We present a simple quantum many-body system---a two-dimensional lattice of qubits with a Hamiltonian composed of nearest-neighbor two-body interactions---such that the ground state is a universal resource for quantum computation using single-qubit measurements. This ground state approximates a cluster state that is encoded into a larger number of physical qubits. The Hamiltonian we use is motivated by the projected entangled pair states, which provide a transparent mechanism to produce such approximate encoded cluster states on square or other lattice structures (as well as a variety of other quantum states) as the ground state. We show that the error in this approximation takes the form of independent errors on bonds occurring with a fixed probability. The energy gap of such a system, which in part determines its usefulness for quantum computation, is shown to be independent of the size of the lattice. In addition, we show that the scaling of this energy gap in terms of the coupling constants of the Hamiltonian is directly determined by the lattice geometry. As a result, the approximate encoded cluster state obtained on a hexagonal lattice (a resource that is also universal for quantum computation) can be shown to have a larger energy gap than one on a square lattice with an equivalent Hamiltonian.
We present a simple quantum many-body system---a two-dimensional lattice of qubits with a Hamiltonian composed of nearest-neighbor two-body interactions---such that the ground state is a universal resource for quantum computation using single-qubit measurements. This ground state approximates a cluster state that is encoded into a larger number of physical qubits. The Hamiltonian we use is motivated by the projected entangled pair states, which provide a transparent mechanism to produce such approximate encoded cluster states on square or other lattice structures (as well as a variety of other quantum states) as the ground state. We show that the error in this approximation takes the form of independent errors on bonds occurring with a fixed probability. The energy gap of such a system, which in part determines its usefulness for quantum computation, is shown to be independent of the size of the lattice. In addition, we show that the scaling of this energy gap in terms of the coupling constants of the Hamiltonian is directly determined by the lattice geometry. As a result, the approximate encoded cluster state obtained on a hexagonal lattice (a resource that is also universal for quantum computation) can be shown to have a larger energy gap than one on a square lattice with an equivalent Hamiltonian.
We present a general procedure for constructing lattices of qubits with a Hamiltonian composed of nearest-neighbor two-body interactions such that the ground state encodes a cluster state. We give specific details for lattices in one, two, and three dimensions, investigating both periodic and fixed boundary conditions, as well as present a proof for the applicability of this procedure to any graph. We determine the energy gap of these systems, which is shown to be independent of the size of the lattice but dependent on the type of lattice (in particular, the coordination number), and investigate the scaling of this gap in terms of the coupling constants of the Hamiltonian. We provide a comparative analysis of the different lattice types with respect to their usefulness for measurement-based quantum computation.
Many-body entangled quantum states studied in condensed matter physics can be primary resources for quantum information, allowing any quantum computation to be realized using measurements alone, on the state. Such a universal state would be remarkably valuable, if only it were thermodynamically stable and experimentally accessible, by virtue of being the unique ground state of a physically reasonable Hamiltonian made of two-body, nearest-neighbor interactions. We introduce such a state, composed of six-state particles on a hexagonal lattice, and describe a general method for analyzing its properties based on its projected entangled pair state representation.
Entanglement between the constituents of a quantum system is an essential resource in the implementation of many quantum processes and algorithms. Indeed, universal quantum computation is possible by measuring individual qubits that constitute highly entangled ``cluster states.'' In this work, it is shown that the unique gapped ground state of noninteracting fermions hopping on a specially prepared lattice is equivalent to a cluster state, where the entanglement between qubits results from fermionic indistinguishability and antisymmetry. A deterministic strategy for universal measurement-based quantum computation with this resource is described. Because most matter is composed of fermions, these results suggest that resources for quantum information processing might be generic in nature.
In this paper, we build a framework allowing for a systematic investigation of the fundamental issue: ' Which quantum states serve as universal resources for measurement-based (one-way) quantum computation? ' We start our study by re-examining what is exactly meant by 'universality' in quantum computation, and what the implications are for universal one-way quantum computation. Given the framework of a measurement-based quantum computer, where quantum information is processed by local operations only, we find that the most general universal one-way quantum computer is one which is capable of accepting arbitrary classical inputs and producing arbitrary quantum outputs—we refer to this property as CQ-universality. We then show that a systematic study of CQ-universality in one-way quantum computation is possible by identifying entanglement features that are required to be present in every universal resource. In particular, we find that a large class of entanglement measures must reach its supremum on every universal resource. These insights are used to identify several families of states as being not universal, such as one-dimensional (1D) cluster states, Greenberger–Horne–Zeilinger (GHZ) states, W states, and ground states of non-critical 1D spin systems. Our criteria are strengthened by considering the efficiency of a quantum computation, and we find that entanglement measures must obey a certain scaling law with the system size for all efficient universal resources. This again leads to examples of non-universal resources, such as, e.g. ground states of critical 1D spin systems. On the other hand, we provide several examples of efficient universal resources, namely graph states corresponding to hexagonal, triangular and Kagome lattices. Finally, we consider the more general notion of encoded CQ-universality, where quantum outputs are allowed to be produced in an encoded form. Again we provide entanglement-based criteria for encoded universality. Moreover, we present a general procedure to construct encoded universal resources.
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 gap-protected 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 or 4.8 kHz, respectively.
We show that a local Hamiltonian of spin-23 particles with only two-body nearest-neighbor Affleck-Kennedy-Lieb-Tasaki and exchange-type interactions has a unique ground state, which can be used to implement universal quantum computation merely with single-spin measurements. We prove that the Hamiltonian is gapped, independent of the system size. Our result provides a further step toward utilizing systems with condensed-matter-type interactions for measurement-based quantum computation.