We find that multidimensional determinants "hyperdeterminants", related to entanglement measures (the so-called concurrence or 3-tangle for the 2 or 3 qubits, respectively), are derived from a duality between entangled states and separable states. By means of the hyperdeterminant and its singularities, the single copy of multipartite pure entangled states is classified into an onion structure of every closed subset, similar to that by the local rank in the bipartite case. This reveals how inequivalent multipartite entangled classes are partially ordered under local actions. In particular, the generic entangled class of the maximal dimension, distinguished as the nonzero hyperdeterminant, does not include the maximally entangled states in Bell's inequalities in general (e.g., in the n ≥ 4 qubits), contrary to the widely known bipartite or 3-qubit cases. It suggests that not only are they never locally interconvertible with the majority of multipartite entangled states, but they would have no grounds for the canonical n-partite entangled states. Our classification is also useful for the mixed states.

Let G(A, B ) denote the two-qubit gate that acts as the one-qubit SU(2) gates A and B in the even and odd parity subspaces, respectively, of two qubits. Using a Clifford algebra formalism, we show that arbitrary uniform families of circuits of these gates, restricted to act only on nearest neighbour (n.n.) qubit lines, can be classically efficiently simulated. This reproduces a result originally proved by Valiant using his matchgate formalism, and subsequently related by others to free fermionic physics. We further show that if the n.n. condition is slightly relaxed, to allow the same gates to act only on n.n. and next n.n. qubit lines, then the resulting circuits can efficiently perform universal quantum computation. From this point of view, the gap between efficient classical and quantum computational power is bridged by a very modest use of a seemingly innocuous resource (qubit swapping). We also extend the simulation result above in various ways. In particular, by exploiting properties of Clifford operations in conjunction with the Jordan-Wigner representation of a Clifford algebra, we show how one may generalize the simulation result above to provide further classes of classically efficiently simulatable quantum circuits, which we call Gaussian quantum circuits.

We investigate which entanglement resources allow universal measurement-based quantum computation via single-qubit operations. We find that any entanglement feature exhibited by the 2D cluster state must also be present in any other universal resource. We obtain a powerful criterion to assess universality of graph states, by introducing an entanglement measure which necessarily grows unboundedly with the system size for all universal resource states. Furthermore, we prove that graph states associated with 2D lattices such as the hexagonal and triangular lattice are universal, and obtain the first example of a universal non-graph state.PACS numbers: 03.67. Lx, 03.65.Ta, 03.67.Mn Introduction.-Quantum computation is a promising attempt to utilize the laws of quantum physics for novel applications. Indeed, it was shown that problems such as factoring or database search can be performed much faster on a quantum computer than on any known classical device. Despite of these exciting perspectives, the question: what are the essential resources that give quantum computers their additional power over classical devices? is still poorly understood. Various models for a quantum computer exist, each based on different concepts, which indicates that there may not be a straightforward answer to this difficult question. The new paradigm of measurement-based quantum computation (MQC) [1,2,3,4], with the one-way quantum computer [1] and the teleportation-based model [2,3] as the most prominent examples, has lead to a new and fresh perspective in these respects. In particular, these and other studies [5] highlight the central role of entanglement in quantum computation.In MQC, quantum information stored in a quantum state is processed by performing sequences of adaptive measurements. This is in striking contrast to the quantum circuit model, where unitary operations are realized via coherent evolution. While the teleportation-based models [2,3] use joint (i.e. entangling) measurements on two or more qubits, thereby performing sequences of teleportation-based gates, the one-way model [1] uses a highly entangled state, the cluster state [6], as a universal resource which is processed by single-qubit measurements. Unified descriptions of all measurement-based models have recently been proposed in Refs. [7,8].Here we focus on the one-way model, where the resource character of entanglement is particularly highlighted, as it is clearly separated from the processing via local measurements which do not act as additional source for entanglement. The distinct features of the one-way model also allow us to cast the introductory question into a much more concise form, viz. what are the essential properties of the cluster state that make it a universal resource? In this letter we will investigate this question.

We propose a scheme for a ground-code measurement-based quantum computer, which enjoys two major advantages. First, every logical qubit is encoded in the gapped degenerate ground subspace of a spin-1 chain with nearest-neighbor two-body interactions, so that it equips built-in robustness against noise. Second, computation is processed by single-spin measurements along multiple chains dynamically coupled on demand, so as to keep teleporting only logical information into a gapprotected ground state of the residual chains after the interactions with spins to be measured are turned off. We describe implementations using trapped atoms or polar molecules in an optical lattice, where the gap is expected to be as large as 0.2 kHz or 4.8 kHz respectively. Introduction.-Reliable quantum computers require hardware with low error rates and sufficient resources to perform software-based error correction. One appealing approach to reduce the massive overhead for error correction is to process quantum information in the gapped ground states of some many-body interaction. This is the tactic used in topological quantum computation and adiabatic quantum computation. Yet, the hardware demands for the former are substantial, and the fault tolerance of the later, especially when restricted to two-body interactions, is unclear [1]. On the other hand, measurement-based quantum computation (MQC), in particular one-way computation on the 2D cluster state [2], runs by beginning with a highly entangled state dynamically generated from nearest-neighbor two-body interactions and performing computation by only single-qubit measurements and feed forward of their outcomes. However, its bare implementation may suffer decoherence of physical qubits waiting for their round of measurements in the far future, and that severely damages a prominent capability to parallelize computation. Although its fault-tolerant method by error correction has been well established [3], it is clearly advantageous if some gap-protection is provided on the hardware level.