Efficient methods for one-shot quantum communication

We address the question of efficient implementation of quantum protocols, with small communication and entanglement, and short depth circuit for encoding or decoding. We introduce two new methods to achieve this, the first method involving two new versions of convex-split lemma that use small amount of additional resource (in comparison to prior version) and the second method being inspired by the technique of classical correlated sampling in computer science. These lead to a series of new consequences, as follows. First, we consider the task of quantum decoupling, where the aim is to apply an operation on a n-qubit register so as to make it independent of an inaccessible quantum system. Many previous works achieve decoupling with the aid of a random unitary. It is known that random unitaries can be replaced by random circuits of size O(n\log n) and depth poly(\log n), or unitary 2-designs based on Clifford circuits of similar size and depth. 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 and additionally uses O(n) catalytic qubits in maximally mixed quantum state. Our unitary performs addition and multiplication modulo a prime and hence achieves a circuit size of O(n\log n) and logarithmic depth. This shows that the circuit complexity of integer multiplication (modulo a prime) is lower bounded by the optimal circuit complexity of decoupling. 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 pre-shared entanglement is exponentially smaller than that in the previous protocol near-optimal in communication. We also achieve similar results for one-shot quantum state redistribution.