Entanglement and Causal Relation in distributed quantum computation

Abstract

In this thesis, we investigate two different aspects of entanglement and classical communication in distributed quantum computation (DQC). In the first part, we analyze implementable computation over a given quantum network resource by introducing a new concept, quantum network coding for quantum computation. We consider a setting of networks where quantum communication for each edge of a network is restricted to sending one-qubit, but classical communication is unrestricted. Specifically, we analyze implementable $k$-qubit unitary operations over a certain class of networks, called cluster networks. We show that any two-qubit unitary operation is implementable over the butterfly network and the grail network, which are fundamental primitive networks for network coding. We also obtain necessary and sufficient conditions for the probabilistic implementability of unitary operations over cluster networks. In the second part, we investigate resource tradeoffs in DQC. First, we show that entanglement required for local state discrimination can be substituted by less entanglement by increasing the rounds of classical communication. Second, we develop a new framework of deterministic joint quantum operations by using a causal relation between the outputs and inputs of the local operations without predefined causal order, called "classical communication" without predefined causal order (CC*). We show that local operations and CC* (LOCC*) is equivalent to the separable operation (SEP). This result indicates that entanglement-assisted LOCC implementing SEP can be simulated by LOCC* without entanglement. We also investigate the relationship between LOCC* and another formalism for deterministic joint quantum operations without assuming predefined causal order based on the quantum process formalism. As a result, we construct an example of non-LOCC SEP by using LOCC*.