We address the question of efficient implementation of quantum protocols, with short depth circuits and small additional resource. We introduce two new methods in this direction. The first method, inspired by the technique of classical correlated sampling, is to unitarily extend a given quantum state into a quantum state uniform in a subspace. The second method involves two new versions of the convex-split lemma that use exponentially small amount of additional resource in comparison to the previous quantum version. Using these methods, we obtain the following results.
First, we consider the task of quantum decoupling on n qubits. Most previous works achieve decoupling with the aid of a random unitary followed by discarding some qubits. The random unitaries can be replaced by random quantum circuits or unitary 2 designs of size O(nlogn) and depth poly(logn). We show that given any choice of basis such as the computational basis, decoupling can be achieved by a unitary that takes basis vectors to basis vectors. Thus, the circuit acts in a `classical' manner. Our unitary performs addition and multiplication modulo a prime and hence achieves O(nlogn) circuit size and logarithmic depth.
Next, we construct a new one-shot entanglement-assisted protocol for quantum channel coding that achieves near-optimal communication through a given channel. The number of qubits of entanglement used in this protocol is proportional to the number of qubits input to the channel. Previous one-shot works were either near-optimal in communication but required exponentially more entanglement, or required small amount of entanglement but did not achieve near-optimal communication. We also achieve similar exponential improvement in the entanglement required for one-shot quantum state redistribution, while keeping the communication similar to the best known achievable communication.