Zhaohui Wei

Efficient Protocols for Generating Bipartite Classical Distributions and Quantum States

We investigate the fundamental problem of generating bipartite classical distributions or quantum states. By designing efficient communication protocols and proving their optimality, we establish a number of intriguing connections to fundamental measures in optimization, convex geometry, and information theory. 1) To generate a classical distribution P(x,y), we tightly characterize the minimum amount of quantum communication needed by the psd-rank of P (as a matrix), a measure recently proposed by Fiorini et al. (Proc. 44th ACM Symp. Theory Comput., pp. 95-106, 2012) in studies of the minimum size of extended formulations of optimization problems such as TSP. This echos the previous characterization for the optimal classical communication cost by the nonnegative rank of P. The result is obtained via investigating the more general case of bipartite quantum state generation and designing an optimal protocol for it. 2) When an approximation ϵ is allowed to generate a distribution (X,Y)~P, we present a classical protocol of the communication cost O((C(X,Y)+1)/ϵ, where C(X,Y) is common information, a well-studied measure in information theory introduced by Wyner (IEEE Trans. Inf. Theory, 21 (2):163-179, 1975). This also links nonnegative rank and common information, two seemingly unrelated quantities in different fields. 3) For approximately generating a quantum pure state |ψ〉, we completely characterize the minimum cost by a corresponding approximate rank, closing a possibly exponential gap left in Ambainis etal. (SIAM J. Comput., 32 (6):1570-1585, 2003).

A modified quantum adiabatic evolution for the Deutsch-Jozsa problem

Abstract Deutsch–Jozsa algorithm has been implemented via a quantum adiabatic evolution by S. Das et al. [S. Das, R. Kobes, G. Kunstatter, Phys. Rev. A 65 (2002) 062310]. This adiabatic algorithm gives rise to a quadratic speed up over classical algorithms. We show that a modified version of the adiabatic evolution in that paper can improve the performance to constant time.

On the minimum dimension of a Hilbert space needed to generate a quantum correlation

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this Letter, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on many well-known correlations and discuss how it can rule out correlations of having a finite-dimensional quantum representation. We show that our bound is multiplicative under product correlations and also that it can witness the nonconvexity of certain restricted-dimensional quantum correlations.

Experimental demonstration of quantum gain in a zero-sum game

We propose and experimentally demonstrate a zero-sum game that is in a fair Nash equilibrium for classical players, but has the property that a quantum player can always win using an appropriate strategy. The gain of the quantum player is measured experimentally for different quantum strategies and input states. It is found that the quantum gain is maximized by a maximally entangled state, but does not decrease to zero when entanglement disappears. Instead, it links with another kind of quantum correlation described by discord for the qubit case and the connection is demonstrated both theoretically and experimentally.

Some upper and lower bounds on PSD-rank

Positive semidefinite rank (PSD-rank) is a relatively new complexity measure on matrices, with applications to combinatorial optimization and communication complexity. We first study several basic properties of PSD-rank, and then develop new techniques for showing lower bounds on the PSD-rank. All of these bounds are based on viewing a positive semidefinite factorization of a matrix M as a quantum communication protocol. These lower bounds depend on the entries of the matrix and not only on its support (the zero/nonzero pattern), overcoming a limitation of some previous techniques. We compare these new lower bounds with known bounds, and give examples where the new ones are better. As an application we determine the PSD-rank of (approximations of) some common matrices.

Quantum adiabatic computation and adiabatic conditions

Recently, quantum adiabatic computation has attracted more and more attention in the literature. It is a novel quantum computation model based on adiabatic approximation, and the analysis of a quantum adiabatic algorithm depends highly on the adiabatic conditions. However, it has been pointed out that the traditional adiabatic conditions are problematic. Thus, results obtained previously should be checked and sufficient adiabatic conditions applicable to adiabatic computation should be proposed. Based on a result of Tong et al. [Phys. Rev. Lett. 98, 150402 (2007)], we propose a modified adiabatic criterion which is more applicable to the analysis of adiabatic algorithms. As an example, we prove the validity of the local adiabatic search algorithm by employing our criterion.

Completely positive semidefinite rank

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Efficient one-way quantum computations for quantum error correction

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