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 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.
Communicational complexity problem among three parties for the calculation of a three party inner product modular function is discussed, where each party possess some of the function’s input. Classical communicational complexity of this function can be evaluated by three classical bits. In classical theory, the three party modular function can’t be evaluated by two classical bits, but using quantum entanglement in quantum theory two classical bits are sufficient to calculate the three party problem.
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.
Abstract. We give a direct product theorem for the entanglement-assisted interactive quantum communication complexity of an [Formula: see text]-player predicate [Formula: see text]. In particular, we show that for a distribution [Formula: see text] that is product across the input sets of the [Formula: see text] players, the success probability of any entanglement-assisted quantum communication protocol for computing [Formula: see text] copies of [Formula: see text], whose communication is [Formula: see text], goes down exponentially in [Formula: see text]. Here [Formula: see text] is a distributional version of the quantum efficiency or partition bound introduced in [S. Laplante, V. Lerays, and J. Roland, Classical and quantum partition bound and detector inefficiency, in Automata, Languages, and Programming, Springer, Berlin, Heidelberg, 2012, pp. 617–628], which is a lower bound on the distributional quantum communication complexity of computing a single copy of [Formula: see text] with respect to [Formula: see text]. Applying our direct product theorem for small communication, and techniques related to [Formula: see text], we show that it is possible to do device-independent (DI) quantum cryptography without the assumption that devices do not leak any information. We analyze parallel and sequential versions of the DI quantum key distribution protocol given in [R. Jain, C. A. Miller, and Y. Shi [ IEEE Trans. Inform. Theory, 66 (2020), pp. 5567–5584], and show that it is possible to extract [Formula: see text] bits of key from it, even in the presence of [Formula: see text] bits of leakage. Finally, we show that proofs of quantumness with two entangled provers are resistant to leakage, i.e., classical players who communicate [Formula: see text] bits with each other cannot convince the verifier that they share entanglement.
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as ``quantum teleportation'' and ``superdense coding.'' The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.
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.
Private simultaneous quantum messages (PSQM) provide a minimal model of noninteractive secure multiparty quantum computation, yet their communication complexity for concrete functions remains poorly understood. We show that familiar total functions already exhibit both genuine quantum advantages and strong structural limitations. For three-party equality on n-bit inputs, which determines whether the players' n-bit strings are identical, we construct an entanglement-assisted protocol with asymptotic communication (1.633…)n qubits, whereas every perfectly correct protocol without prior entanglement requires at least 3n qubits. For three-bit majority, one shared GHZ state and one transmitted qubit per player suffice for perfect correctness and perfect privacy, reducing the communication from the optimal classical cost of six bits to three qubits. Three-bit AND behaves differently. No perfectly correct and perfectly private protocol exists when the players share an arbitrary three-qubit state, apply arbitrary local unitary encodings, and transmit one qubit each. However, increasing the local dimension overcomes this obstruction: one shared three-qutrit GHZ state and one transmitted qutrit per player yield a one-shot protocol with total communication 3log23. Together with the six-bit classical lower bound, this gives a strict quantum communication advantage. More generally, for every k≥3, we construct a perfectly correct and perfectly private one-shot protocol for k-bit AND using one qudit of dimension at most k per player, and hence at most klog2k qubits in total.