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 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.

Figure 1

Clean, quantum protocol for calculating ${IP}_n$ in the phase. Here we illustrate the first four rounds of communication. In each round, a player cleans up the message they sent previously, applies the relevant global phase, and communicates the next bit of their input string.

Figure 2

Partitions of the communication matrix into rectangles. Note that knowledge of $y$, together with knowledge of which rectangle the players’ input pair belongs to, allows Bob to correctly deduce the value of $f(x,y)$. (a) As there exists a protocol for computing $f$ that partitions $M^f$ into large rectangles, the Kolmogorov complexity of $M^f$ is low. (b) For $M^f$ to have high Kolmogorov complexity, all protocols for computing $f$ must partition $M^f$ into either very narrow or very thin rectangles. To produce the bound in Eq. (9), we take a distribution over the shaded rectangles.

Figure 3

Schematic of a classical communication protocol. Here we show how the random variables held by each player change during round $i$ of a communication protocol. Primed variables denote local memories, while nonprimed variables are communication. Each player uses a deterministic, reversible function ($S_i$ and $T_i$) to determine their next message and update their local memory.