signals to Alice. This method requires $O(\sqrt{K})$ qubits transmitted forward and $O(\sqrt{K})$ qubits transmitted backwards to disentangle.

We remark that the shifting operation can be done rather easily in distributed quantum computation between Alice and Bob. This is a non-trivial observation because the number of qubits to be shifted from Alice to Bob is itself a quantum mechanical variable. This, however, does not create much problem. Bob can always communicate with Alice using a bus of fixed length. For example, he applies local operations to swap the desired quantum superposition of various numbers of qubits from his tape to the bus, sends such a bus to Alice, etc.

The result is the following theorem:

Alice and Bob may truncate a communication session after the transmission of $m$ encoded signals, retaining high fidelity with the cost of $O(\sqrt{m})$ additional qubits transmitted.

In the above discussion, we have focused on the simple case when Bob would like to perform a measurement on the whole set of the first $m$ signals. Suppose Bob is interested only in a particular signal, say the $m$-th one, but not the others. There exists a more efficient scheme for doing it. We shall skip the discussion here.

