36,548 research outputs found
A subexponential-time quantum algorithm for the dihedral hidden subgroup problem
We present a quantum algorithm for the dihedral hidden subgroup problem with
time and query complexity . In this problem an oracle
computes a function on the dihedral group which is invariant under a
hidden reflection in . By contrast the classical query complexity of DHSP
is . The algorithm also applies to the hidden shift problem for an
arbitrary finitely generated abelian group.
The algorithm begins with the quantum character transform on the group, just
as for other hidden subgroup problems. Then it tensors irreducible
representations of and extracts summands to obtain target
representations. Finally, state tomography on the target representations
reveals the hidden subgroup.Comment: 11 pages. Revised in response to referee reports. Early sections are
more accessible; expanded section on other hidden subgroup problem
Quantum algorithm for the Boolean hidden shift problem
The hidden shift problem is a natural place to look for new separations
between classical and quantum models of computation. One advantage of this
problem is its flexibility, since it can be defined for a whole range of
functions and a whole range of underlying groups. In a way, this distinguishes
it from the hidden subgroup problem where more stringent requirements about the
existence of a periodic subgroup have to be made. And yet, the hidden shift
problem proves to be rich enough to capture interesting features of problems of
algebraic, geometric, and combinatorial flavor. We present a quantum algorithm
to identify the hidden shift for any Boolean function. Using Fourier analysis
for Boolean functions we relate the time and query complexity of the algorithm
to an intrinsic property of the function, namely its minimum influence. We show
that for randomly chosen functions the time complexity of the algorithm is
polynomial. Based on this we show an average case exponential separation
between classical and quantum time complexity. A perhaps interesting aspect of
this work is that, while the extremal case of the Boolean hidden shift problem
over so-called bent functions can be reduced to a hidden subgroup problem over
an abelian group, the more general case studied here does not seem to allow
such a reduction.Comment: 10 pages, 1 figur
On the Quantum Complexity of the Continuous Hidden Subgroup Problem
The Hidden Subgroup Problem (HSP) aims at capturing all problems that are susceptible to be solvable in quantum polynomial time following the blueprints of Shor's celebrated algorithm. Successful solutions to this problems over various commutative groups allow to efficiently perform number-theoretic tasks such as factoring or finding discrete logarithms.
The latest successful generalization (Eisentrager et al. STOC 2014) considers the problem of finding a full-rank lattice as the hidden subgroup of the continuous vector space Rm
, even for large dimensions m
. It unlocked new cryptanalytic algorithms (Biasse-Song SODA 2016, Cramer et al. EUROCRYPT 2016 and 2017), in particular to find mildly short vectors in ideal lattices.
The cryptanalytic relevance of such a problem raises the question of a more refined and quantitative complexity analysis. In the light of the increasing physical difficulty of maintaining a large entanglement of qubits, the degree of concern may be different whether the above algorithm requires only linearly many qubits or a much larger polynomial amount of qubits.
This is the question we start addressing with this work. We propose a detailed analysis of (a variation of) the aforementioned HSP algorithm, and conclude on its complexity as a function of all the relevant parameters. Incidentally, our work clarifies certain claims from the extended abstract of Eisentrager et al
Quantum algorithms for hidden nonlinear structures
Attempts to find new quantum algorithms that outperform classical computation
have focused primarily on the nonabelian hidden subgroup problem, which
generalizes the central problem solved by Shor's factoring algorithm. We
suggest an alternative generalization, namely to problems of finding hidden
nonlinear structures over finite fields. We give examples of two such problems
that can be solved efficiently by a quantum computer, but not by a classical
computer. We also give some positive results on the quantum query complexity of
finding hidden nonlinear structures.Comment: 13 page
On the Quantum Complexity of the Continuous Hidden Subgroup Problem
The Hidden Subgroup Problem (HSP) aims at capturing all problems that are susceptible to be solvable in quantum polynomial time following the blueprints of Shor\u27s celebrated algorithm. Successful solutions to this problems over various commutative groups allow to efficiently perform number-theoretic tasks such as factoring or finding discrete logarithms.
The latest successful generalization (Eisentraeger et al. STOC 2014) considers the problem of finding a full-rank lattice as the hidden subgroup of the continuous vector space , even for large dimensions . It unlocked new cryptanalytic algorithms (Biasse-Song SODA 2016, Cramer et al. EUROCRYPT 2016 and 2017), in particular to find mildly short vectors in ideal lattices.
The cryptanalytic relevance of such a problem raises the question of a more refined and quantitative complexity analysis. In the light of the increasing physical difficulty of maintaining a large entanglement of qubits, the degree of concern may be different whether the above algorithm requires only linearly many qubits or a much larger polynomial amount of qubits.
This is the question we start addressing with this work. We propose a detailed analysis of (a variation of) the aforementioned HSP algorithm, and conclude on its complexity as a function of all the relevant parameters. Our modular analysis is tailored to support the optimization of future specialization to cases of cryptanalytic interests. We suggest a few ideas in this direction
On the Quantum Complexity of the Continuous Hidden Subgroup Problem
The Hidden Subgroup Problem (HSP) aims at capturing all problems that are susceptible to be solvable in quantum polynomial time following the blueprints of Shor’s celebrated algorithm. Successful solutions to this problems over various commutative groups allow to efficiently perform number-theoretic tasks such as factoring or finding discrete logarithms. The latest successful generalization (Eisenträger et al. STOC 2014) considers the problem of finding a full-rank lattice as the hidden subgroup of the continuous vector space Rm, even for large dimensions m. It unlocked new cryptanalytic algorithms (Biasse-Song SODA 2016, Cramer et al. EUROCRYPT 2016 and 2017), in particular to find mildly short vectors in ideal lattices. The cryptanalytic relevance of such a problem raises the question of a more refined and quantitative complexity analysis. In the light of the increasing physical difficulty of maintaining a large entanglement of qubits, the degree of concern may be different whether the above algorithm requires only linearly many qubits or a much larger polynomial amount of qubits. This is the question we start addressing with this work. We propose a detailed analysis of (a variation of) the aforementioned HSP algorithm, and conclude on its complexity as a function of all the relevant parameters. Our modular analysis is tailored to support the optimization of future specialization to cases of cryptanalytic interests. We suggest a few ideas in this direction
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