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We construct families of translationally invariant, nearest-neighbour Hamiltonians on a 2D square lattice of d -level 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...
Citations
... (Moore [20]). Examples of physical problems where this kind of complexity has been shown to arise include 3D billiards [20], 3D optical systems [23], neural networks [24] or the spectral gap in quantum many-body physics [10]. In a series of works we addressed the following related question posed by Moore [20] and revisited by Tao [27,25,26]: Is hydrodynamics capable of universal computation? ...
In 1991, Moore [20] raised a question about whether hydrodynamics is capable of performing computations. Similarly, in 2016, Tao [25] asked whether a mechanical system, including a fluid flow, can simulate a universal Turing machine. In this expository article, we review the construction in [8] of a "Fluid computer" in dimension 3 that combines techniques in symbolic dynamics with the connection between steady Euler flows and contact geometry unveiled by Etnyre and Ghrist. In addition, we argue that the metric that renders the vector field Beltrami cannot be critical in the Chern-Hamilton sense [9]. We also sketch the completely different construction for the Euclidean metric in $\mathbb R^3$ as given in [7]. These results reveal the existence of undecidable fluid particle paths. We conclude the article with a list of open problems.
... Under the assumptions that h is polynomial and Red d semialgebraic over R, the Tarski-Seidenberg theorem guarantees that the set defined in Eq. (9) is semialgebraic since all equations are polynomial and all quantifiers run over semialgebraic sets. The fact that the quantifiers in Eq. (9) are not at the beginning is not an obstacle since every first-order formula can be brought to prenex normal form, which is used in Theorem 1. Consequently, I γ → (γ ) is a semialgebraic function and therefore piecewise algebraic over R. Hence, the proof is completed by any example whose ground state energy density is transcendental over R (despite the fact that γ → h(γ ) is polynomial). The next section will provide such an example for the case d = 2. ...
... Proof By the results of [2,9] there is a d ∈ N and a family of interaction Hamiltonians h on C d ⊗ C d with matrix entries in Q[ √ 2] (and thus expressible in R exp ) such that the ground state energy density problem ...
The set of two-body reduced states of translation invariant, infinite quantum spin chains can be approximated from inside and outside using matrix product states and marginals of finite systems, respectively. These lead to hierarchies of algebraic approximations that become tight only in the limit of infinitely many auxiliary variables. We show that this is necessarily so for any algebraic ansatz by proving that the set of reduced states is not semialgebraic. We also provide evidence that additional elementary transcendental functions cannot lead to a finitary description.
... Proof. By the results of[BCLPG20,CPGW22] there is a d ∈ N and a family of interaction Hamiltonians h on C d ⊗ C d with matrix entries in Q[ thus expressible in R exp ) such that the ground state energy density problem ∀ρ ∈ Red d : tr [hρ] > 0 (14) is undecidable. Assuming that Red d was definable in the first-order language of R exp , then Eq.(14) would be expressible as a sentence in the first-order language of R exp . ...
The set of two-body reduced states of translation invariant, infinite quantum spin chains can be approximated from inside and outside using matrix product states and marginals of finite systems, respectively. These lead to hierarchies of algebraic approximations that become tight only in the limit of infinitely many auxiliary variables. We show that this is necessarily so for any algebraic ansatz by proving that the set of reduced states is not semialgebraic. We also provide evidence that additional elementary transcendental functions cannot lead to a finitary description.
... The decidability of the spectral gap was answered for quantum spin models on Γ = Z D , D ≥ 2, with translation invariant, nearest neighbor, frustration free interactions in (Cubitt et al., 2015). Specifically, a quantum spin interaction was constructed for which the problem of determining if the model is gapped could be translated into a halting problem of a Turing machine. ...
This work provides an overview of gapped quantum spin systems, including concepts, techniques, properties, and results. The basic framework and objects of interest for quantum spin systems are introduced, and the main ideas behind methods for proving spectral gaps for frustration-free models are outlined. After reviewing recent progress on several spectral gap conjectures, we discuss quasi-locality of the Heisenberg dynamics and its utility in proving properties of gapped quantum spin systems. Lieb-Robinson bounds have played a central role in establishing exponential decay of ground state correlations, an area law for one-dimensional systems, a many-body adiabatic theorem, and spectral gap stability. They also aided in the development of the quasi-adiabatic continuation, which is a useful for investigating gapped ground state phases, both of which are also discussed.
... (Moore [19]). Given the Hamiltonian of a quantum many-body system, does there exist an algorithm to check whether it has a spectral gap? (this is known as the spectral gap problem, recently proved to be undecidable [10]). And last but not least, can a mechanical system (including a fluid flow) simulate a universal Turing machine? ...
... Therefore the question of whether a system with the type of Hamiltonian they defined is gapped or gapless also is undecidable. Note that this is subject to the logical limitation that, as defined by the authors [54], in the original proof "gapped" is not defined strictly as the negation of "gapless." ...
Though calculations based on density functional theory (DFT) are used remarkably widely in chemistry, physics, materials science, and biomolecular research and though the modern form of DFT has been studied for almost 60 years, some mathematical problems remain. From a physical science perspective, it is far from clear whether those problems are of major import. For context, we provide an outline of the basic structure of DFT as it is presented and used conventionally in physical sciences, note some unresolved mathematical difficulties with those conventional demonstrations, then pose several questions regarding both the time-independent and time-dependent forms of DFT that could benefit from attention in applied mathematics. Progress on any of these would aid in development of better approximate functionals and in interpretation of DFT.
... Note also the very interesting work[75,76] which shows that for spin chains, the presence of a gap can be undecidable! These are limits of 0d systems with infinitely many fields, so are excluded by our definitions. ...
Tame geometry originated in mathematical logic and implements strong finiteness properties by defining the notion of tame sets and functions. In part I we argued that observables in a wide class of quantum field theories are tame functions and that the tameness of a theory relies on its UV definition. The aims of this work are (1) to formalize the connection between quantum field theories and logical structures, and (2) to investigate the tameness of conformal field theories. To address the first aim, we start from a set of quantum field theories and explain how they define a logical structure that is subsequently extended to a second structure by adding physical observables. Tameness, or o-minimality, of the two structures is then a well-defined property, and sharp statements can be made by identifying these with known examples in mathematics. For the second aim we quantify our expectations on the tameness of the set of conformal field theories and effective theories that can be coupled to quantum gravity. We formulate tameness conjectures about conformal field theory observables and propose universal constraints that render spaces of conformal field theories to be tame sets. We test these conjectures in several examples and highlight first implications.
... For example, the quantum gap spectral problem states that given a quantum many-body Hamiltonian, is the system that it describes gapped or gapless? In fact, as recently as 2015, this question was proved to be undecidable (in other words, not answerable) [13]. The notion of the undecidability of a system goes back to the times of the Russell paradox -a paradox inherent in the statement "This statement is false" formulated by Bertrand Russell. ...
This article attempts to use the ideas from the field of complexity sciences to revisit the classical field of fluid mechanics. For almost a century, the mathematical self-consistency of Navier-Stokes equations has remained elusive to the community of functional analysts, who posed the Navier-Stokes problem as one of the seven millennium problems in the dawn of 21st century. This article attempts to trace back the historical developments of fluid mechanics as a discipline and explain the consequences of not rationalising one of the commonly agreed upon tenets - continuum hypothesis - in the community. The article argues that 'fluids' can be treated as 'emergent' in nature, in that the atoms and molecules in the nanometre length scale can likely be correlated with the continuum physics at the microscale. If this is the case, then one might start trying to find a theoretical framework that models the emergence of fluids from atoms, effectively solving the multi-scale problem using a single abstract framework. Cantor set with layers $N$ (N can have up to two orders of magnitude) is presented as a potential contender (analytical framework) for connecting the energy in molecular level $C_{1}$ at length scale $l_{cut}$ to the energy at continuum level $C_N$ with length scale $L$. Apart from fluid mechanics, Cantor set is shown to represent the conceptual understanding of VLSI hardware design ($N=5$). Apart from Cantor set, an experimental technique of architecting metafluids is also shown to solve emergence experimentally (i.e. connect physics at specific lower scales to higher scales).
... The Hamiltonian for this construction is given by = =1 ℎ ( , +1) where is the number of spins in the simulator system, and ℎ is a two-body interaction of the form [43,Theorem 32]: ...
... where is a fixed Hermitian matrix and , are fixed non-Hermitian matrices. For a detailed construction of the terms in the Hamiltonian we refer the interested reader to [43,Section 4]. ...
... However, it is important to note that using the construction from [39] it is possible to encode a computation with exponential runtime into a Hamiltonian on polynomially many spins. Details of the construction are given in [43,Section 4.5] (in particular the relevant scaling is discussed on [43,Page 81]). We will not give the details of the construction here, but note that it encodes a Turing machine which runs for O( exp( )) time steps in a Hamiltonian acting on spins [43,Proposition 45]. ...
Recent work has demonstrated the existence of universal Hamiltonians – simple spin lattice models that can simulate any other quantum many body system. These universal Hamiltonians have applications for developing quantum simulators, as well as for Hamiltonian complexity, quantum computation, and fundamental physics. In this thesis we extend the theory of universal Hamiltonians. We begin by developing a new method for proving that a given family of Hamiltonians is indeed universal. We then use this method to construct two new universal models – both of which consist of translationally invariant interactions acting on a 1D spin chain. But the benefit of our method doesn’t just lie in the simple universal models it allows us to construct. It also gives deeper insight into the origins of universality – and demonstrates a link between the universality and complexity. We make this insight rigorous, and derive a complexity theoretic classification of universal Hamiltonians which encompasses all known universal models. This classification provides a new, simplified route to checking whether a particular family of Hamiltonians meets the conditions to be a universal simulator. We also consider the practical use of analogue Hamiltonian simulation. Under- standing the effect of noise on Hamiltonian simulation is a key issue in practical implementations. The first step to tackling this issue is characterising the noise processes affecting near term quantum devices. Motivated by this, we develop and numerically benchmark an algorithm which fits noise models to tomographic data from quantum devices to enable this process. This algorithm has applicability beyond analogue simulators, and could be used to investigate the physical noise processes in any quantum computing device. Finally, we apply the theory of universal Hamiltonians to high energy physics by using them to construct toy models of holographic duality which capture more of the expected features of the AdS/CFT correspondence.
... As a matter of fact, unsolvability arise in many different (mathematical) areas: if a mathematical system allows to store and process information in sufficiently complicated ways, then it will be computationally universal, and therefore even simple questions about these systems will be unsolvable. Examples of unsolvability occur even outside mathematics [5]. (For the record, computability theory doesn't stop after discovering that some problems are unsolvable. ...
Artificial intelligence plays an important role in contemporary medicine. In this short note, we emphasize that philosophy played a role in the development of artificial intelligence. We argue that research in computability theory, the theoretical foundation of modern computer science, was motivated by a philosophical question: can we characterize precisely the class of problems that can be solved by algorithms? We suggest that reflecting on the connection between philosophy and artificial intelligence helps realize that philosophical and scientific progress are connected.
