized queries and n/2
quantum queries are needed to compute parity on all inputs. But surprisingly,
we give a randomized algorithm for Weak Parity that makes only
O(n/log^0.246(1/eps)) queries, as well as a quantum algorithm that makes only
O(n/sqrt(log(1/eps))) queries. We also prove a lower bound of
Omega(n/log(1/eps)) in both cases; and using extremal combinatorics, prove
lower bounds of Omega(log n) in the randomized case and Omega(sqrt(log n)) in
the quantum case for any eps>0. We show that improving our lower bounds is
intimately related to two longstanding open problems about Boolean functions:
the Sensitivity Conjecture, and the relationships between query complexity and
polynomial degree.Comment: 18 page
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
The quantum complexity of approximating the frequency moments
The 'th frequency moment of a sequence of integers is defined as , where is the number of times that occurs in the
sequence. Here we study the quantum complexity of approximately computing the
frequency moments in two settings. In the query complexity setting, we wish to
minimise the number of queries to the input used to approximate up to
relative error . We give quantum algorithms which outperform the best
possible classical algorithms up to quadratically. In the multiple-pass
streaming setting, we see the elements of the input one at a time, and seek to
minimise the amount of storage space, or passes over the data, used to
approximate . We describe quantum algorithms for , and
in this model which substantially outperform the best possible
classical algorithms in certain parameter regimes.Comment: 22 pages; v3: essentially published versio
Claw Finding Algorithms Using Quantum Walk
The claw finding problem has been studied in terms of query complexity as one
of the problems closely connected to cryptography. For given two functions, f
and g, as an oracle which have domains of size N and M (N<=M), respectively,
and the same range, the goal of the problem is to find x and y such that
f(x)=g(y). This paper describes an optimal algorithm using quantum walk that
solves this problem. Our algorithm can be generalized to find a claw of k
functions for any constant integer k>1, where the domains of the functions may
have different size.Comment: 12 pages. Introduction revised. A reference added. Weak lower bound
delete
An optimal quantum algorithm for the oracle identification problem
In the oracle identification problem, we are given oracle access to an
unknown N-bit string x promised to belong to a known set C of size M and our
task is to identify x. We present a quantum algorithm for the problem that is
optimal in its dependence on N and M. Our algorithm considerably simplifies and
improves the previous best algorithm due to Ambainis et al. Our algorithm also
has applications in quantum learning theory, where it improves the complexity
of exact learning with membership queries, resolving a conjecture of Hunziker
et al.
The algorithm is based on ideas from classical learning theory and a new
composition theorem for solutions of the filtered -norm semidefinite
program, which characterizes quantum query complexity. Our composition theorem
is quite general and allows us to compose quantum algorithms with
input-dependent query complexities without incurring a logarithmic overhead for
error reduction. As an application of the composition theorem, we remove all
log factors from the best known quantum algorithm for Boolean matrix
multiplication.Comment: 16 pages; v2: minor change
- …