On the probability of generating a lattice

Fontein, Felix; Wocjan, Paweł

doi:10.1016/j.jsc.2013.12.002

Cited by 5 publications

(4 citation statements)

References 22 publications

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“…not relatively prime to N ; this takes O(log log N ) trials to succeed with high probability. A similar approach works, in general, for sampling generating-sets of uniquely-encoded finite Abelian groups [140]. 8 As discussed in [3], the states |p are unnormalizable and, strictly speaking, do not form a "basis" in the usual sense (they are not elements of H Z ) but they can nevertheless used as a basis for our practical purposes.…”

Section: The Integers Zmentioning

confidence: 99%

The computational power of normalizer circuits over black-box groups

Bermejo-Vega,

Lin,

Nest

2014

Preprint

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This work presents a precise connection between Clifford circuits, Shor's factoring algorithm and several other famous quantum algorithms with exponential quantum speed-ups for solving Abelian hidden subgroup problems. We show that all these different forms of quantum computation belong to a common new restricted model of quantum operations that we call black-box normalizer circuits. To define these, we extend the previous model of normalizer circuits [1-3], which are built of quantum Fourier transforms, group automorphism and quadratic phase gates associated with an Abelian group G. In [1-3], the group is always given in an explicitly decomposed form. In our model, we remove this assumption and allow G to be a black-box group [4]. While standard normalizer circuits were shown to be efficiently classically simulable [1-3], we find that normalizer circuits are powerful enough to factorize and solve classically-hard problems in the black-box setting. We further set upper limits to their computational power by showing that decomposing finite Abelian groups is complete for the associated complexity class. In particular, solving this problem renders black-box normalizer circuits efficiently classically simulable by exploiting the generalized stabilizer formalism in [1][2][3]. Lastly, we employ our connection to draw a few practical implications for quantum algorithm design: namely, we give a no-go theorem for finding new quantum algorithms with black-box normalizer circuits, a universality result for low-depth normalizer circuits, and identify two other complete problems.

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Section: The Integers Zmentioning

confidence: 99%

The computational power of normalizer circuits over black-box groups

Bermejo-Vega,

Lin,

Nest

2014

Preprint

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“…These are limitations of Theorem 1. A similar situation occurs in a somewhat 'dual' case [2], where the probability of that m integer vectors with bounded entries generate a same lattice of rank n was studied. In [2], the ideal choice is m = n + 1 but a constant probability only for the case of m ≥ 2n+1 was rigorously proven.…”

Section: Introductionmentioning

confidence: 86%

“…A similar situation occurs in a somewhat 'dual' case [2], where the probability of that m integer vectors with bounded entries generate a same lattice of rank n was studied. In [2], the ideal choice is m = n + 1 but a constant probability only for the case of m ≥ 2n+1 was rigorously proven. However, based on an extensive experimental study, we conjecture that a constant probability exists as well for 0 ≤ s ≤ 2; see Conjecture 1 for detail.…”

Section: Introductionmentioning

confidence: 86%

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On the probability of generating a primitive matrix

Chen¹,

Feng²,

Liu³

et al. 2021

Preprint

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Given a k × n integer primitive matrix A (i.e., a matrix can be extended to an n × n unimodular matrix over the integers) with size of entries bounded by λ, we study the probability that the m × n matrix extended from A by choosing other m − k vectors uniformly at random from {0, 1, . . . , λ − 1} is still primitive. We present a complete and rigorous proof that the probability is at least a constant for the case of m ≤ n − 4. Previously, only the limit case for λ → ∞ with k = 0 was analysed in Maze et al. (2011), known as the natural density. As an application, we prove that there exists a fast Las Vegas algorithm that completes a k × n primitive matrix A to an n × n unimodular matrix within expected O(n ω log A ) bit operations, where O is big-O but without log factors, ω is the exponent on the arithmetic operations of matrix multiplication and A is the maximal absolute value of entries of A.

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