12 research outputs found
Two Results about Quantum Messages
We show two results about the relationship between quantum and classical
messages. Our first contribution is to show how to replace a quantum message in
a one-way communication protocol by a deterministic message, establishing that
for all partial Boolean functions we
have . This bound was previously
known for total functions, while for partial functions this improves on results
by Aaronson, in which either a log-factor on the right hand is present, or the
left hand side is , and in which also no entanglement is
allowed.
In our second contribution we investigate the power of quantum proofs over
classical proofs. We give the first example of a scenario, where quantum proofs
lead to exponential savings in computing a Boolean function. The previously
only known separation between the power of quantum and classical proofs is in a
setting where the input is also quantum.
We exhibit a partial Boolean function , such that there is a one-way
quantum communication protocol receiving a quantum proof (i.e., a protocol of
type QMA) that has cost for , whereas every one-way quantum
protocol for receiving a classical proof (protocol of type QCMA) requires
communication
New Bounds for the Garden-Hose Model
We show new results about the garden-hose model. Our main results include
improved lower bounds based on non-deterministic communication complexity
(leading to the previously unknown bounds for Inner Product mod 2
and Disjointness), as well as an upper bound for the
Distributed Majority function (previously conjectured to have quadratic
complexity). We show an efficient simulation of formulae made of AND, OR, XOR
gates in the garden-hose model, which implies that lower bounds on the
garden-hose complexity of the order will be
hard to obtain for explicit functions. Furthermore we study a time-bounded
variant of the model, in which even modest savings in time can lead to
exponential lower bounds on the size of garden-hose protocols.Comment: In FSTTCS 201
Quantum Query Complexity of Subgraph Isomorphism and Homomorphism
Let be a fixed graph on vertices. Let iff the input
graph on vertices contains as a (not necessarily induced) subgraph.
Let denote the cardinality of a maximum independent set of . In
this paper we show:
where
denotes the quantum query complexity of .
As a consequence we obtain a lower bounds for in terms of several
other parameters of such as the average degree, minimum vertex cover,
chromatic number, and the critical probability.
We also use the above bound to show that for any
, improving on the previously best known bound of . Until
very recently, it was believed that the quantum query complexity is at least
square root of the randomized one. Our bound for
matches the square root of the current best known bound for the randomized
query complexity of , which is due to Gr\"oger.
Interestingly, the randomized bound of for
still remains open.
We also study the Subgraph Homomorphism Problem, denoted by , and
show that .
Finally we extend our results to the -uniform hypergraphs. In particular,
we show an bound for quantum query complexity of the Subgraph
Isomorphism, improving on the previously known bound. For the
Subgraph Homomorphism, we obtain an bound for the same.Comment: 16 pages, 2 figure
Communication Memento: Memoryless Communication Complexity
We study the communication complexity of computing functions
in the memoryless
communication model. Here, Alice is given , Bob is given and their goal is to compute F(x,y) subject to the following
constraint: at every round, Alice receives a message from Bob and her reply to
Bob solely depends on the message received and her input x; the same applies to
Bob. The cost of computing F in this model is the maximum number of bits
exchanged in any round between Alice and Bob (on the worst case input x,y). In
this paper, we also consider variants of our memoryless model wherein one party
is allowed to have memory, the parties are allowed to communicate quantum bits,
only one player is allowed to send messages. We show that our memoryless
communication model capture the garden-hose model of computation by Buhrman et
al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13)
and the overlay communication complexity by Papakonstantinou et al. (CCC'14).
Thus the memoryless communication complexity model provides a unified framework
to study space-bounded communication models. We establish the following: (1) We
show that the memoryless communication complexity of F equals the logarithm of
the size of the smallest bipartite branching program computing F (up to a
factor 2); (2) We show that memoryless communication complexity equals
garden-hose complexity; (3) We exhibit various exponential separations between
these memoryless communication models.
We end with an intriguing open question: can we find an explicit function F
and universal constant c>1 for which the memoryless communication complexity is
at least ? Note that would imply a
lower bound for general formula size, improving
upon the best lower bound by Ne\v{c}iporuk in 1966.Comment: 30 Pages; several improvements to the presentation
EXPLORING DIFFERENT MODELS OF QUERY COMPLEXITY AND COMMUNICATION COMPLEXITY
Ph.DDOCTOR OF PHILOSOPH
On the Fine-Grained Query Complexity of Symmetric Functions
This paper explores a fine-grained version of the Watrous conjecture,
including the randomized and quantum algorithms with success probabilities
arbitrarily close to . Our contributions include the following:
i) An analysis of the optimal success probability of quantum and randomized
query algorithms of two fundamental partial symmetric Boolean functions given a
fixed number of queries. We prove that for any quantum algorithm computing
these two functions using queries, there exist randomized algorithms using
queries that achieve the same success probability as the
quantum algorithm, even if the success probability is arbitrarily close to 1/2.
ii) We establish that for any total symmetric Boolean function , if a
quantum algorithm uses queries to compute with success probability
, then there exists a randomized algorithm using queries to
compute with success probability on a
fraction of inputs, where can be arbitrarily small
positive values. As a corollary, we prove a randomized version of
Aaronson-Ambainis Conjecture for total symmetric Boolean functions in the
regime where the success probability of algorithms can be arbitrarily close to
1/2.
iii) We present polynomial equivalences for several fundamental complexity
measures of partial symmetric Boolean functions. Specifically, we first prove
that for certain partial symmetric Boolean functions, quantum query complexity
is at most quadratic in approximate degree for any error arbitrarily close to
1/2. Next, we show exact quantum query complexity is at most quadratic in
degree. Additionally, we give the tight bounds of several complexity measures,
indicating their polynomial equivalence.Comment: accepted in ISAAC 202
Decision Tree Complexity versus Block Sensitivity and Degree
Relations between the decision tree complexity and various other complexity
measures of Boolean functions is a thriving topic of research in computational
complexity. It is known that decision tree complexity is bounded above by the
cube of block sensitivity, and the cube of polynomial degree. However, the
widest separation between decision tree complexity and each of block
sensitivity and degree that is witnessed by known Boolean functions is
quadratic. In this work, we investigate the tightness of the existing cubic
upper bounds.
We improve the cubic upper bounds for many interesting classes of Boolean
functions. We show that for graph properties and for functions with a constant
number of alternations, both of the cubic upper bounds can be improved to
quadratic. We define a class of Boolean functions, which we call the zebra
functions, that comprises Boolean functions where each monotone path from 0^n
to 1^n has an equal number of alternations. This class contains the symmetric
and monotone functions as its subclasses. We show that for any zebra function,
decision tree complexity is at most the square of block sensitivity, and
certificate complexity is at most the square of degree.
Finally, we show using a lifting theorem of communication complexity by
G{\"{o}}{\"{o}}s, Pitassi and Watson that the task of proving an improved upper
bound on the decision tree complexity for all functions is in a sense
equivalent to the potentially easier task of proving a similar upper bound on
communication complexity for each bi-partition of the input variables, for all
functions. In particular, this implies that to bound the decision tree
complexity it suffices to bound smaller measures like parity decision tree
complexity, subcube decision tree complexity and decision tree rank, that are
defined in terms of models that can be efficiently simulated by communication
protocols
Graph Properties in Node-Query Setting: Effect of Breaking Symmetry
The query complexity of graph properties is well-studied when queries are on the edges. We investigate the same when queries are on the nodes. In this setting a graph G = (V,E) on n vertices and a property P are given. A black-box access to an unknown subset S of V is provided via queries of the form "Does i belong to S?". We are interested in the minimum number of queries needed in the worst case in order to determine whether G[S] - the subgraph of G induced on S - satisfies P.
Our primary motivation to study this model comes from the fact that it allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on the hereditary graph properties - properties that are closed under deletion of vertices as well as edges. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n.
We show that in the absence of any symmetry on G it can fall as low as O(n^{1/(d + 1)}) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P. In particular, every hereditary property benefits at least quadratically. The main question left open is: Can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Omega(n^{1/k}) for a constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Omega(log(n)*log(log(n))) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdos and Rado
Secure Software Leasing Without Assumptions
Quantum cryptography is known for enabling functionalities that are
unattainable using classical information alone. Recently, Secure Software
Leasing (SSL) has emerged as one of these areas of interest. Given a target
circuit from a circuit class, SSL produces an encoding of that enables
a recipient to evaluate , and also enables the originator of the software to
verify that the software has been returned -- meaning that the recipient has
relinquished the possibility of any further use of the software. Clearly, such
a functionality is unachievable using classical information alone, since it is
impossible to prevent a user from keeping a copy of the software. Recent
results have shown the achievability of SSL using quantum information for a
class of functions called compute-and-compare (these are a generalization of
the well-known point functions). These prior works, however all make use of
setup or computational assumptions. Here, we show that SSL is achievable for
compute-and-compare circuits without any assumptions.
Our technique involves the study of quantum copy-protection, which is a
notion related to SSL, but where the encoding procedure inherently prevents a
would-be quantum software pirate from splitting a single copy of an encoding
for into two parts, each of which enables a user to evaluate . We show
that point functions can be copy-protected without any assumptions, for a novel
security definition involving one honest and one malicious evaluator; this is
achieved by showing that from any quantum message authentication code, we can
derive such an honest-malicious copy-protection scheme. We then show that a
generic honest-malicious copy-protection scheme implies SSL; by prior work,
this yields SSL for compute-and-compare functions.Comment: 41 pages, 5 figure