Perturbation Gadgets: Arbitrary Energy Scales from a Single Strong Interaction


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jats:titleAbstract</jats:title>jats:pFundamentally, it is believed that interactions between physical objects are two-body. Perturbative gadgets are one way to break up an effective many-body coupling into pairwise interactions: a Hamiltonian with high interaction strength introduces a low-energy space in which the effective theory appearsjats:italick</jats:italic>-body and approximates a target Hamiltonian to within precisionjats:inline-formulajats:alternativesjats:tex-math$$\epsilon $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:miϵ</mml:mi></mml:math></jats:alternatives></jats:inline-formula>. One caveat of existing constructions is that the interaction strength generally scales exponentially in the locality of the terms to be approximated. In this work we propose a many-body Hamiltonian construction which introduces only a single separate energy scale of orderjats:inline-formulajats:alternativesjats:tex-math$$\Theta (1/N^{2+\delta })$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mrowmml:miΘ</mml:mi>mml:mo(</mml:mo>mml:mn1</mml:mn>mml:mo/</mml:mo>mml:msupmml:miN</mml:mi>mml:mrowmml:mn2</mml:mn>mml:mo+</mml:mo>mml:miδ</mml:mi></mml:mrow></mml:msup>mml:mo)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>, for a small parameterjats:inline-formulajats:alternativesjats:tex-math$$\delta >0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mrowmml:miδ</mml:mi>mml:mo></mml:mo>mml:mn0</mml:mn></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>, and forjats:italicN</jats:italic>terms in the target Hamiltonianjats:inline-formulajats:alternativesjats:tex-math$$\mathbf H_\mathrm {t}=\sum _{i=1}^N \mathbf h_i$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mrowmml:msubmml:miH</mml:mi>mml:mit</mml:mi></mml:msub>mml:mo=</mml:mo>mml:msubsupmml:mo∑</mml:mo>mml:mrowmml:mii</mml:mi>mml:mo=</mml:mo>mml:mn1</mml:mn></mml:mrow>mml:miN</mml:mi></mml:msubsup>mml:msubmml:mih</mml:mi>mml:mii</mml:mi></mml:msub></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>to be simulated: in its low-energy subspace, our constructed system can approximate any such target Hamiltonianjats:inline-formulajats:alternativesjats:tex-math$$\mathbf H_t$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:msubmml:miH</mml:mi>mml:mit</mml:mi></mml:msub></mml:math></jats:alternatives></jats:inline-formula>with norm ratiosjats:inline-formulajats:alternativesjats:tex-math$$r=\max _{i,j\in {1,\ldots ,N}}\Vert \mathbf h_i\Vert / \Vert \mathbf h_j \Vert ={{,\mathrm{O},}}(\exp (\exp ({{,\mathrm{poly},}}N)))$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mrowmml:mir</mml:mi>mml:mo=</mml:mo>mml:msubmml:momax</mml:mo>mml:mrowmml:mii</mml:mi>mml:mo,</mml:mo>mml:mij</mml:mi>mml:mo∈</mml:mo>mml:mo{</mml:mo>mml:mn1</mml:mn>mml:mo,</mml:mo>mml:mo…</mml:mo>mml:mo,</mml:mo>mml:miN</mml:mi>mml:mo}</mml:mo></mml:mrow></mml:msub>mml:mrowmml:mo‖</mml:mo></mml:mrow>mml:msubmml:mih</mml:mi>mml:mii</mml:mi></mml:msub>mml:mrowmml:mo‖</mml:mo>mml:mo/</mml:mo>mml:mo‖</mml:mo></mml:mrow>mml:msubmml:mih</mml:mi>mml:mij</mml:mi></mml:msub>mml:mrowmml:mo‖</mml:mo>mml:mo=</mml:mo>mml:mrow<mml:mspace />mml:miO</mml:mi><mml:mspace /></mml:mrow>mml:mrowmml:mo(</mml:mo>mml:moexp</mml:mo>mml:mrowmml:mo(</mml:mo>mml:moexp</mml:mo>mml:mrowmml:mo(</mml:mo>mml:mrow<mml:mspace />mml:mipoly</mml:mi><mml:mspace /></mml:mrow>mml:miN</mml:mi>mml:mo)</mml:mo></mml:mrow>mml:mo)</mml:mo></mml:mrow>mml:mo)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>to withinjats:italicrelative</jats:italic>precisionjats:inline-formulajats:alternativesjats:tex-math$${{,\mathrm{O},}}(N^{-\delta })$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">mml:mrowmml:mrow<mml:mspace />mml:miO</mml:mi><mml:mspace /></mml:mrow>mml:mo(</mml:mo>mml:msupmml:miN</mml:mi>mml:mrowmml:mo-</mml:mo>mml:miδ</mml:mi></mml:mrow></mml:msup>mml:mo)</mml:mo></mml:mrow></mml:math></jats:alternatives></jats:inline-formula>. This comes at the expense of increasing the locality by at most one, and adding an at most poly-sized ancillary system for each coupling; interactions on the ancillary system are geometrically local, and can be translationally invariant. In order to prove this claim, we borrow a technique from high energy physics—where matter fields obtain effective properties (such as mass) from interactions with an exchange particle—and employ a tiling Hamiltonian to discard all cross-terms at higher expansion orders of a Feynman–Dyson series expansion. As an application, we discuss implications for QMA-hardness of thejats:scLocal Hamiltonian</jats:sc>problem, and argue that “almost” translational invariance—defined as arbitrarily small relative variations of the strength of the local terms—is as good as non-translational invariance in many of the constructions used throughout Hamiltonian complexity theory. We furthermore show that the choice of geared limit of many-body systems, where e.g. width and height of a lattice are taken to infinity in a specific relation, can have different complexity-theoretic implications: even for translationally invariant models, changing the geared limit can vary the hardness of finding the ground state energy with respect to a given promise gap from computationally trivial, to QMAjats:subEXP</jats:sub>-, or even BQEXPSPACE-complete.</jats:p>

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Journal Title
Annales Henri Poincaré
Conference Name
Journal ISSN
1424-0637
1424-0661
Volume Title
21
Publisher
Springer Science and Business Media LLC
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All rights reserved
Sponsorship
Pembroke College (JRF)