具有高斯和的NMR试验因子数

An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group

The construction of unitary operator bases in a finite-dimensional Hilbert space is reviewed through a nonstandard approach combinining angular momentum theory and representation theory of SU(2). A single formula for the bases is obtained from a polar decomposition of SU(2) and analysed in terms of cyclic groups, quadratic Fourier transforms, Hadamard matrices and generalized Gauss sums. Weyl pairs, generalized Pauli operators and their application to the unitary group and the Pauli group naturally arise in this approach.

New factorization algorithm based on a continuous representation of truncated Gauss sums

In this paper, we will describe a new factorization algorithm based on the continuous representation of Gauss sums, generalizable to orders j>2. Such an algorithm allows one, for the first time, to find all the factors of a number N in a single run without precalculating the ratio N/l, where l are all the possible trial factors. Continuous truncated exponential sums turn out to be a powerful tool for distinguishing factors from non-factors (we also suggest, with regard to this topic, to read an interesting paper by S. Woelk et al. also published in this issue [Woelk, Feiler, Schleich, J. Mod. …

Equientangled Bases in Arbitrary Dimensions and Quadratic Gauss Sums

Recently [Karimipour and Memarzadeh, PhysRevA 73, 012329 (2006)] posed the problem of finding a continuous family of orthonormal bases in a bipartite space of two identical systems with the following properties: i) in each basis, all states have to be equally entangled, and ii) the family continuously interpolate between the product basis and the maximally entangled basis. The authors provided a necessary condition and simple examples for relatively small dimensions, but questioned the existence of a general solution for arbitrary dimensions. Employing the properties of quadratic Gauss sums, w…

Factorization of numbers with Gauss sums: II. Suggestions for implementations with chirped laser pulses

We propose three implementations of the Gauss sum factorization schemes discussed in part I of this series: (i) a two-photon transition in a multi-level ladder system induced by a chirped laser pulse, (ii) a chirped one-photon transition in a two-level atom with a periodically modulated excited state, and (iii) a linearly chirped one-photon transition driven by a sequence of ultrashort pulses. For each of these quantum systems we show that the excitation probability amplitude is given by an appropriate Gauss sum. We provide rules how to encode the number N to be factored in our system and how …

Classical simulation of quantum circuits by half Gauss sums

We give an efficient algorithm to evaluate a certain class of exponential sums, namely the periodic, quadratic, multivariate half Gauss sums. We show that these exponential sums become $\#\mathsf{P}$-hard to compute when we omit either the periodicity or quadraticity condition. We apply our results about these exponential sums to the classical simulation of quantum circuits, and give an alternative proof of the Gottesman-Knill theorem. We also explore a connection between these exponential sums and the Holant framework. In particular, we generalize the existing definition of affine signatures …

相互无偏的相位状态、相位不确定性和高斯和

Mutually unbiased phase states, phase uncertainties, and Gauss sums

互无偏基（MUB）是使得在不同正交基中的两个向量之间的内积是等于1/sqrt（d）的常数，其中d是有限希尔伯特空间的维数，对于诸如量子层析成像和密码学的应用以及与量子光学的纠缠态和海森堡-韦伊群的关系，互无偏基（MUB）正变得越来越多地被研究。对于素数幂维数d=p^m，我们利用抽象代数的工具导出了基数为d+1的MUB的完备集。这里我们将MUB重新解释为量子相态，即厄米相位算符的本征向量，推广了Pegg & Barnett在1989年提出的那些。我们涉及MUB状态加性特征的伽罗瓦域（奇特征p）和伽罗瓦环（特征2）。量子傅里叶变换的基础向量中的分量定义了一个更一般的类MUB的乘法字符和加法的。我们研究了上述相位算子关于数算子的互补性质。我们还研究了一般纯量子电磁态的相位概率分布和方差，发现它们与高斯和有关，高斯和是场（或环）的乘性和加性特征的乘积的所有元素的和。最后，我们介绍了相互无偏性和最大纠缠的概念。这使得使用良好的研究代数概念作为有效的工具，在研究纠缠和它的信息方面

循环矩阵、高斯和与互无偏。质数的情况

Circulant matrices, gauss sums and mutually unbiased I. The prime number case

本文研究了素数维$d$中的互无偏基问题。已知精确地提供$d+1$相互无偏的碱基。我们使用一类循环$d \times d$矩阵重新讨论这个问题。建设性的证明一组$d+1$相互无偏基地如下，连同一组性质的高斯和，双幺模序列。

估计高斯和的有效量子算法

Efficient Quantum Algorithms for Estimating Gauss Sums

我们提出了一个有效的量子算法来估计有限域和有限环上的高斯和。这是一个自然的问题，因为高斯和的描述可以在不参考黑盒函数的情况下完成。通过将离散对数问题转化为高斯和估计问题，证明了该问题对于经典算法是困难的。量子算法的工作依赖于傅里叶变换的加性特征与高斯和的乘性特征之间的相互作用。

Chirped quasi-phase-matching with Gauss sums for production of biphotons

We study the theory of linearly chirped biphoton wave-packets produced in two basic quasi-phase-matching configurations: chirped photonic-like crystals and aperiodically poled crystals. The novelty is that these structures are considered as definite assembles of nonlinear layers that leads to detailed description of spontaneous parametric down-conversion (SPDC) processes through the discrete Gauss sums. We demonstrate that biphoton spectra for chirped photonic crystals involving a small number of layers consist from definite well-resolved spectral lines. We also discuss the forming of broadban…

高斯和与量子力学

Gauss Sums and Quantum Mechanics

将Feynman路径求和方法应用于相空间为环面的量子力学系统，给出了二次高斯和的Landsberg-Schaar恒等式的一个新证明。在现有的非初等证明，其中使用无限的总和和一个限制过程或轮廓积分，只有有限的总和。经典相空间的环形性质导致了离散的位置和动量，因此也导致了离散的时间。相应的“路径积分”是有限的总和，其归一化导出，并显示出相互干扰的循环性和离散性，以给出开尔文镜像法的有限版本。

Superpositions of coherent states determined by Gauss sums

We describe a family of quantum states of the Schrödinger cat type as superpositions of the harmonic oscillator coherent states with coefficients defined by the quadratic Gauss sums. These states emerge as eigenfunctions of the lowering operators obtained after canonical transformations of the Heisenberg-Weyl algebra associated with the ordinary and fractional Fourier transformation. The first member of this family is given by the well known Yurke-Stoler coherent state.

Construction of Equientangled Bases in Arbitrary Dimensions via Quadratic Gauss Sums and Graph States

Recently [Karimipour and Memarzadeh, Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space each of dimension $D$ with the following properties: (i) The family continuously interpolates between the product basis and the maximally entangled basis as some parameter $t$ is varied, and (ii) for a fixed $t$, all basis states have the same amount of entanglement. The authors derived a necessary condition and provided explicit solutions for $D \leq 5$ but the existence of a solution for arbitrary dimensions remained an open problem. We prove …

互无偏测量与高斯和中的量子相位不确定性

Quantum phase uncertainty in mutually unbiased measurements and Gauss sums

互无偏基（MUB），这是这样的两个向量之间的内积在不同的正交基是常数等于逆$1/\sqrt{d}$，与$d$的维数有限希尔伯特空间，正变得越来越多的研究应用，如量子断层扫描和密码学，并与纠缠态和量子光学的海森堡-韦伊群。使用抽象代数的工具，已经导出了素幂维数$d=p^m$的基数$d+1$的MUB的完整集合（Wootters in 1989，Klappenecker in 2003）。可以推测，对于非素维，基数要小得多。基可以被重新解释为量子相位态，即厄米相位算符的本征向量，推广了Pegg & Barnett在1989年提出的那些。MUB态与伽罗瓦域（奇特征p）和伽罗瓦环（特征2）的加性特征有关。量子傅里叶变换的基础向量中的分量定义了一个更一般的类MUB的乘法字符和加法的。我们研究了上述相位算子关于数算子的互补性质。我们还研究了物理状态的相位概率分布和方差，并发现它们与高斯和有关，高斯和是乘法和加法特征的乘积的域（或环）的所有元素的和。最后，我们介绍了相互无偏性和最大纠缠的概念。这使得我们可以使用良好的研究代数概念作为有效的工具，在我们的追求最小的不确定性，在量子信息原语。