The unsolvable problem in scientific american of october 2018. Undecidability of the spectral gap quantum field theory. 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. It is to be emphasized, however, that all suggested formalizations have turned out to be equivalent and, moreover, the existence of undecidable. However, the shape of their amplitude spectrum remains unmodi. We will first relate undecidability of the spectral gap to undecidability of another important physical quantity, the ground state energy density, which for a 2d lattice is given by e. Recent work has characterised rigorously what it means for one quantum system to simulate another, and demonstrated the existence of universal hamiltonianssimple spin lattice hamiltonians that can replicate the entire physics of any other quantum many body system. This is the same sense in which the halting problem is undecidable. Undecidability of the spectral gap toby cubitt 1, david perezgarciay2, and michael m. On the other extreme, for two or higher dimensional. Assistant vice president of production and manufacturing. Undecidability of the spectral gap full version core.
In quantum manybody systems, the existence of a spectral gap above the ground state has farreaching consequences. The authors used an aperiodic tiling of quantum turing. A hamiltonian with a spectral gap is called a gapped hamiltonian, and those that do not are called gapless in solidstate physics, the most important spectral gap is for the manybody. Spectral analysis of signalspetre stoica and randolph moses p. The title a paradox at the heart of mathematics makes a physics problem unanswerable is that of the nature news article, written by davide castelvecci though i dont know if he wrote the title. In this work we construct simple examples of 2d quantum spinlattice models with small.
Dec 09, 2015 a small spectral gap the energy needed to transfer an electron from a lowenergy state to an excited state is the central property of semiconductors. Why some physicists are excited about the undecidability of the. Spectral gaps, additive energy, and a fractal uncertainty. In the simplest case of nearestneighbour frustrationfree qubit interactions, there is a complete classification. For all system sizes smaller than some threshold n, the lowenergy physics are classical. This cited by count includes citations to the following articles in scholar. Quantum complexity theory siam journal on computing vol. The spectral gap the energy difference between the ground state and first excited state of a system is central to quantum manybody physics. The spectral gap the energy difference between the ground state and first excited state is central to quantum manybody physics. Ask an unbounded question, get an uncomputable answer. Recently, this important problem was shown to be undecidable for quantum systems in two or more spatial dimensions. Nature just published a paper by cubitt, perezgarcia and wolf titled undecidability of the spectral gap, there is an extended version on arxiv which is 146 pages long.
A short version of this paper was published in nature cpw15a. This additive energy can in turn be estimated in terms of the constants in ahlforsdavid regularity of. Determining whether a turing machine is a busy beaver champion i. What happens to undecidability in the quantum computing paradigm.
Wolf submitted on 16 feb 2015, last revised 18 apr 2018 this version, v3 abstract. Undecidability of the spectral gap full version authors. We explain the spectral gap problem, its importance for physics and possi ble consequences of this exciting new result. The spectral gap problem determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a. On the other hand, ztr and lst preserve the absolute level of amplitude but lead to greatly increased spectral noise for increasing gap size. These and other problems are particular cases of the general.
Specifically, we construct families of translationallyinvariant. A small spectral gap the energy needed to transfer an electron from a lowenergy state to an excited state is the central property of semiconductors. Spectral theory in hilbert spaces eth zuric h, fs 09. We show that the spectral gap problem is undecidable. A hamiltonian with a spectral gap is called a gapped hamiltonian, and those that do not are called gapless. Related properties, like ergodicity roughly, the equivalence between averages over time and averages over the state space in a markov chain, also fall under the umbrella of mixing but we will not address them. The spectral gap problemwhether the hamiltonian of a quantum manybody problem is gapped or gaplessis rigorously proved to be undecidable. The most important step in proving undecidability of the spectral gap is to prove undecidability of another relevant quantity. In quantum mechanics, the spectral gap of a system is the energy difference between its ground state and its first excited state. It is sucient to prove undecidability of the ground state energy with constant promise gap, i. These and other problems are particular cases of the general spectral gap problem. We then show how to transform the halting problem into a question about ground state energy densities. In 2015 it was shown that the problem of determining the existence of a spectral gap is undecidable. Full text also available in the acm digital library as pdf html digital edition.
Undecidability of the spectral gap short version arxiv. Reproduced below is the abstract to the paper in nature, which gives a better summary of the proof than i could presume to do. 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. The most common among such formalizations is a turing machine.
Previous universality results have required proofs involving complicated chains of perturbative gadgets. This algorithmic undecidability is well known to imply axiomatic independence of the spectral gap problem. 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. For multiple small data gaps, dft, ztr and lst can, unlike fft. The spectral gapthe energy difference between the ground state and first excited stateis central to quantum manybody physics. Wolfz5 1department of computer science, university college london, gower street, london wc1e 6ea, united kingdom 2damtp, university of cambridge, centre for mathematical sciences, wilberforce road, cambridge cb3 0wa, united kingdom.
The size of the gap is expressed using the additive energy of stereographic projections of the limit set. Spectral gaps of frustrationfree spin systems with. Conceptually, the process x t may be described as follows. 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, existence of gapped topological spin liquid phases, or the yangmills gap conjecture, concern spectral gaps. The mass gap is the spectral gap between the vacuum and the lightest particle. The spectral gap problem is algorithmically 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. More precisely, we give a reduction from the ground state energy density problem to the. Once we have this, it is relatively easy to lift it to undecidability of the spectral gap. Godel and turing enter quantum physics 9 december 2015 a mathematical problem underlying fundamental questions in particle and quantum physics is.
Lloyd, on the uncomputability of the spectral gap, arxiv. The spectral gap problem consist in deciding, given a local interaction, whether the corresponding translationally invariant hamiltonian on a lattice has a spectral gap independent of the system size or not. The halting problem determining whether a turing machine halts on a given input and the mortality problem determining whether it halts for every starting configuration. In this case, estimating the spectral gap becomes undecidable. The undecidability of a problem means that an algorithm is impossible in principle not only that no algorithm is presently known. The aim of this short note is to clarify some of the claims made in the comparison made in s. Specifically, we construct families of translationally. The spectral gap problem is axiomatically independent. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. Many challenging open problems, such as the haldane conjecture, existence of gapped topological spin liquid phases.
We now state a result bounding the pmixing times in terms of the spectral gap. Undecidability of the spectral gap in one dimension inspire. The spectral gap problem determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a continuous range of lowenergy excitations pervades quantum manybody physics. Reproduced below is the abstract to the paper in nature, which gives a better summary of the proof than i. Undecidability of the spectral gap short version core. A recent preprint as well as a refereed paper in nature describe the situation more completely. This is somewhat similar to the undecidable tiling problems in which one has to decide whether it is possible to tile an in. Undecidability of the spectral gap eprints complutense. Undecidability of the spectral gap in one dimension. The spectral gap is one of the most important physical properties of a quantum manybody system, determining much of its low energy physics the theorem states that the sgp is algorithmically undecidable i.
The spectral gap the energy difference between the ground state and first excited state of a systemis central to quantum manybody physics. The question of the spectral gap is connected with the question of quantization because the existence of a spectral gap involves the presence of a discontinuous distribution of energy inside complex system while the absence of a spectral gap involves a continuous. Review of spectral theory and compact operators 16 2. Thermally induced metallic phase in a gapped quantum spin liquid a monte carlo study of the kitaev model with parity projection.
Undecidability of the spectral gap free download as pdf file. In solidstate physics, the most important spectral gap is for the manybody system of electrons in a solid material, in which case it is often known as an. Chapter 1 introduction in a 2015 paper by cubitt et al. Markov chain monte carlo, mixing, and the spectral gap.
Undecidability of the spectral gap short version arxiv vanity. Undecidability of the spectral gap toby cubitt 1,2, david perezgarciay3,4, and michael m. In this paper, we discuss finitesize criteria for having a spectral gap in frustrationfree spin systems and their applications. In fact, undecidability of the ground state energy density is stronger than we really need to prove undecidability of the spectral gap. Other articles where turings undecidability theorem is discussed. Just to be clear, the title to our nature article is undecidability of the spectral gap. The one i know about is the undecidability of the spectral gap in quantum manybody physics.