31 research outputs found
Average-Case Quantum Query Complexity
We compare classical and quantum query complexities of total Boolean
functions. It is known that for worst-case complexity, the gap between quantum
and classical can be at most polynomial. We show that for average-case
complexity under the uniform distribution, quantum algorithms can be
exponentially faster than classical algorithms. Under non-uniform distributions
the gap can even be super-exponential. We also prove some general bounds for
average-case complexity and show that the average-case quantum complexity of
MAJORITY under the uniform distribution is nearly quadratically better than the
classical complexity.Comment: 14 pages, LaTeX. Some parts rewritten. This version to appear in the
Journal of Physics
Almost-Everywhere Superiority for Quantum Computing
Simon as extended by Brassard and H{\o}yer shows that there are tasks on
which polynomial-time quantum machines are exponentially faster than each
classical machine infinitely often. The present paper shows that there are
tasks on which polynomial-time quantum machines are exponentially faster than
each classical machine almost everywhere.Comment: 16 page
Exact Quantum Query Algorithm for Error Detection Code Verification
Quantum algorithms can be analyzed in a query model to compute Boolean functions.
Function input is provided in a black box, and the aim is to compute the function value using as few queries to the black box as possible.
A repetition code is an error detection scheme that repeats each bit of the original message r times.
After a message with redundant bits is transmitted via a communication channel, it must be verified.
If the received message consists of r-size blocks of equal bits, the conclusion is that there were no errors.
The verification procedure can be interpreted as an application of a query algorithm, where input is a message to be checked.
Classically, for N-bit message, values of all N variables must be queried. We demonstrate an exact quantum algorithm that uses only N/2 queries
Quantum walk speedup of backtracking algorithms
We describe a general method to obtain quantum speedups of classical
algorithms which are based on the technique of backtracking, a standard
approach for solving constraint satisfaction problems (CSPs). Backtracking
algorithms explore a tree whose vertices are partial solutions to a CSP in an
attempt to find a complete solution. Assume there is a classical backtracking
algorithm which finds a solution to a CSP on n variables, or outputs that none
exists, and whose corresponding tree contains T vertices, each vertex
corresponding to a test of a partial solution. Then we show that there is a
bounded-error quantum algorithm which completes the same task using O(sqrt(T)
n^(3/2) log n) tests. In particular, this quantum algorithm can be used to
speed up the DPLL algorithm, which is the basis of many of the most efficient
SAT solvers used in practice. The quantum algorithm is based on the use of a
quantum walk algorithm of Belovs to search in the backtracking tree. We also
discuss how, for certain distributions on the inputs, the algorithm can lead to
an exponential reduction in expected runtime.Comment: 23 pages; v2: minor changes to presentatio
Provably secure key establishment against quantum adversaries
At Crypto 2011, some of us had proposed a family of cryptographic protocols
for key establishment capable of protecting quantum and classical legitimate
parties unconditionally against a quantum eavesdropper in the query complexity
model. Unfortunately, our security proofs were unsatisfactory from a
cryptographically meaningful perspective because they were sound only in a
worst-case scenario. Here, we extend our results and prove that for any e > 0,
there is a classical protocol that allows the legitimate parties to establish a
common key after O(N) expected queries to a random oracle, yet any quantum
eavesdropper will have a vanishing probability of learning their key after
O(N^{1.5-e}) queries to the same oracle. The vanishing probability applies to a
typical run of the protocol. If we allow the legitimate parties to use a
quantum computer as well, their advantage over the quantum eavesdropper becomes
arbitrarily close to the quadratic advantage that classical legitimate parties
enjoyed over classical eavesdroppers in the seminal 1974 work of Ralph Merkle.
Along the way, we develop new tools to give lower bounds on the number of
quantum queries required to distinguish two probability distributions. This
method in itself could have multiple applications in cryptography. We use it
here to study average-case quantum query complexity, for which we develop a new
composition theorem of independent interest.Comment: 22 pages, no figures, fixes a problem with arXiv:1108.2316v2. Will
appear in the Proceedings of the 12th Conference on Theory of Quantum
Computation, Communication and Cryptography (TQC), Paris, June 2017. The only
change in v2 is that there was a problem with the affiliations in v