We consider a quantum and a classical version of a multi-party function computation problem with n players, where players 2,…,n need to communicate appropriate information to player 1 so that a “generalized” inner product function with an appropriate promise can be calculated. In the quantum version of the protocol, the players have access to entangled qudits but the communication is still classical. The communication complexity of a given protocol is the total number of classical bits that need to be communicated. When n is prime, and for our chosen function, we exhibit a quantum protocol (with complexity (n−1)(logn) bits) and a classical protocol (with complexity ((n−1)2(logn2) bits)). Furthermore, we present an integer linear programming formulation for determining a lower bound on the classical communication complexity. This demonstrates that our quantum protocol is strictly better than classical protocols.
We consider a variation of the multi-party communication complexity scenario where the parties are supplied with an extra resource: particles in an entangled quantum state. We show that, although a prior quantum entanglement cannot be used to simulate a communication channel, it can reduce the communication complexity of functions in some cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of the function with only three bits of communication occurring among the parties, whereas, without quantum entanglement, four bits of communication are necessary. We also show that, for a particular two-party probabilistic communication complexity problem, quantum entanglement results in less communication than is required with only classical random correlations (instead of quantum entanglement). These results are a noteworthy contrast to the well-known fact that quantum entanglement cannot be used to actually simulate communication among remote parties.
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.
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.
By how much must the communication complexity of a function increase if we demand that the parties not only correctly compute the function but also return all registers (other than the one containing the answer) to their initial states at the end of the communication protocol? Protocols that achieve this are referred to as clean and the associated cost as the clean communication complexity. Here we present clean protocols for calculating the inner product of two n-bit strings, showing that (in the absence of preshared entanglement) at most n+3 qubits or n+O(sqrt[n]) bits of communication are required. The quantum protocol provides inspiration for obtaining the optimal method to implement distributed cnot gates in parallel while minimizing the amount of quantum communication. For more general functions, we show that nearly all Boolean functions require close to 2n bits of classical communication to compute and close to n qubits if the parties have access to preshared entanglement. Both of these values are maximal for their respective paradigms.
In two-party quantum communication complexity, Alice and Bob receive some classical inputs and wish to compute some function that depends on both these inputs, while minimizing the communication. This model has found numerous applications in many areas of computer science. One notion that has received a lot of attention recently is the information cost of the protocol, namely how much information the players reveal about their inputs when they run the protocol. In the quantum world, it is not straightforward to define a notion of quantum information cost. We study two different notions and analyze their relation. We also provide new quantum protocols for the Inner Product function and for Private Information Retrieval, and show that protocols for Private Information Retrieval whose classical or quantum information cost for the user is zero can have exponentially different information cost for the server.