Communication Cost of Quantum Processes

A common scenario in distributed computing involves a client who asks a server to perform a computation on a remote computer. An important problem is to determine the minimum amount of communication needed to specify the desired computation. Here we extend this problem to the quantum domain, analyzing the total amount of (classical and quantum) communication needed by a server in order to accurately execute a quantum process chosen by a client from a parametric family of quantum processes. We derive a general lower bound on the communication cost, establishing a relation with the precision limits of quantum metrology: if a $\nu$-dimensional family of processes can be estimated with mean squared error $n^{-\beta}$ by using $n$ parallel queries, then the communication cost for $n$ parallel executions of a process in the family is at least $(\beta\nu/2-\epsilon)\log n$ qubits at the leading order in $n$, for every $\epsilon>0$. For a class of quantum processes satisfying the standard quantum limit ($\beta=1$), we show that the bound can be attained by transmitting an approximate classical description of the desired process. For quantum processes satisfying the Heisenberg limit ($\beta=2$), our bound shows that the communication cost is at least twice as the cost of communicating standard quantum limited processes with the same number of parameters.