Spectral analysis of signalspetre stoica and randolph moses p. More precisely, we give a reduction from the ground state energy density problem to the. 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 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. 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. The one i know about is the undecidability of the spectral gap in quantum manybody physics.
In the simplest case of nearestneighbour frustrationfree qubit interactions, there is a complete classification. The most important step in proving undecidability of the spectral gap is to prove undecidability of another relevant quantity. 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. Spectral gaps of frustrationfree spin systems with. It is to be emphasized, however, that all suggested formalizations have turned out to be equivalent and, moreover, the existence of undecidable. The spectral gap the energy difference between the ground state and first excited state is central to quantum manybody physics. Undecidability of the spectral gap in one dimension.
In this paper, we discuss finitesize criteria for having a spectral gap in frustrationfree spin systems and their applications. This algorithmic undecidability is well known to imply axiomatic independence of the spectral gap problem. Quantum complexity theory siam journal on computing vol. Once we have this, it is relatively easy to lift it to undecidability of the spectral gap. The ones marked may be different from the article in the profile. This is the same sense in which the halting problem is undecidable. It is sucient to prove undecidability of the ground state energy with constant promise gap, i. This is somewhat similar to the undecidable tiling problems in which one has to decide whether it is possible to tile an in.
The mass gap is the spectral gap between the vacuum and the lightest particle. Undecidability of the spectral gap in one dimension inspire. 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. Specifically, we construct families of translationally. Full text also available in the acm digital library as pdf html digital edition. Related properties, like ergodicity roughly, the equivalence between averages over time and averages over the state space in a markov chain, also fall. 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 spectral gap problem is algorithmically undecidable.
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. What happens to undecidability in the quantum computing paradigm. However, the shape of their amplitude spectrum remains unmodi. The undecidability of a problem means that an algorithm is impossible in principle not only that no algorithm is presently known.
Many challenging open problems, such as the haldane conjecture, existence of gapped topological spin liquid phases. The spectral gap problem determining whether the energy spectrum of a system has an energy gap above ground state, or if there is a. 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. Undecidability of the spectral gap eprints complutense. Other articles where turings undecidability theorem is discussed. Undecidability of the spectral gap short version core. 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. 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. 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.
These and other problems are particular cases of the general. 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. Conceptually, the process x t may be described as follows. Ask an unbounded question, get an uncomputable answer. Just to be clear, the title to our nature article is undecidability of the spectral gap. Undecidability of the spectral gap toby cubitt 1, david perezgarciay2, and michael m. A hamiltonian with a spectral gap is called a gapped hamiltonian, and those that do not are called gapless. The spectral gap problemwhether the hamiltonian of a quantum manybody problem is gapped or gaplessis rigorously proved to be undecidable. 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.
We then show how to transform the halting problem into a question about ground state energy densities. For multiple small data gaps, dft, ztr and lst can, unlike fft. Previous universality results have required proofs involving complicated chains of perturbative gadgets. On the other extreme, for two or higher dimensional. Quantum complexity theory siam journal on computing. In 2015 it was shown that the problem of determining the existence of a spectral gap is undecidable. On the other hand, ztr and lst preserve the absolute level of amplitude but lead to greatly increased spectral noise for increasing gap size. Undecidability of the spectral gap short version arxiv vanity.
The spectral gapthe energy difference between the ground state and first excited stateis central to quantum manybody physics. In this case, estimating the spectral gap becomes undecidable. Wolf submitted on 16 feb 2015, last revised 18 apr 2018 this version, v3 abstract. Undecidability of the spectral gap full version core. The spectral gap problem is axiomatically independent. A recent preprint as well as a refereed paper in nature describe the situation more completely. Dec 10, 2015 the spectral gap the energy difference between the ground state and first excited state of a systemis central to quantum manybody physics.
Chapter 1 introduction in a 2015 paper by cubitt et al. Undecidability of the spectral gap toby cubitt 1,2, david perezgarciay3,4, and michael m. The unsolvable problem in scientific american of october 2018. Undecidability of the spectral gap free download as pdf file. 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. 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 translationallyinvariant, nearestneighbour hamiltonians on a 2d square lattice of dlevel quantum systems d constant, for which determining whether the system is. The size of the gap is expressed using the additive energy of stereographic projections of the limit set. Review of spectral theory and compact operators 16 2. Undecidability of the spectral gap quantum field theory.
For all system sizes smaller than some threshold n, the lowenergy physics are classical. Markov chain monte carlo, mixing, and the spectral gap. A short version of this paper was published in nature cpw15a. Spectral theory in hilbert spaces eth zuric h, fs 09. 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. This cited by count includes citations to the following articles in scholar. Spectral gaps, additive energy, and a fractal uncertainty. Specifically, we construct families of translationallyinvariant. The authors used an aperiodic tiling of quantum turing. This additive energy can in turn be estimated in terms of the constants in ahlforsdavid regularity of.
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. Assistant vice president of production and manufacturing. 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. We explain the spectral gap problem, its importance for physics and possi ble consequences of this exciting new result. The churchturing theorem of undecidability, combined with the related result of the polishborn american mathematician alfred tarski 190283 on undecidability of truth, eliminated the possibility of a purely mechanical device replacing mathematicians. 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. Lloyd, on the uncomputability of the spectral gap, arxiv. In quantum manybody systems, the existence of a spectral gap above the ground state has farreaching consequences. Reproduced below is the abstract to the paper in nature, which gives a better summary of the proof than i could presume to do. The most common among such formalizations is a turing machine.
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. In fact, undecidability of the ground state energy density is stronger than we really need to prove undecidability of the spectral gap. Undecidability of the spectral gap full version authors. Determining whether a turing machine is a busy beaver champion i. This algorithmic undecidability relates to solving the spectral gap for a family of hamiltonians, as we might have if we want to map out the phase diagram of a system as we vary some external parameters. 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 of a systemis central to quantum manybody physics. We now state a result bounding the pmixing times in terms of the spectral gap. Reproduced below is the abstract to the paper in nature, which gives a better summary of the proof than i. Thermally induced metallic phase in a gapped quantum spin liquid a monte carlo study of the kitaev model with parity projection. Godel and turing enter quantum physics 9 december 2015 a mathematical problem underlying fundamental questions in particle and quantum physics is. The spectral gapthe energy difference between the ground state and first excited state of a systemis central to quantum manybody physics.