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Toby S Cubitt  One of the best experts on this subject based on the ideXlab platform.

Undecidability of the spectral gap
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum manybody physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum manybody system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a twodimensional lattice with translationally invariant, nearestneighbour interactions, for which the spectral gap problem is undecidable. This result extends to Undecidability of other lowenergy properties, such as the existence of algebraically decaying groundstate correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phaseestimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics. The spectral gap problem—whether the Hamiltonian of a quantum manybody problem is gapped or gapless—is rigorously proved to be undecidable; there exists no algorithm to determine whether an arbitrary quantum manybody model is gapped or gapless, and there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics. In quantum manybody physics, the spectral gap is the energy difference between the ground state of a system and its first excited state. Establishing whether it is possible to make a decision about the system being gapped or gapless, given a specific model Hamiltonian, is a longstanding problem in physics known as the spectral gap problem. Here, Toby Cubitt et al. prove that the spectral gap problem is undecidable. Although it had been known before that deciding about the existence of a spectral gap is difficult, this result proves the strongest possible form of algorithmic difficulty for a core problem of manybody physics.

Undecidability of the spectral gap
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The spectral gapthe energy difference between the ground state and first excited state of a systemis central to quantum manybody physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the YangMills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum manybody system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a twodimensional lattice with translationally invariant, nearestneighbour interactions, for which the spectral gap problem is undecidable. This result extends to Undecidability of other lowenergy properties, such as the existence of algebraically decaying groundstate correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phaseestimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

Undecidability of the spectral gap full version
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:We show that the spectral gap problem is undecidable. Specifically, we construct families of translationallyinvariant, nearestneighbour Hamiltonians on a 2D square lattice of dlevel quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lowerbounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.
Michael M Wolf  One of the best experts on this subject based on the ideXlab platform.

Undecidability of the spectral gap
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum manybody physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum manybody system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a twodimensional lattice with translationally invariant, nearestneighbour interactions, for which the spectral gap problem is undecidable. This result extends to Undecidability of other lowenergy properties, such as the existence of algebraically decaying groundstate correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phaseestimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics. The spectral gap problem—whether the Hamiltonian of a quantum manybody problem is gapped or gapless—is rigorously proved to be undecidable; there exists no algorithm to determine whether an arbitrary quantum manybody model is gapped or gapless, and there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics. In quantum manybody physics, the spectral gap is the energy difference between the ground state of a system and its first excited state. Establishing whether it is possible to make a decision about the system being gapped or gapless, given a specific model Hamiltonian, is a longstanding problem in physics known as the spectral gap problem. Here, Toby Cubitt et al. prove that the spectral gap problem is undecidable. Although it had been known before that deciding about the existence of a spectral gap is difficult, this result proves the strongest possible form of algorithmic difficulty for a core problem of manybody physics.

Undecidability of the spectral gap
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The spectral gapthe energy difference between the ground state and first excited state of a systemis central to quantum manybody physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the YangMills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum manybody system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a twodimensional lattice with translationally invariant, nearestneighbour interactions, for which the spectral gap problem is undecidable. This result extends to Undecidability of other lowenergy properties, such as the existence of algebraically decaying groundstate correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phaseestimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

Undecidability of the spectral gap full version
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:We show that the spectral gap problem is undecidable. Specifically, we construct families of translationallyinvariant, nearestneighbour Hamiltonians on a 2D square lattice of dlevel quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lowerbounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.
David Perezgarcia  One of the best experts on this subject based on the ideXlab platform.

Undecidability of the spectral gap
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The spectral gap—the energy difference between the ground state and first excited state of a system—is central to quantum manybody physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the Yang–Mills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum manybody system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a twodimensional lattice with translationally invariant, nearestneighbour interactions, for which the spectral gap problem is undecidable. This result extends to Undecidability of other lowenergy properties, such as the existence of algebraically decaying groundstate correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phaseestimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding ‘halting problem’. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics. The spectral gap problem—whether the Hamiltonian of a quantum manybody problem is gapped or gapless—is rigorously proved to be undecidable; there exists no algorithm to determine whether an arbitrary quantum manybody model is gapped or gapless, and there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics. In quantum manybody physics, the spectral gap is the energy difference between the ground state of a system and its first excited state. Establishing whether it is possible to make a decision about the system being gapped or gapless, given a specific model Hamiltonian, is a longstanding problem in physics known as the spectral gap problem. Here, Toby Cubitt et al. prove that the spectral gap problem is undecidable. Although it had been known before that deciding about the existence of a spectral gap is difficult, this result proves the strongest possible form of algorithmic difficulty for a core problem of manybody physics.

Undecidability of the spectral gap
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:The spectral gapthe energy difference between the ground state and first excited state of a systemis central to quantum manybody physics. Many challenging open problems, such as the Haldane conjecture, the question of the existence of gapped topological spin liquid phases, and the YangMills gap conjecture, concern spectral gaps. These and other problems are particular cases of the general spectral gap problem: given the Hamiltonian of a quantum manybody system, is it gapped or gapless? Here we prove that this is an undecidable problem. Specifically, we construct families of quantum spin systems on a twodimensional lattice with translationally invariant, nearestneighbour interactions, for which the spectral gap problem is undecidable. This result extends to Undecidability of other lowenergy properties, such as the existence of algebraically decaying groundstate correlations. The proof combines Hamiltonian complexity techniques with aperiodic tilings, to construct a Hamiltonian whose ground state encodes the evolution of a quantum phaseestimation algorithm followed by a universal Turing machine. The spectral gap depends on the outcome of the corresponding 'halting problem'. Our result implies that there exists no algorithm to determine whether an arbitrary model is gapped or gapless, and that there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics.

Undecidability of the spectral gap full version
Nature, 2015CoAuthors: Toby S Cubitt, David Perezgarcia, Michael M WolfAbstract:We show that the spectral gap problem is undecidable. Specifically, we construct families of translationallyinvariant, nearestneighbour Hamiltonians on a 2D square lattice of dlevel quantum systems (d constant), for which determining whether the system is gapped or gapless is an undecidable problem. This is true even with the promise that each Hamiltonian is either gapped or gapless in the strongest sense: it is promised to either have continuous spectrum above the ground state in the thermodynamic limit, or its spectral gap is lowerbounded by a constant in the thermodynamic limit. Moreover, this constant can be taken equal to the local interaction strength of the Hamiltonian.
Steffen Kopecki  One of the best experts on this subject based on the ideXlab platform.

transducer descriptions of dna code properties and Undecidability of antimorphic problems
Information & Computation, 2017CoAuthors: Lila Kari, Stavros Konstantinidis, Steffen KopeckiAbstract:This work concerns formal descriptions of DNA code properties and related (un)decidability questions. This line of research allows us to give a property as input to an algorithm, in addition to any regular language, which can then answer questions about the language and the property. Here we define DNA code properties via transducers and show that this method is strictly more expressive than that of regular trajectories, without sacrificing the efficiency of deciding the satisfaction question. We also show that the maximality question can be undecidable. Our Undecidability results hold not only for the fixed DNA involution but also for any fixed antimorphic permutation. Moreover, we also show the Undecidability of the antimorphic version of the Post correspondence problem, for any fixed antimorphic permutation.

transducer descriptions of dna code properties and Undecidability of antimorphic problems
arXiv: Formal Languages and Automata Theory, 2015CoAuthors: Lila Kari, Stavros Konstantinidis, Steffen KopeckiAbstract:This work concerns formal descriptions of DNA code properties, and builds on previous work on transducer descriptions of classic code properties and on trajectory descriptions of DNA code properties. This line of research allows us to give a property as input to an algorithm, in addition to any regular language, which can then answer questions about the language and the property. Here we define DNA code properties via transducers and show that this method is strictly more expressive than that of trajectories, without sacrificing the efficiency of deciding the satisfaction question. We also show that the maximality question can be undecidable. Our Undecidability results hold not only for the fixed DNA involution but also for any fixed antimorphic permutation. Moreover, we also show the Undecidability of the antimorphic version of the Post Corresponding Problem, for any fixed antimorphic permutation.
Lila Kari  One of the best experts on this subject based on the ideXlab platform.

transducer descriptions of dna code properties and Undecidability of antimorphic problems
Information & Computation, 2017CoAuthors: Lila Kari, Stavros Konstantinidis, Steffen KopeckiAbstract:This work concerns formal descriptions of DNA code properties and related (un)decidability questions. This line of research allows us to give a property as input to an algorithm, in addition to any regular language, which can then answer questions about the language and the property. Here we define DNA code properties via transducers and show that this method is strictly more expressive than that of regular trajectories, without sacrificing the efficiency of deciding the satisfaction question. We also show that the maximality question can be undecidable. Our Undecidability results hold not only for the fixed DNA involution but also for any fixed antimorphic permutation. Moreover, we also show the Undecidability of the antimorphic version of the Post correspondence problem, for any fixed antimorphic permutation.

transducer descriptions of dna code properties and Undecidability of antimorphic problems
arXiv: Formal Languages and Automata Theory, 2015CoAuthors: Lila Kari, Stavros Konstantinidis, Steffen KopeckiAbstract:This work concerns formal descriptions of DNA code properties, and builds on previous work on transducer descriptions of classic code properties and on trajectory descriptions of DNA code properties. This line of research allows us to give a property as input to an algorithm, in addition to any regular language, which can then answer questions about the language and the property. Here we define DNA code properties via transducers and show that this method is strictly more expressive than that of trajectories, without sacrificing the efficiency of deciding the satisfaction question. We also show that the maximality question can be undecidable. Our Undecidability results hold not only for the fixed DNA involution but also for any fixed antimorphic permutation. Moreover, we also show the Undecidability of the antimorphic version of the Post Corresponding Problem, for any fixed antimorphic permutation.