Physical systems, characterized by an ensemble of interacting constituents, can be represented and studied by different algebras of operators ͑observables͒. For example, a fully polarized electronic system can be studied by means of the algebra generated by the usual fermionic creation and annihilation operators or by the algebra of Pauli ͑spin-1/2͒ operators. The Jordan-Wigner isomorphism gives the correspondence between the two algebras. As we previously noted, similar isomorphisms enable one to represent any physical system in a quantum computer. In this paper we evolve and exploit this fundamental observation to simulate generic physical phenomena by quantum networks. We give quantum circuits useful for the efficient evaluation of the physical properties ͑e.g., the spectrum of observables or relevant correlation functions͒ of an arbitrary system with Hamiltonian H.

Phase transitions and critical phenomena have consistently been among the principal subjects of active studies in statistical physics. The simple act of transforming one state of matter or phase into another, for instance by changing the temperature, has always captivated the curious mind. This book provides an introductory account on the theory of phase transitions and critical phenomena, a subject now recognized to be indispensable for students and researchers from many fields of physics and related disciplines. The first five chapters are very basic and quintessential, and cover standard topics such as mean-field theories, the renormalization group and scaling, universality, and statistical field theory methods. The remaining chapters develop more advanced concepts, including conformal field theory, the Kosterlitz-Thouless transition, the effects of randomness, percolation, exactly solvable models, series expansions, duality transformations, and numerical techniques. Moreover, a comprehensive series of appendices expand and clarify several issues not developed in the main text. The important role played by symmetry and topology in understanding the competition between phases and the resulting emergent collective behaviour, giving rise to rigidity and soft elementary excitations, is stressed throughout the book. Serious attempts have been directed toward a self-contained modular approach so that the reader does not have to refer to other sources for supplementary information. Accordingly, most of the concepts and calculations are described in detail, sometimes with additional/auxiliary descriptions given in appendices and exercises. The latter are presented as the topics develop with solutions found at the end of the book, thus giving the text a self-learning character.

We introduce a generalization of entanglement based on the idea that entanglement is relative to a distinguished subspace of observables rather than a distinguished subsystem decomposition. A pure quantum state is entangled relative to such a subspace if its expectations are a proper mixture of those of other states. Many information-theoretic aspects of entanglement can be extended to the general setting, suggesting new ways of measuring and classifying entanglement in multipartite systems. By going beyond the distinguishable-subsystem framework, generalized entanglement also provides novel tools for probing quantum correlations in interacting many-body systems.PACS numbers: 03.67.Mn, 03.65.Ud, Entanglement is a uniquely quantum phenomenon whereby a pure state of a composite quantum system may cease to be determined by the states of its constituent subsystems [1]. Entangled pure states are those that have mixed subsystem states. To determine an entangled state requires knowledge of the correlations between the subsystems. As no pure state of a classical system can be correlated, such correlations are intrinsically non-classical, as strikingly manifested by the violation of local realism and Bell's inequalities [2]. In the science of quantum information processing (QIP), entanglement is regarded as the defining resource for quantum communication and an essential feature needed for unlocking the power of quantum computation. However, in spite of intensive investigation, a complete understanding of entanglement is far from being reached.To unambiguously define entanglement requires a preferred partition of the overall system into subsystems. In conventional QIP scenarios, subsystems are associated with spatially separated "local" parties, which legitimates the distinguishability assumption implicit in standard entanglement theory. However, because quantum correlations are at the heart of many physical phenomena, it would be desirable for a notion of entanglement to be useful in contexts other than QIP. Strongly interacting quantum systems offer compelling examples of situations where the usual subsystem-based view is inadequate. Whenever indistinguishable particles are sufficiently close to each other, quantum statistics forces the accessible state space to be a proper subspace of the full tensor product space, and exchange correlations arise that are not a usable resource in the usual QIP sense. Thus, the natural identification of particles with preferred subsystems becomes problematic. Even if a distinguishable-subsystem structure may be associated to degrees of freedom different from the original particles (such as a set of modes [3]), inequivalent factorizations may occur on the same footing. Finally, the introduction of quasiparticles, or the purposeful transformation of the algebraic language used to analyze the system [4], may further complicate the choice of preferred subsystems.While efforts are under way to obtain entanglement-like notions for bosons and fermions [3,5] and to study entanglement in quant...

We investigate the simulation of fermionic systems on a quantum computer.We show in detail how quantum computers avoid the dynamical sign problem present in classical simulations of these systems, therefore reducing a problem believed to be of exponential complexity into one of polynomial complexity.The key to our demonstration is the spin-particle connection (or generalized Jordan-Wigner transformation) that allows exact algebraic invertible mappings of operators with different statistical properties. We give an explicit implementation of a simple problem using a quantum computer based on standard qubits.

We present a unifying framework to study physical systems which exhibit topological quantum order (TQO). The major guiding principle behind our approach is that of symmetries and entanglement. To this end, we introduce the concept of low-dimensional Gauge-Like Symmetries (GLSs), and the physical conservation laws (including topological terms, fractionalization, and the absence of quasi-particle excitations) which emerge from them. We prove then sufficient conditions for TQO at both zero and finite temperatures. The physical engine for TQO are topological defects associated with the restoration of GLSs. These defects propagate freely through the system and enforce TQO. Our results are strongest for gapped systems with continuous GLSs. At zero temperature, selection rules associated with the GLSs enable us to systematically construct general states with TQO; these selection rules do not rely on the existence of a finite gap between the ground states to all other excited states. Indices associated with these symmetries correspond to different topological sectors. All currently known examples of TQO display GLSs. Other systems exhibiting such symmetries include Hamiltonians depicting orbital-dependent spin-exchange and Jahn-Teller effects in transition metal orbital compounds, short-range frustrated Klein spin models, and p+ip superconducting arrays. The symmetry based framework discussed herein allows us to go beyond standard topological field theories and systematically engineer new physical models with finite temperature TQO (both Abelian and non-Abelian). Furthermore, we analyze the insufficiency of entanglement entropy (we introduce SU (N ) Klein models on small world networks to make the argument even sharper), spectral structures, maximal string correlators, and fractionalization in establishing TQO. We show that Kitaev's Toric code model and Wen's plaquette model are equivalent and reduce, by a duality mapping, to an Ising chain, demonstrating that despite the spectral gap in these systems the toric operator expectation values may vanish once thermal fluctuations are present. This illustrates the fact that the quantum states themselves in a particular (operator language) representation encode TQO and that the duality mappings, being non-local in the original representation, disentangle the order. We present a general algorithm for the construction of long-range string and brane orders in general systems with entangled ground states; this algorithm relies on general ground states selection rules and becomes of the broadest applicability in gapped systems in arbitrary dimensions. We discuss relations to problems in graph theory.