210 research outputs found
Spectral Norm of Symmetric Functions
The spectral norm of a Boolean function is the sum
of the absolute values of its Fourier coefficients. This quantity provides
useful upper and lower bounds on the complexity of a function in areas such as
learning theory, circuit complexity, and communication complexity. In this
paper, we give a combinatorial characterization for the spectral norm of
symmetric functions. We show that the logarithm of the spectral norm is of the
same order of magnitude as where ,
and and are the smallest integers less than such that
or is constant for all with . We mention some applications to the decision tree and communication
complexity of symmetric functions
On the Composition of Randomized Query Complexity and Approximate Degree
For any Boolean functions f and g, the question whether R(f?g) = ??(R(f) ? R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg?(f?g) = ??(deg?(f)?deg?(g)). These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily.
It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg? compose.
A recent landmark result (Ben-David and Blais, 2020) showed that R(f?g) = ?(noisyR(f)? R(g)). This implies that composition holds whenever noisyR(f) = ??(R(f)). We show two results:
1. When R(f) = ?(n), then noisyR(f) = ?(R(f)). In other words, composition holds whenever the randomized query complexity of the outer function is full.
2. If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg?(f?g) = ?(M(f) ? deg?(g)) (for some non-trivial complexity measure M(?)) was known to the best of our knowledge. We prove that deg?(f?g) = ??(?{bs(f)} ? deg?(g)), where bs(f) is the block sensitivity of f. This implies that deg? composes when deg?(f) is asymptotically equal to ?{bs(f)}.
It is already known that both R and deg? compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function
Approximate degree in classical and quantum computing
In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over the real numbers.FG-2022-18482 - Alfred P. Sloan Foundation; CNS-2046425 - National Science Foundation; CCF-1947889 - National Science FoundationAccepted manuscrip
On the Composition of Randomized Query Complexity and Approximate Degree
For any Boolean functions and , the question whether , is known as the composition question for the
randomized query complexity. Similarly, the composition question for the
approximate degree asks whether . These questions are
two of the most important and well-studied problems, and yet we are far from
answering them satisfactorily.
It is known that the measures compose if one assumes various properties of
the outer function (or inner function ). This paper extends the class of
outer functions for which and compose.
A recent landmark result (Ben-David and Blais, 2020) showed that . This implies that composition holds whenever
noisyR(f) = \Tilde{\Theta}(R(f)). We show two results:
(1)When , then .
(2) If composes with respect to an outer function, then
also composes with respect to the same outer function. On the
other hand, no result of the type (for some non-trivial complexity measure )
was known to the best of our knowledge. We prove that
where is the block sensitivity of . This implies that
composes when is
asymptotically equal to .
It is already known that both and compose
when the outer function is symmetric. We also extend these results to weaker
notions of symmetry with respect to the outer function
Quantum Query-To-Communication Simulation Needs a Logarithmic Overhead
Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean
function and the two-party bounded-error quantum communication complexity of is , where is the bounded-error quantum query
complexity of . Note that the bounded-error randomized communication
complexity of is bounded by , where denotes
the bounded-error randomized query complexity of . Thus, the BCW simulation
has an extra factor appearing that is absent in classical
simulation. A natural question is if this factor can be avoided. H{\o}yer and
de Wolf (STACS'02) showed that for the Set-Disjointness function, this can be
reduced to for some constant , and subsequently Aaronson and
Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the
quantum communication complexity of the Set-Disjointness function (which is
) is .
Perhaps somewhat surprisingly, we show that when , then
the extra factor in the BCW simulation is unavoidable. In other words,
we exhibit a total function such that .
To the best of our knowledge, it was not even known prior to this work
whether there existed a total function and 2-bit function , such
that
On Query-To-Communication Lifting for Adversary Bounds
We investigate query-to-communication lifting theorems for models related to the quantum adversary bounds. Our results are as follows:
1) We show that the classical adversary bound lifts to a lower bound on randomized communication complexity with a constant-sized gadget. We also show that the classical adversary bound is a strictly stronger lower bound technique than the previously-lifted measure known as critical block sensitivity, making our lifting theorem one of the strongest lifting theorems for randomized communication complexity using a constant-sized gadget.
2) Turning to quantum models, we show a connection between lifting theorems for quantum adversary bounds and secure 2-party quantum computation in a certain "honest-but-curious" model. Under the assumption that such secure 2-party computation is impossible, we show that a simplified version of the positive-weight adversary bound lifts to a quantum communication lower bound using a constant-sized gadget. We also give an unconditional lifting theorem which lower bounds bounded-round quantum communication protocols.
3) Finally, we give some new results in query complexity. We show that the classical adversary and the positive-weight quantum adversary are quadratically related. We also show that the positive-weight quantum adversary is never larger than the square of the approximate degree. Both relations hold even for partial functions
Quantum Distinguishing Complexity, Zero-Error Algorithms, and Statistical Zero Knowledge
We define a new query measure we call quantum distinguishing complexity, denoted QD(f) for a Boolean function f. Unlike a quantum query algorithm, which must output a state close to |0> on a 0-input and a state close to |1> on a 1-input, a "quantum distinguishing algorithm" can output any state, as long as the output states for any 0-input and 1-input are distinguishable.
Using this measure, we establish a new relationship in query complexity: For all total functions f, Q_0(f)=O~(Q(f)^5), where Q_0(f) and Q(f) denote the zero-error and bounded-error quantum query complexity of f respectively, improving on the previously known sixth power relationship.
We also define a query measure based on quantum statistical zero-knowledge proofs, QSZK(f), which is at most Q(f). We show that QD(f) in fact lower bounds QSZK(f) and not just Q(f). QD(f) also upper bounds the (positive-weights) adversary bound, which yields the following relationships for all f: Q(f) >= QSZK(f) >= QD(f) = Omega(Adv(f)). This sheds some light on why the adversary bound proves suboptimal bounds for problems like Collision and Set Equality, which have low QSZK complexity.
Lastly, we show implications for lifting theorems in communication complexity. We show that a general lifting theorem for either zero-error quantum query complexity or for QSZK would imply a general lifting theorem for bounded-error quantum query complexity
Certificate games
We introduce and study Certificate Game complexity, a measure of complexity
based on the probability of winning a game where two players are given inputs
with different function values and are asked to output such that (zero-communication setting).
We give upper and lower bounds for private coin, public coin, shared
entanglement and non-signaling strategies, and give some separations. We show
that complexity in the public coin model is upper bounded by Randomized query
and Certificate complexity. On the other hand, it is lower bounded by
fractional and randomized certificate complexity, making it a good candidate to
prove strong lower bounds on randomized query complexity. Complexity in the
private coin model is bounded from below by zero-error randomized query
complexity.
The quantum measure highlights an interesting and surprising difference
between classical and quantum query models. Whereas the public coin certificate
game complexity is bounded from above by randomized query complexity, the
quantum certificate game complexity can be quadratically larger than quantum
query complexity. We use non-signaling, a notion from quantum information, to
give a lower bound of on the quantum certificate game complexity of the
function, whose quantum query complexity is , then go on
to show that this ``non-signaling bottleneck'' applies to all functions with
high sensitivity, block sensitivity or fractional block sensitivity.
We consider the single-bit version of certificate games (inputs of the two
players have Hamming distance ). We prove that the single-bit version of
certificate game complexity with shared randomness is equal to sensitivity up
to constant factors, giving a new characterization of sensitivity. The
single-bit version with private randomness is equal to , where
is the spectral sensitivity.Comment: 43 pages, 1 figure, ITCS202
- …