2,172 research outputs found
Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs
A strong direct product theorem says that if we want to compute k independent
instances of a function, using less than k times the resources needed for one
instance, then our overall success probability will be exponentially small in
k. We establish such theorems for the classical as well as quantum query
complexity of the OR function. This implies slightly weaker direct product
results for all total functions. We prove a similar result for quantum
communication protocols computing k instances of the Disjointness function.
Our direct product theorems imply a time-space tradeoff T^2*S=Omega(N^3) for
sorting N items on a quantum computer, which is optimal up to polylog factors.
They also give several tight time-space and communication-space tradeoffs for
the problems of Boolean matrix-vector multiplication and matrix multiplication.Comment: 22 pages LaTeX. 2nd version: some parts rewritten, results are
essentially the same. A shorter version will appear in IEEE FOCS 0
A New Quantum Lower Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs
We give a new version of the adversary method for proving lower bounds on
quantum query algorithms. The new method is based on analyzing the eigenspace
structure of the problem at hand. We use it to prove a new and optimal strong
direct product theorem for 2-sided error quantum algorithms computing k
independent instances of a symmetric Boolean function: if the algorithm uses
significantly less than k times the number of queries needed for one instance
of the function, then its success probability is exponentially small in k. We
also use the polynomial method to prove a direct product theorem for 1-sided
error algorithms for k threshold functions with a stronger bound on the success
probability. Finally, we present a quantum algorithm for evaluating solutions
to systems of linear inequalities, and use our direct product theorems to show
that the time-space tradeoff of this algorithm is close to optimal.Comment: 16 pages LaTeX. Version 2: title changed, proofs significantly
cleaned up and made selfcontained. This version to appear in the proceedings
of the STOC 06 conferenc
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
Improved Low-qubit Hidden Shift Algorithms
Hidden shift problems are relevant to assess the quantum security of various
cryptographic constructs. Multiple quantum subexponential time algorithms have
been proposed. In this paper, we propose some improvements on a polynomial
quantum memory algorithm proposed by Childs, Jao and Soukharev in 2010. We use
subset-sum algorithms to significantly reduce its complexity. We also propose
new tradeoffs between quantum queries, classical time and classical memory to
solve this problem
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
Quantum and classical strong direct product theorems and optimal time-space tradeoffs
A strong direct product theorem says that if we want to compute
independent instances of a function, using less than times
the resources needed for one instance, then our overall success
probability will be exponentially small in .
We establish such theorems for the classical as well as quantum
query complexity of the OR-function. This implies slightly
weaker direct product results for all total functions.
We prove a similar result for quantum communication
protocols computing instances of the disjointness function.
Our direct product theorems imply a time-space tradeoff
T^2S=\Om{N^3} for sorting items on a quantum computer, which
is optimal up to polylog factors. They also give several tight
time-space and communication-space tradeoffs for the problems of
Boolean matrix-vector multiplication and matrix multiplication
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