Capacity approaching coding for low noise interactive quantum communication

Abstract

We consider the problem of implementing two-party interactive quantum communication over noisy channels, a necessary endeavor if we wish to fully reap quantum advantages for communication. For an arbitrary protocol with n messages, designed for a noiseless qudit channel over a poly (n) size alphabet, our main result is a simulation method that fails with probability less than 2 −Θ(n ) and uses a qudit channel over the same alphabet n(1 + Θ( √ )) times, of which an fraction can be corrupted adversarially. The simulation is thus capacity achieving to leading order, and we conjecture that it is optimal up to a constant factor in the √ term. Furthermore, the simulation is in a model that does not require pre-shared resources such as randomness or entanglement between the communicating parties. Our work improves over the best previously known quantum result where the overhead is a non-explicit large constant [Brassard et al., FOCS'14] for low . Introduction Motivation The main questions.Quantum communication offers the possibility of distributed computation with extraordinary provable savings in communication as compared with classical communication (see, e.g., [RK11] and the references therein). Most often, if not always, the savings are achieved by protocols that assume access to noiseless communication channels. In practice, though, imperfection in channels is inevitable. Is it possible to make the protocols robust to noise while maintaining the advantages offered by quantum communication? If so, what is the cost of making the protocols robust, and how much noise can be tolerated? In this article, we address these questions in the context of quantum communication protocols involving two parties, in the low noise regime. Following convention, we call the two parties Alice and Bob. 1.1.2 Channel coding theory as a special case. Proposition 2.3. The Bell states φ j,k 0≤j,k≤d−1 form an orthonormal basis in A ⊗ B. Proposition 2.4. For any unitary operator U on register A, it holds that Quantum Communication ModelThe definitions for the noiseless and noisy quantum communication models are copied from Ref.[BNT + 19]. We refer the reader there for a more formal definition of the noisy quantum communication model, as well as the relationship of the noiseless quantum communication model to well-studied quantum communication complexity models such a Yao's model and the Cleve-Buhrman model. 6 Noiseless Communication ModelIn the noiseless quantum communication model that we want to simulate, there are five quantum registers: the A register held by Alice, the B register held by Bob, the C register, which is the communication register exchanged back-and-forth between Alice and Bob and initially held by Alice, the E register held by a potential adversary Eve, and finally the R register, a reference system which purifies the state of the ABCE registers throughout the protocol. The initial stateis chosen arbitrarily from the set of possible inputs, and is fixed at the outset of the protocol, but possibly unknown (totally or partial...

Noisy searching: simple, fast and correct

“…The theory of coding schemes for noisy communication is out of scope of this survey and we point to some recent works [8,18,19,21,28].…”