A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a {\em total} Boolean function in the two-party interactive model. Razborov's result ({\em Izvestiya: Mathematics}, 67(1):145--159, 2002) implies the conjectured negative answer for functions
F of the following form:
F(x,y)=fn(x1⋅y1,x2⋅y2,...,xn⋅yn), where
fn is a {\em symmetric} Boolean function on
n Boolean inputs, and
xi,
yi are the
i'th bit of
x and
y, respectively. His proof critically depends on the symmetry of
fn. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions
F is the ``block-composition'' of a ``building block''
gk:{0,1}k×{0,1}k→{0,1}, and an
fn:{0,1}n→{0,1}, such that
F(x,y)=fn(gk(x1,y1),gk(x2,y2),...,gk(xn,yn)), where
xi and
yi are the
i'th
k-bit block of
x,y∈{0,1}nk, respectively. We show that as long as g_k itself is "hard'' enough, its block-composition with an arbitrary f_n has polynomially related quantum and classical communication complexities. For example, when g_k is the Inner Product function with k=\Omega(\log n), the deterministic communication complexity of its block-composition with any f_n is asymptotically at most the quantum complexity to the power of 7.
A major open problem in quantum communication complexity is whether quantum protocols can be exponentially more efficient than classical protocols for computing total Boolean functions; the prevailing conjecture is that they cannot be so. In a seminal work, Razborov (2002) resolved this question for AND-functions of the form
F(x,y)=f(x1∧y1,…,xn∧yn), when the outer function
f is symmetric, by proving that their bounded-error quantum and classical communication complexities are polynomially related. Since then, extending this result to all AND-functions has remained open and has been posed by several authors.
In this work, we settle this problem in a strong way. We show that for every Boolean function
f, the bounded-error quantum and classical deterministic communication complexities of the function
f∘AND2 are polynomially related, up to polylogarithmic factors in
n. We prove this by showing that both are characterized--up to polynomial loss--by the logarithm of the De Morgan sparsity of
f.
Our results build on the recent work of Chattopadhyay, Dahiya, and Lovett (2025) on structural characterizations of non-sparse Boolean functions, which we extend to resolve the conjecture for general AND-functions.
This dissertation is motivated by the following fundamental questions: (a) are there any exponential gaps between quantum and classical communication complexities? (b) what is the role of entanglement in assisting quantum communications? (c) how to characterize the nonlocality of quantum operations? We study four specific problems below. 1. The communication complexity of the Hamming Distance problem. The Hamming Distance problem is for two parties to determine whether or not the Hamming distance between two n-bit strings is more than a given threshold. We prove tighter quantum lower bounds in the general two-party, interactive communication model. We also construct an efficient classical protocol in the more restricted Simultaneous Message Passing model, improving previous results. 2. The Log-Equivalence Conjecture. A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the two-party, interactive model. The answer is believed to be No. Razborov proved this conjecture for the most general class of functions so far. We prove this conjecture for a broader class of functions that we called block-composed functions. Our proof appears to be the first demonstration of the dual approach of the polynomial method in proving new results. 3. Classical simulations of bipartite quantum measurement. We define a new ix concept that measures the nonlocality of bipartite quantum operations. From this measure, we derive an upper bound that shows the limitation of entanglement in reducing communication costs. 4. The maximum tensor norm of bipartite superoperators. We define a maximum tensor norm to quantify the nonlocality of bipartite superoperators. We show that a bipartite physically realizable superoperator is bi-local if and only if its maximum tensor norm is 1. Furthermore, the estimation of the maximum tensor norm can also be used to prove quantum lower bounds on communication complexities.
I show that a simple multiparty communication task can be performed more efficiently with quantum communication than with classical communication, even with low detection efficiency $\ensuremath{\eta}.$ The task is a communication complexity problem in which distant parties need to compute a function of the distributed inputs, while minimizing the amount of communication between them. A realistic quantum optical setup is suggested that can demonstrate a five-party quantum protocol with higher-than-classical performance, provided \ensuremath{\eta}>0.33.
Quantum entanglement cannot be used to achieve direct communication between remote parties, but it can reduce the communication needed for some problems. Let each of
k parties hold some partial input data to some fixed
k-variable function
f. The communication complexity of
f is the minimum number of classical bits required to be broadcasted for every party to know the value of
f on their inputs. We construct a function
G such that for the one-round communication model and three parties,
G can be computed with
n+1 bits of communication when the parties share prior entanglement. We then show that without entangled particles, the one-round communication complexity of
G is
(3/2)n+1. Next we generalize this function to a function
F. We show that if the parties share prior quantum entanglement, then the communication complexity of
F is exactly
k. We also show that, if no entangled particles are provided, then the communication complexity of
F is roughly
klog2k. These two results prove communication complexity separations better than a constant number of bits.
One of the most intriguing facts about communication using quantum states is that these states cannot be used to transmit more classical bits than the number of qubits used, yet there are ways of conveying information with exponentially fewer qubits than possible classically. Moreover, these methods have a very simple structure---they involve little interaction between the communicating parties. We look more closely at the ways in which information encoded in quantum states may be manipulated, and consider the question as to whether every classical protocol may be transformed to a ``simpler'' quantum protocol of similar efficiency. By a simpler protocol, we mean a protocol that uses fewer message exchanges. We show that for any constant k, there is a problem such that its k+1 message classical communication complexity is exponentially smaller than its k message quantum communication complexity, thus answering the above question in the negative. Our result builds on two primitives, local transitions in bi-partite states (based on previous work) and average encoding which may be of significance in other applications as well.