Wasserstein barycenters correspond to optimal solutions of transportation problems for several marginals, and as such have a wide range of applications ranging from economics to statistics and computer science. When the marginal probability measures are absolutely continuous (or vanish on small sets) the theory of Wasserstein barycenters is well-developed (see the seminal paper [1]). However, exact continuous computation of Wasserstein barycenters in this setting is tractable in only a small number of specialized cases. Moreover, in many applications data is given as a set of probability measures with finite support. In this paper, we develop theoretical results for Wasserstein barycenters in this discrete setting. Our results rely heavily on polyhedral theory which is possible due to the discrete structure of the marginals.Our results closely mirror those in the continuous case with a few exceptions. In this discrete setting we establish that Wasserstein barycenters must also be discrete measures and there is always a barycenter which is provably sparse. Moreover, for each Wasserstein barycenter there exists a nonmass-splitting optimal transport to each of the discrete marginals. Such non-mass-splitting transports do not generally exist between two discrete measures unless special mass balance conditions hold. This makes Wasserstein barycenters in this discrete setting special in this regard.We illustrate the results of our discrete barycenter theory with a proof-of-concept computation for a hypothetical transportation problem with multiple marginals: distributing a fixed set of goods when the demand can take on different distributional shapes characterized by the discrete marginal distributions. A Wasserstein barycenter, in this case, represents an optimal distribution of inventory facilities which minimize the squared distance/transportation cost totaled over all demands.

Measurement-based quantum computation (MQC) is a paradigm for studying quantum computation using many-body entanglement and single-qubit measurements. Although MQC has inspired wide-ranging discoveries throughout quantum information, our understanding of the general principles underlying MQC seems to be biased by its historical reliance upon the archetypal 2D cluster state. Here we utilise recent advances in the subject of symmetry-protected topological order (SPTO) to introduce a novel MQC resource state, whose physical and computational behaviour differs fundamentally from that of the cluster state. We show that, in sharp contrast to the cluster state, our state enables universal quantum computation using only measurements of single-qubit Pauli X, Y, and Z operators. This novel computational feature is related to the 'genuine' 2D SPTO possessed by our state, and which is absent in the cluster state. Our concrete connection between the latent computational complexity of many-body systems and macroscopic quantum orders may find applications in quantum many-body simulation for benchmarking classically intractable complexity. INTRODUCTIONThe idea of measurement-based quantum computation (MQC), where computation is carried out solely through single-qubit measurements on a fixed many-body resource state and classical feed-forward of measurement outcomes, 1-3 is quite surprising. This is because it highlights the origins of quantum advantage in terms of entanglement and non-commutative measurements, uniquely quantum effects without counterparts in classical mechanics. In particular, so-called universal resource states, the states that are capable of efficiently implementing universal MQC, represent a class of maximal entanglement in the classification of many-body entanglement, 4 so that the structure and complexity of their entanglement is of great interest for advancing the understanding of quantum computation. Following the canonical example of the two-dimensional (2D) cluster state, 5 many other universal resource states have been found, including cluster states defined on various lattices, 4 some tensor network states, 4,6-10 and model ground states in condensed matter physics such as 2D Affleck-Kennedy-Lieb-Tasaki (AKLT) states. [10][11][12][13][14][15] Given the existence of these various known universal resource states, a natural question is whether we might be able to find any common key feature, so as to explore more their variety in fundamental structures as well as in practical applications. Although the earliest resource states for MQC were found in short-range correlated states described as somewhat artificial tensor network states, 4,6-10 a new insight has been that a class of short-ranged entangled states structured by symmetry, endowed with so-called symmetry-protected topological order (SPTO), [16][17][18][19][20][21][22][23][24] make excellent candidate resource states systematically. Indeed, in the setting of 1D spin chains, the ground states of several SPTO phases have already been shown to poss...

We investigate entanglement naturally present in the 1D topologically ordered phase protected with the on-site symmetry group of an octahedron as a potential resource for teleportation-based quantum computation. We show that, as long as certain characteristic lengths are finite, all its ground states have the capability to implement any unit-fidelity one-qubit gate operation asymptotically as a key computational building block. This feature is intrinsic to the entire phase, in that perfect gate fidelity coincides with perfect string order parameters under a state-insensitive renormalization procedure. Our approach may pave the way toward a novel program to classify quantum many-body systems based on their operational use for quantum information processing. Introduction.-Entanglement is ubiquitous in quantum many-body systems, and its complexity has drawn attention from interdisciplinary research fields, such as condensed-matter physics [1][2][3][4], quantum information processing (QIP) [5][6][7], and quantum simulation of quantum many-body systems [8][9][10][11][12]. A primary example is exotic ground states of topologically ordered phases [13][14][15], which arise from underlying nonlocal entanglement. It is widely known that braiding their excitations, known as anyons, could be used for topological quantum computation [16], and their intrinsic insensitivity against local noise could be used for quantum error correction [16,17]. Many-body entanglement can be harnessed in a more direct way, and certain many-body states like 2D cluster states [18] and certain tensor network states [19][20][21][22][23][24][25] are quantum resources for measurement-based (or teleportation-based) quantum computation, in that universal quantum computation can be implemented on these states using only single-spin measurements.Having in hand a long list of many-body entanglement useful for QIP, however, one may wonder "Is such computational usefulness robust in the same way that collective phenomena of quantum many-body systems do not depend on their microscopic details?" Phrased differently, "Can we define quantum phases useful for certain QIP tasks in the same way we define phase diagrams in condensed matter physics, which are typically characterized by order parameters?" There have been several attempts [26][27][28][29][30][31][32][33][34] to answer this affirmatively, but they unfortunately, with a few exceptions [30], were largely based on a limited class of states, using rather artificial Hamiltonians from a condensed matter physics perspective.Here we tackle this challenge using the 1D counterpart of topologically ordered phases as a key building block for measurement-based quantum computation, taking advantage of recent characterizations of symmetry