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24
The power of quantum systems on a line
, 2009
"... We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a onedimensional quantum system (with 9 states per particle). This might have practical implicati ..."
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Cited by 46 (7 self)
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We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a onedimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMAcomplete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, one dimensional MAX2SAT with nearest neighbor constraints, is in P. The proof of the QMAcompleteness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Since it is unlikely that quantum computers can efficiently solve QMA problems, our construction gives a onedimensional system which takes an exponential time to relax to its ground state at any temperature. This makes it a candidate for a onedimensional spin glass.
On the localization transition of random copolymers near selective interfaces
 J. Statist. Phys
"... Abstract. In this note we consider the (de)localization transition for random directed (1 + 1)–dimensional copolymers in the proximity of an interface separating selective solvents. We derive a rigorous lower bound on the free energy. This yields a substantial improvement on the bounds from below on ..."
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Cited by 35 (14 self)
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Abstract. In this note we consider the (de)localization transition for random directed (1 + 1)–dimensional copolymers in the proximity of an interface separating selective solvents. We derive a rigorous lower bound on the free energy. This yields a substantial improvement on the bounds from below on the critical line that were known so far. Our result implies that the critical curve does not lie below the critical curve conjectured by Monthus [11] on the base of a renormalization group analysis. We discuss this result in the light of the (rigorous and non rigorous) approaches present in the literature and, by making an analogy with a particular asymptotics of a disordered wetting model, we propose a simplified framework in which the question of identifying the critical curve, as well as understanding the nature of the transition, may be approached.
On the irrelevant disorder regime of pinning models
, 2007
"... Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that, albeit still partial, is now rigorous. In this wor ..."
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Cited by 14 (4 self)
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Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that, albeit still partial, is now rigorous. In this work we exploit interpolation and replica coupling methods to get sharper results on the irrelevant disorder regime of pinning models. In particular, we compute in this regime the first order term in the expansion of the free energy close to criticality, which coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and the quenched averaged correlation length exponents coincide, while in general they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.
A numerical approach to copolymers at selective interfaces
 J. Statist. Phys
"... Abstract. We consider a model of a random copolymer at a selective interface which undergoes a localization/delocalization transition. In spite of the several rigorous results available for this model, the theoretical characterization of the phase transition has remained elusive and there is still n ..."
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Cited by 12 (5 self)
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Abstract. We consider a model of a random copolymer at a selective interface which undergoes a localization/delocalization transition. In spite of the several rigorous results available for this model, the theoretical characterization of the phase transition has remained elusive and there is still no agreement about several important issues, for example the behavior of the polymer near the phase transition line. From a rigorous viewpoint non coinciding upper and lower bounds on the critical line are known. In this paper we combine numerical computations with rigorous arguments to get to a better understanding of the phase diagram. Our main results include: – Various numerical observations that suggest that the critical line lies strictly in between the two bounds. – A rigorous statistical test based on concentration inequalities and super–additivity, for determining whether a given point of the phase diagram is in the localized phase. This is applied in particular to show that, with a very low level of error, the lower bound does not coincide with the critical line. – An analysis of the precise asymptotic behavior of the partition function in the delocalized phase, with particular attention to the effect of rare atypical stretches in the disorder sequence and on whether or not in the delocalized regime the polymer path has a Brownian scaling. – A new proof of the lower bound on the critical line. This proof relies on a characterization of the localized regime which is more appealing for interpreting the numerical data.
Spacetime percolation
"... The contact model for the spread of disease may be viewed as a directed percolation model on Z×R in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The c ..."
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Cited by 8 (0 self)
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The contact model for the spread of disease may be viewed as a directed percolation model on Z×R in which the continuum axis is oriented in the direction of increasing time. Techniques from percolation have enabled a fairly complete analysis of the contact model at and near its critical point. The corresponding process when the timeaxis is unoriented is an undirected percolation model to which now standard techniques may be applied. One may construct in similar vein a randomcluster model on Z × R, with associated continuum Ising and Potts models. These models are of independent interest, in addition to providing a pathintegral representation of the quantum Ising model with transverse field. This representation may be used to obtain a bound on the entanglement of a finite set of spins in the quantum Ising model on Z, where this entanglement is measured via the entropy of the reduced density matrix. The meanfield version of the quantum Ising model gives rise to a randomcluster model on Kn × R, thereby extending the Erdős–Rényi random graph on the complete graph Kn. 1
Integrable models in statistical mechanics: The hidden field with unsolved problems
, 1999
"... In the past 30 years there have been extensive discoveries in the theory of integrable statistical mechanical models including the discovery of nonlinear differential equations for Ising model correlation functions, the theory of random impurities, level crossing transitions in the chiral Potts mod ..."
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Cited by 5 (0 self)
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In the past 30 years there have been extensive discoveries in the theory of integrable statistical mechanical models including the discovery of nonlinear differential equations for Ising model correlation functions, the theory of random impurities, level crossing transitions in the chiral Potts model and the use of RogersRamanujan identities to generalize our concepts of Bose/Fermi statistics. Each of these advances has led to the further discovery of major unsolved problems of great mathematical and physical interest. I will here discuss the mathematical advances, the physical insights and extraordinary lack of visibility of this field of physics.
Magnetic properties of strongly disordered electronic systems
 Rev. Lett
, 1998
"... We present a unified, global perspective on the magnetic properties of strongly disordered electronic systems, with special emphasis on the case where the ground state is metallic. We review the arguments for the instability of the disordered Fermi liquid state towards the formation of local magneti ..."
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Cited by 3 (0 self)
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We present a unified, global perspective on the magnetic properties of strongly disordered electronic systems, with special emphasis on the case where the ground state is metallic. We review the arguments for the instability of the disordered Fermi liquid state towards the formation of local magnetic moments, and argue that their singular low temperature thermodynamics are the “quantum Griffiths” precursors of the quantum phase transition to a metallic spin glass; the local moment formation is therefore not directly related to the metalinsulator transition. We also review the the meanfield theory of the disordered Fermi liquid to metallic spin glass transition and describe the separate regime of “nonFermi liquid ” behavior at higher temperatures near the quantum critical point. The relationship to experimental results on doped semiconductors and heavyfermion compounds is noted. 1.
The power of quantum . . .
, 2009
"... We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a onedimensional quantum system (with 9 states per particle). This might have practical implicati ..."
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Cited by 1 (0 self)
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We study the computational strength of quantum particles (each of finite dimensionality) arranged on a line. First, we prove that it is possible to perform universal adiabatic quantum computation using a onedimensional quantum system (with 9 states per particle). This might have practical implications for experimentalists interested in constructing an adiabatic quantum computer. Building on the same construction, but with some additional technical effort and 12 states per particle, we show that the problem of approximating the ground state energy of a system composed of a line of quantum particles is QMAcomplete; QMA is a quantum analogue of NP. This is in striking contrast to the fact that the analogous classical problem, namely, onedimensional MAX2SAT with nearest neighbor constraints, is in P. The proof of the QMAcompleteness result requires an additional idea beyond the usual techniques in the area: Not all illegal configurations can be ruled out by local checks, so instead we rule out such illegal configurations because they would, in the future, evolve into a state which can be seen locally to be illegal. Our construction implies (assuming the quantum ChurchTuring thesis and that quantum computers cannot efficiently solve QMAcomplete problems) that there are onedimensional systems which take an exponential time to relax to their ground states at any temperature, making them candidates for being onedimensional spin glasses.
nonequilibrium
, 2012
"... The quantum transversefield Ising chain in circuit quantum electrodynamics: effects of disorder on the ..."
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The quantum transversefield Ising chain in circuit quantum electrodynamics: effects of disorder on the