Complexity of Strong Approximation on the Sphere

Sardari, Naser T.

doi:10.1093/imrn/rnz233

Cited by 8 publications

(9 citation statements)

References 10 publications

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“…The second named author formulated a conjecture on the optimal exponent for strong approximation for quadratic forms in 4 variables [Tal15b, Conjecture 1.9] that implies the diameter of LPS Ramanujan graphs is asymptotic to 4/3 log d−1 (n). More recently, he developed and implemented a polynomial time algorithm for navigating LPS Ramanujan graphs under a polynomial time algorithm for factoring integers and an arithmetic conjecture on the distribution of numbers representable as a sum of two squares [Sar17]. The numerical results from that algorithm strongly support that the diameter of LPS Ramanujan graphs is asymptotic to 4/3 log d−1 (n); see [Sar17].…”

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confidence: 98%

Quantum Chaos on Random Cayley Graphs of SL 2[Z/pZ]

Rivin

Sardari²

2017

Experimental Mathematics

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We investigate the statistical behavior of the eigenvalues and diameter of random Cayley graphs of SL 2 [Z/pZ] as the prime number p goes to infinity. We prove a density theorem for the number of exceptional eigenvalues of random Cayley graphs i.e. the eigenvalues with absolute value bigger than the optimal spectral bound. Our numerical results suggest that random Cayley graphs of SL 2 [Z/pZ] and the explicit LPS Ramanujan projective graphs of P 1 (Z/pZ) have optimal spectral gap and diameter as the prime number p goes to infinity.

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confidence: 98%

Quantum Chaos on Random Cayley Graphs of SL 2[Z/pZ]

Rivin

Sardari²

2017

Experimental Mathematics

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“…The main motivation for studying w p (a) for a ∈ S d−2 (Z/qZ) ⊂ S d (Z/qZ) comes from the navigation algorithms for the LPS Ramanujan graphs X p,q , and its archimedean analogue which is the Ross and Selinger algorithm for navigating P SU (2) with the golden quantum gates; see [LPS88], [Mar88], [PLQ08], [Sar17a], and also [RS16] and [PS18].…”

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confidence: 99%

“…The numerical results of Ross and Selinger [RS16] and the third author [Sar17a,Sar18] suggests that for all but tiny fractions of a ∈ S 1 (Z/qZ) ⊂ S 3 (Z/qZ), we have w p (a) = 1 + o q (1). It is also observed that max a (w p (a)) = 4/3 + o q (1).…”

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confidence: 99%

The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$

Hassan¹,

Mao²,

Sardari³

et al. 2018

Preprint

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Assume a polynomial-time algorithm for factoring integers, Conjecture 1.1, d ≥ 3, and q and p are prime numbers, where p ≤ q A for some A > 0. We develop a polynomial-time algorithm in log(q) that lifts every Z/qZ point of S d−2 ⊂ S d to a Z[1/p] point of S d with the minimum height. We implement our algorithm for d = 3 and 4. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the Z/qZ points of S d−2 ⊂ S d .

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“…Giving sharp upper bound of this form on the least prime or the least integer representable by a sum of two squares is crucial in the analysis of the complexity of some algorithms in quantum compiling. In particular, Ross and Selinger's algorithm for the optimal navigation of z-axis rotations in SU (2) by quantum gates [RS14] and its p-adic analogue for finding the shortest path between two diagonal vertices of LPS Ramanujan graphs [Sar17]. In [Sar17], we proved that these heuristic algorithms run in polynomial time under a Cramér type conjecture on the distribution of the inverse image of the integers representable as a sum of two squares by a binary quadratic from; see [Sar17, Conjecture 1.4.].…”

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confidence: 99%

“…In particular, Ross and Selinger's algorithm for the optimal navigation of z-axis rotations in SU (2) by quantum gates [RS14] and its p-adic analogue for finding the shortest path between two diagonal vertices of LPS Ramanujan graphs [Sar17]. In [Sar17], we proved that these heuristic algorithms run in polynomial time under a Cramér type conjecture on the distribution of the inverse image of the integers representable as a sum of two squares by a binary quadratic from; see [Sar17, Conjecture 1.4.]. In this paper, we fix the fundamental discriminant D < 0, and prove that by assuming the generalized Riemann hypothesis a form of this Cramér type Date: May 7, 2018.…”

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confidence: 99%

The least prime ideal in a given ideal class

Sardari¹

2018

Preprint

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Let K be a number field with the discriminant D K and the class number h K , which has bounded degree over Q. By assuming GRH, we prove that every ideal class of K contains a prime ideal with norm less than h 2 K log(D K ) 2 and also all but o(h K ) of them have a prime ideal with norm less than h K log(D K ) 2+ǫ . For imaginary quadratic fields K = Q( √ D), by assuming Conjecture 1.2 (a weak version of the pair correlation conjecure), we improve our bounds by removing a factor of log(D) from our bounds and show that these bounds are optimal.

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Complexity of Strong Approximation on the Sphere

Cited by 8 publications

References 10 publications

Quantum Chaos on Random Cayley Graphs of SL 2[Z/pZ]

Quantum Chaos on Random Cayley Graphs of SL 2[Z/pZ]

The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$

The least prime ideal in a given ideal class

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