2,009 research outputs found
How to Quantize Outputs of a Binary Symmetric Channel to Bits?
Suppose that is obtained by observing a uniform Bernoulli random vector
through a binary symmetric channel with crossover probability .
The "most informative Boolean function" conjecture postulates that the maximal
mutual information between and any Boolean function is
attained by a dictator function. In this paper, we consider the "complementary"
case in which the Boolean function is replaced by
, namely, an bit
quantizer, and show that
for any such . Thus, in this case, the optimal function is of the form
.Comment: 5 pages, accepted ISIT 201
Streaming Hardness of Unique Games
We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by p, the alphabet size of the Unique Game, gives a trivial p-approximation that can be computed in O(log n) space. Meanwhile, with high probability, a sample of O~(n) constraints suffices to estimate the optimal value to (1+epsilon) accuracy. We prove that any single-pass streaming algorithm that achieves a (p-epsilon)-approximation requires Omega_epsilon(sqrt n) space. Our proof is via a reduction from lower bounds for a communication problem that is a p-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to downstream hardness results for streaming approximability for other CSP-like problems
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
Guessing with a Bit of Help
What is the value of a single bit to a guesser? We study this problem in a
setup where Alice wishes to guess an i.i.d. random vector, and can procure one
bit of information from Bob, who observes this vector through a memoryless
channel. We are interested in the guessing efficiency, which we define as the
best possible multiplicative reduction in Alice's guessing-moments obtainable
by observing Bob's bit. For the case of a uniform binary vector observed
through a binary symmetric channel, we provide two lower bounds on the guessing
efficiency by analyzing the performance of the Dictator and Majority functions,
and two upper bounds via maximum entropy and Fourier-analytic /
hypercontractivity arguments. We then extend our maximum entropy argument to
give a lower bound on the guessing efficiency for a general channel with a
binary uniform input, via the strong data-processing inequality constant of the
reverse channel. We compute this bound for the binary erasure channel, and
conjecture that Greedy Dictator functions achieve the guessing efficiency
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