Hamiltonian Groups with Perfect Order Classes

The Hidden Subgroup Problem (HSP) has been widely studied in the context of quantum computing and is known to be efficiently solvable for Abelian groups, yet appears to be difficult for many non-Abelian ones. An efficient algorithm for the HSP over a group \f G\ runs in time polynomial in \f{n\deq\log|G|.} For any subgroup \f H\ of \f G, let \f{N(H)} denote the normalizer of \f H. Let \MG\ denote the intersection of all normalizers in \f G (i.e., \f{\MG=\cap_{H\leq G}N(H)}). \MG\ is always a subgroup of \f G and the index \f{[G:\MG]} can be taken as a measure of ``how non-Abelian'' \f G is (\f{[G:\MG] = 1} for Abelian groups). This measure was considered by Grigni, Schulman, Vazirani and Vazirani, who showed that whenever \f{[G:\MG]\in\exp(O(\log^{1/2}n))} the corresponding HSP can be solved efficiently (under certain assumptions). We show that whenever \f{[G:\MG]\in\poly(n)} the corresponding HSP can be solved efficiently, under the same assumptions (actually, we solve a slightly more general case of the HSP and also show that some assumptions may be relaxed).

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On the classification of generalized Hamiltonian groups

Groups St Andrews 1997 in Bath ◽

10.1017/cbo9780511666148.025 ◽

1999 ◽

pp. 624-632 ◽

Cited By ~ 1

Author(s):

Denise M. Reboli

Keyword(s):

Hamiltonian Groups

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Pearl Thieves and Perfect Order

A World Trimmed with Fur ◽

10.2307/j.ctvqsdv0c.7 ◽

2017 ◽

pp. 55-92

Keyword(s):

Perfect Order

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On the Perfect State

The Proper Order of Things ◽

10.11126/stanford/9781503603561.003.0007 ◽

2018 ◽

pp. 235-276

Author(s):

Heather L. Ferguson

Keyword(s):

Seventeenth Century ◽

Perfect State ◽

Record Keeping ◽

Intellectual Engagement ◽

Political Critique ◽

Political Analysis ◽

Advice Giving ◽

Perfect Order ◽

Political Theories ◽

Mobile Population

This chapter focuses on the link between medieval political theories and a flourishing Ottoman intellectual engagement with ideas concerning a perfect order of governance from within a sense of crisis. This crisis was driven by an increasingly mobile population, regional rebellions, and global climactic and monetary shifts that together challenged the “fundamentals” of Ottoman administrative order. It traces examples of a mode of political analysis, distinct from advice-giving, that linked justice to proper governance rather than to religion or to the sultan. The chapter demonstrates that Ottoman literary producers of the seventeenth century, while apprehensive of change, became innovators themselves and revived rational modes of political critique in the process. It further highlights how seventeenth-century scholar-bureaucrats came to focus on the archival past of the state itself and located Ottoman power in methods of record keeping. Ultimately they sought to restore a commitment to textual transparency.

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Pearl Thieves and Perfect Order

A World Trimmed with Fur ◽

10.11126/stanford/9780804799966.003.0003 ◽

2017 ◽

Author(s):

Jonathan Schlesinger

Keyword(s):

Freshwater Mussels ◽

Ecological Crisis ◽

Local Administration ◽

Qing Empire ◽

Administrative Structure ◽

Perfect Order ◽

Tribute System

Something strange happened in Manchuria under Qing rule: its freshwater mussels disappeared. Stranger still, the Qing empire did everything in its power to preserve them: draft soldiers; fortify passes; patrol rivers; send boats and horses and silver and men. It streamlined the bureaucracy and revamped the local administration. “Nurture the mussels and let them grow,” the emperor ordered; let Manchuria have mussels. Chapter explores what happened: the collapse of the pearl fishery the attempts, in the language of the Qing court, to “nurture the mussels.” The court put its full weight behind efforts to create a long-term sustainability: it reorganized the administrative structure, empowered territorial governors, and created militarized off-limits areas. Poachers were arrested; the mussels allowed to rest. Through a detailed description of the tribute system, the ecological crisis, and the court’s response, the chapter documents how a reinvented Manchuria came to be.

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