Undecidability of the Spectral Gap in One Dimension

Bausch, Johannes; Cubitt, Toby S.; Lucia, Ángelo; Pérez-Garcı́a, David

doi:10.1103/physrevx.10.031038

Cited by 37 publications

(58 citation statements)

References 61 publications

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“…It is known that estimating the spectral gap ∆ is a difficult task [3,5,18]. Our algorithm for finding ground energy, as discussed in Theorem 8, does not depend on knowing the spectral gap.…”

Section: Low-energy State Preparationmentioning

confidence: 99%

Near-optimal ground state preparation

Lin

2020

Quantum

106

105

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Preparing the ground state of a given Hamiltonian and estimating its ground energy are important but computationally hard tasks. However, given some additional information, these problems can be solved efficiently on a quantum computer. We assume that an initial state with non-trivial overlap with the ground state can be efficiently prepared, and the spectral gap between the ground energy and the first excited energy is bounded from below. With these assumptions we design an algorithm that prepares the ground state when an upper bound of the ground energy is known, whose runtime has a logarithmic dependence on the inverse error. When such an upper bound is not known, we propose a hybrid quantum-classical algorithm to estimate the ground energy, where the dependence of the number of queries to the initial state on the desired precision is exponentially improved compared to the current state-of-the-art algorithm proposed in [Ge et al. 2019]. These two algorithms can then be combined to prepare a ground state without knowing an upper bound of the ground energy. We also prove that our algorithms reach the complexity lower bounds by applying it to the unstructured search problem and the quantum approximate counting problem.

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Section: Low-energy State Preparationmentioning

confidence: 99%

Near-optimal ground state preparation

Lin

2020

Quantum

106

105

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“…Since the H alting Problem is known to be undecidable, determining whether the Hamiltonian is gapped or gapless is also undecidable. Conceptually, this is how Cubitt, Perez-Garcia & Wolf, and Bausch et al 9 – 11 proved the undecidability of the spectral gap.…”

Section: Resultsmentioning

confidence: 92%

“… In ref. 11 , point 2 is replaced by a so-called Marker Hamiltonian. This in combination with a circuit-to-Hamiltonian construction results in a ground state, which partitions the spin chain into segments just large enough for the UTM to halt, if it halts.…”

Section: Resultsmentioning

confidence: 99%

“…A significant novel technical contribution of this work is then a proof that the Marker Hamiltonian used in ref. 11 does, in fact, allow for some leeway in the precision to which QPE is performed and can be used to provide a correction for the energy penalty picked up as a result of any errors in the QPE.…”

Section: Resultsmentioning

confidence: 99%

“…The ground state can then be designed to be a product state of the form , where is a valid classical tiling configuration—as described in section ‘Tiling and classical computation’—and is a quantum state with the following properties: We use the 1D Marker Hamiltonian from ref. 11 , and couple its negative energy contribution to the size of each grid square in the classical tiling and the placement of the • marker. The negative energy each square contributes is a determined by where the • marker is placed, and thus by the action of the classical TM.…”

Section: Resultsmentioning

confidence: 99%

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Uncomputability of phase diagrams

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The phase diagram of a material is of central importance in describing the properties and behaviour of a condensed matter system. In this work, we prove that the task of determining the phase diagram of a many-body Hamiltonian is in general uncomputable, by explicitly constructing a continuous one-parameter family of Hamiltonians H(φ), where $$\varphi \in {\mathbb{R}}$$ φ ∈ R , for which this is the case. The H(φ) are translationally-invariant, with nearest-neighbour couplings on a 2D spin lattice. As well as implying uncomputablity of phase diagrams, our result also proves that undecidability can hold for a set of positive measure of a Hamiltonian’s parameter space, whereas previous results only implied undecidability on a zero measure set. This brings the spectral gap undecidability results a step closer to standard condensed matter problems, where one typically studies phase diagrams of many-body models as a function of one or more continuously varying real parameters, such as magnetic field strength or pressure.

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Combining dimensional analysis with model based systems engineering

Martinez-Rojas

Fernández‐Sánchez²

2022

Systems Engineering

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The model based systems engineering (MBSE) approach describes a system using consistent views to provide a holistic model as complete as possible. MBSE methodologies end with the physical architecture of the system, but a physical model is clearly incomplete without the study of its associated physical laws and phenomena related to the whole system or its parts. However, the computational demands could be excessive even for modest projects. Dimensional analysis (DA) is common in fluid dynamics and chemical engineering, but its application to systems engineering is still limited. We describe an engineering methodological process, which incorporates DA as a powerful tool to understand the physical constraints of the system without the burden of complex analytical or numerical calculations. A detailed example describing a microantenna is presented showing the benefits of this approach. The selected example describes a problem rarely covered in modern expositions of DA in order to show the wide benefit of these techniques. The information provided by this analysis is very useful to select the best physically realizable architectures, testing design, and conduct trade-off studies. The complexity of modern systems and systems of systems demands new testing procedures in order to comply with increasingly demanding requirements and regulations. This can be accomplished through research in new DA methods. Finally, this article serves as a fairly comprehensive guide to the use of DA in the context of MBSE, detailing its strengths, limitations, and controversial issues.

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Undecidability of the Spectral Gap in One Dimension

Cited by 37 publications

References 61 publications

Near-optimal ground state preparation

Near-optimal ground state preparation

Uncomputability of phase diagrams

Combining dimensional analysis with model based systems engineering

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