Quantum Information and Computation

Finding conjugate stabilizer subgroups in $\PSL$ and related groups

Aaron Denney, Cristopher Moore, Alexander C. Russell

Published Mar 1, 2010• 7 citations

Abstract

We reduce a case of the hidden subgroup problem (HSP) in $\SL$, $\PSL$, and $\PGL$, three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group $\AGL$. These groups act on projective space in an ``almost'' 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as $\PSL$.

DOI: 10.26421/qic10.3-4-8

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Quantum Information and Computation

Finding conjugate stabilizer subgroups in $\PSL$ and related groups

Aaron Denney, Cristopher Moore, Alexander C. Russell

Published Mar 1, 2010• 7 citations

Publisher

Abstract

We reduce a case of the hidden subgroup problem (HSP) in

\SL

\PSL

, and

\PGL

, three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group

\AGL

. These groups act on projective space in an ``almost'' 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as

\PSL

DOI: 10.26421/qic10.3-4-8

Related papers

Finding conjugate stabilizer subgroups in PSL(2; q) and related groups

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We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an almost 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as PSL(2; q).We reduce a case of the hidden subgroup problem (HSP) in SL(2; q), PSL(2; q), and PGL(2; q), three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group AGL(1; q). These groups act on projective space in an almost 3-transitive way, and we use this fact in each group to distinguish conjugates of its Borel (upper triangular) subgroup, which is also the stabilizer subgroup of an element of projective space. Our observation is mainly group-theoretic, and as such breaks little new ground in quantum algorithms. Nonetheless, these appear to be the first positive results on the HSP in finite simple groups such as PSL(2; q).

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We reduce a case of the hidden subgroup problem (HSP) in

\SL

\PSL

, and

\PGL

, three related families of finite groups of Lie type, to efficiently solvable HSPs in the affine group

\AGL

\PSL

G=A\rtimes_{\varphi} \mathbb{Z}_{p^k}

, with

A

finite Abelian,

p

prime,

k\in \mathbb{N}

and the action of

\mathbb Z_{p^k}

is generated by the scalar automorphism

a\mapstoμa

, for some

μ\in\mathbb Z_{\operatorname{Exp}(A)}^\times

, where

\mathrm{Exp}(A)

is the exponent of

A

. Our algorithm is efficient when

A

has bounded generator rank and

\mathrm{Exp}(A)/p=\mathrm{polylog}(|G|)

. This includes the case

A=\mathbb{Z}_N

for

N\in\mathbb{N}

and

k=1

, studied by Bacon, Childs and van Dam (FOCS 2005), and

A=\mathbb{Z}_{q^r}

with

q

prime and

r\in \mathbb{N}

studied by van Dam and Dey (TQC 2014). Second, we give a polynomial-time quantum algorithm for finite quasi-Hamiltonian groups under a mild assumption on the input structure. Quasi-Hamiltonian groups are finite nilpotent groups with modular subgroup lattice, or equivalently the finite groups in which every subgroup is permutable. As far as we know, this is the first quantum algorithm to exploit the modularity of the subgroup lattice for solving the HSP. This extends, under the aforementioned structured input assumption, the quantum algorithm for Dedekind groups given by Hallgren, Russell, and Ta-Shma (SIAM J. Comput. 32, 2003).