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arXiv:1310.3991

Asymptotic Compressibility of Entanglement and Classical Communication in Distributed Quantum Computation

Eyuri Wakakuwa, Mio Murao

Published Oct 15, 2013• 3 citations
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Abstract

We consider implementations of a bipartite unitary on many pairs of unknown input states by local operation and classical communication assisted by shared entanglement. We investigate to what extent the entanglement cost and the classical communication cost can be compressed by allowing nonzero but vanishing error in the asymptotic limit of infinite pairs. We show that a lower bound on the minimal entanglement cost, the forward classical communication cost, and the backward classical communication cost per pair is given by the Schmidt strength of the unitary. We also prove that an upper bound on these three kinds of the cost is given by the amount of randomness that is required to partially decouple a tripartite quantum state associated with the unitary. In the proof, we construct a protocol in which quantum state merging is used. For generalized Clifford operators, we show that the lower bound and the upper bound coincide. We then apply our result to the problem of distributed compression of tripartite quantum states, and derive a lower and an upper bound on the optimal quantum communication rate required therein.

DOI
10.48550/arxiv.1310.3991

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We consider implementations of a bipartite unitary on many pairs of unknown input states by local operation and classical communication assisted by shared entanglement. We investigate to what extent the entanglement cost and the classical communication cost can be compressed by allowing nonzero but vanishing error in the asymptotic limit of infinite pairs. We show that a lower bound on the minimal entanglement cost, the forward classical communication cost, and the backward classical communication cost per pair is given by the Schmidt strength of the unitary. We also prove that an upper bound on these three kinds of the cost is given by the amount of randomness that is required to partially decouple a tripartite quantum state associated with the unitary. In the proof, we construct a protocol in which quantum state merging is used. For generalized Clifford operators, we show that the lower bound and the upper bound coincide. We then apply our result to the problem of distributed compression of tripartite quantum states, and derive a lower and an upper bound on the optimal quantum communication rate required therein.
We analyze implementations of bipartite unitaries by means of local operations and classical communication (LOCC) assisted by shared entanglement. We employ concepts and techniques developed in the quantum Shannon theory to study an asymptotic scenario, in which two distant parties perform the same bipartite unitary on infinitely many pairs of inputs. We analyze minimum cost of entanglement and classical communication per copy. For two-round LOCC protocols, we derive a single-letter formula for the minimum cost of entanglement and classical communication, under an additional requirement that the error converges to zero faster than 1/n4, where n is the number of input pairs. The formula is given by the “Markovianizing cost” of a tripartite state associated with the unitary, which can be computed by a finite-step algorithm. We also derive a lower bound on the minimum cost of resources, which applies for protocols with arbitrary number of rounds.
Distributed quantum computation connects multiple quantum processing units (QPUs) through quantum communication to jointly perform large-scale quantum computations. Since the number of qubits available at each QPU is limited, it is important to reduce quantum communication while keeping the qubit overhead small. To this end, we study an entanglement-assisted model, in which entanglement consumption serves as a measure of quantum communication cost. For exact deterministic implementations in this model, the operator Schmidt rank of the target bipartite unitary provides a general lower bound on entanglement cost when qubit overhead is unrestricted. However, it has remained unclear how closely this bound can be approached with constant qubit overhead. In this work, we show that this lower bound is attainable for every bipartite Clifford unitary using at most two auxiliary qubits per QPU. In addition, for non-Clifford unitaries specified by exact Clifford+TTT decompositions with TTT-count ttt, we construct implementations with the same qubit overhead whose entanglement cost is at most 2t2t2t Bell pairs above this lower bound.
We analyze implementations of bipartite unitaries in a distributed quantum computation setting using local operations and classical communication (LOCC) assisted by shared entanglement. We employ concepts and techniques developed in quantum Shannon theory to study an asymptotic scenario in which the two distant parties perform the same bipartite unitary on infinitely many pairs of input states generated by a completely random i.i.d. (independent and identically distributed) quantum information source. We analyze the minimum costs of resources of entanglement and classical communication per copy. For protocols consisting of two-round LOCC, we prove that an achievable rate tuple of costs of entanglement and classical communication is given by the “Markovianizing cost” of a tripartite state associated with the unitary, which is conjectured to be optimal as well. The Markovianizing cost can be computed by a finite-step algorithm.
Distributed quantum computation is a task in which many distant parties perform a large quantum computation in collaboration with each other, by using classical communication, quantum communication or shared entanglement as resources.In this thesis, we consider one of the simplest nontrivial task of distributed quantum computation, that is, implementation of bipartite unitary by LOCC (local operations and classical communication) assisted by shared entanglement.Here, two input states are distributed over two distant parties.Contrary to previous researches, we consider an asymptotic scenario in which the two parties perform the same bipartite unitary on infinitely many independent input pairs.We analyze the minimum amount of resources required for the task by exploiting concepts and techniques developed in the field of quantum Shannon theory, which has originally been developed to analyze quantum communication.As a tool for analyzing distributed quantum computation, we first introduce a task that we call Markovianization.Markovianization is a task in which a tripartite pure state is transformed to a state called quantum Markov chain, by a randomizing operation on one of the three subsystems.We derive the minimum cost of randomness required for Markovianization (Markovianizing cost) by exploiting a novel decomposition of a Hilbert space introduced in the context of quantum cryptography.We then apply the obtained result to distributed quantum computation.We mainly consider implementation of bipartite unitaries by entanglement-assisted LOCC protocols consisting of three steps, called one-round protocols.This is the first nontrivial case because it is not possible to implement bipartite unitaries by protocols consisting of only one or two steps.Our main result is that we derive the minimum costs of entanglement and classical communication.We show that the minimum costs are given by the Markovianizing cost of a tripartite state associated with the unitary.The result indicates that information about input states of computation is divided into three components: one of which only information of amplitude in a basis is required for computation (information of phase is not), another of which both amplitude and phase information are required, and the other of which no information is involved in computation.Our construction of a protocol that requires the smallest amount of resources is based on this decomposition.We also propose a method to compute the Markovianizing cost of a state associated with the unitary.We also consider two-round protocols, that is, protocols consisting of concatenation of two one-round protocols, for two-qubit controlled unitaries.We show that, for a particular class of two-qubit controlled unitaries, the minimum entanglement cost in two-round protocols is strictly smaller than that in one-round protocols.This result is the first reported case that indicates a property of a trade-off relation between entanglement requirement and the number of turns in entanglement-assisted LOCC tasks.The research in this thesis opens a possibility to build a new bridge between quantum computation theory and quantum communication theory, which are the two largest subfield of quantum information theory, by analyzing a problem in the former using concepts and techniques developed in the latter.
We prove continuity of distillable entanglement, the optimal asymptotic rate at which copies of a bipartite state can be converted into maximally entangled pairs by local operations and classical communication (LOCC) with vanishing error. We establish an explicit asymptotic continuity bound that holds for arbitrary bipartite states and depends logarithmically on the system dimension. This bound has the same structure as known continuity bounds for conditional entropy, entanglement cost, and relative entropy of entanglement. Our proof constructs an explicit LOCC protocol that converts either of two arbitrary bipartite states into the other at an ebit cost that vanishes as their trace distance tends to zero.
We prove that there is a trade-off relation between the entanglement cost and the number of rounds of communication, for two distant parties to accomplish a bidirectional quantum information task by local operations and classical communication (LOCC). We consider an implementation of a class of two-qubit controlled-unitary gate by LOCC assisted by shared entanglement, in an information theoretical scenario of asymptotically many input pairs and vanishingly small error. We prove the trade-off relation by showing that one ebit of entanglement per pair is necessary to be consumed for implementing the unitary by any two-round protocol, whereas the entanglement cost by a four-round protocol is strictly smaller than one ebit per pair.