Clean quantum and classical communication protocols

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 pre-shared 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 whilst 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 pre-shared entanglement. Both of these values are maximal for their respective paradigms.