The Quintic Gauss Sums
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Classical Simulation of Quantum Circuits by Half Gauss Sums
CLASSICAL SIMULATION OF QUANTUM CIRCUITS BY HALF GAUSS SUMS † ‡ KAIFENG BU ∗ AND DAX ENSHAN KOH ABSTRACT. We give an efficient algorithm to evaluate a certain class of expo- nential sums, namely the periodic, quadratic, multivariate half Gauss sums. We show that these exponential sums become #P-hard to compute when we omit either the periodic or quadratic 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 to arbitrary dimensions, and use our results about half Gauss sums to show that the Holant problem for the set of affine signatures is tractable. CONTENTS 1. Introduction 1 2. Half Gausssums 5 3. m-qudit Clifford circuits 11 4. Hardness results and complexity dichotomy theorems 15 5. Tractable signature in Holant problem 18 6. Concluding remarks 20 Acknowledgments 20 Appendix A. Exponential sum terminology 20 Appendix B. Properties of Gauss sum 21 Appendix C. Half Gauss sum for ξd = ω2d with even d 21 Appendix D. Relationship between half− Gauss sums and zeros of a polynomial 22 References 23 arXiv:1812.00224v1 [quant-ph] 1 Dec 2018 1. INTRODUCTION Exponential sums have been extensively studied in number theory [1] and have a rich history that dates back to the time of Gauss [2]. They have found numerous (†) SCHOOL OF MATHEMATICAL SCIENCES, ZHEJIANG UNIVERSITY, HANGZHOU, ZHE- JIANG 310027, CHINA (*) DEPARTMENT OF PHYSICS, HARVARD UNIVERSITY, CAMBRIDGE, MASSACHUSETTS 02138, USA (‡) DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY, CAMBRIDGE, MASSACHUSETTS 02139, USA E-mail addresses: [email protected] (K.Bu), [email protected] (D.E.Koh). -
Cubic and Biquadratic Reciprocity
Rational (!) Cubic and Biquadratic Reciprocity Paul Pollack 2005 Ross Summer Mathematics Program It is ordinary rational arithmetic which attracts the ordinary man ... G.H. Hardy, An Introduction to the Theory of Numbers, Bulletin of the AMS 35, 1929 1 Quadratic Reciprocity Law (Gauss). If p and q are distinct odd primes, then q p 1 q 1 p = ( 1) −2 −2 . p − q We also have the supplementary laws: 1 (p 1)/2 − = ( 1) − , p − 2 (p2 1)/8 and = ( 1) − . p − These laws enable us to completely character- ize the primes p for which a given prime q is a square. Question: Can we characterize the primes p for which a given prime q is a cube? a fourth power? We will focus on cubes in this talk. 2 QR in Action: From the supplementary law we know that 2 is a square modulo an odd prime p if and only if p 1 (mod 8). ≡± Or take q = 11. We have 11 = p for p 1 p 11 ≡ (mod 4), and 11 = p for p 1 (mod 4). p − 11 6≡ So solve the system of congruences p 1 (mod 4), p (mod 11). ≡ ≡ OR p 1 (mod 4), p (mod 11). ≡− 6≡ Computing which nonzero elements mod p are squares and nonsquares, we find that 11 is a square modulo a prime p = 2, 11 if and only if 6 p 1, 5, 7, 9, 19, 25, 35, 37, 39, 43 (mod 44). ≡ q Observe that the p with p = 1 are exactly the primes in certain arithmetic progressions. -
Appendices A. Quadratic Reciprocity Via Gauss Sums
1 Appendices We collect some results that might be covered in a first course in algebraic number theory. A. Quadratic Reciprocity Via Gauss Sums A1. Introduction In this appendix, p is an odd prime unless otherwise specified. A quadratic equation 2 modulo p looks like ax + bx + c =0inFp. Multiplying by 4a, we have 2 2ax + b ≡ b2 − 4ac mod p Thus in studying quadratic equations mod p, it suffices to consider equations of the form x2 ≡ a mod p. If p|a we have the uninteresting equation x2 ≡ 0, hence x ≡ 0, mod p. Thus assume that p does not divide a. A2. Definition The Legendre symbol a χ(a)= p is given by 1ifa(p−1)/2 ≡ 1modp χ(a)= −1ifa(p−1)/2 ≡−1modp. If b = a(p−1)/2 then b2 = ap−1 ≡ 1modp,sob ≡±1modp and χ is well-defined. Thus χ(a) ≡ a(p−1)/2 mod p. A3. Theorem a The Legendre symbol ( p ) is 1 if and only if a is a quadratic residue (from now on abbre- viated QR) mod p. Proof.Ifa ≡ x2 mod p then a(p−1)/2 ≡ xp−1 ≡ 1modp. (Note that if p divides x then p divides a, a contradiction.) Conversely, suppose a(p−1)/2 ≡ 1modp.Ifg is a primitive root mod p, then a ≡ gr mod p for some r. Therefore a(p−1)/2 ≡ gr(p−1)/2 ≡ 1modp, so p − 1 divides r(p − 1)/2, hence r/2 is an integer. But then (gr/2)2 = gr ≡ a mod p, and a isaQRmodp. -
Sums of Gauss, Jacobi, and Jacobsthai* One of the Primary
JOURNAL OF NUMBER THEORY 11, 349-398 (1979) Sums of Gauss, Jacobi, and JacobsthaI* BRUCE C. BERNDT Department of Mathematics, University of Illinois, Urbana, Illinois 61801 AND RONALD J. EVANS Department of Mathematics, University of California at San Diego, L.a Jolla, California 92093 Received February 2, 1979 DEDICATED TO PROFESSOR S. CHOWLA ON THE OCCASION OF HIS 70TH BIRTHDAY 1. INTRODUCTION One of the primary motivations for this work is the desire to evaluate, for certain natural numbers k, the Gauss sum D-l G, = c e277inkl~, T&=0 wherep is a prime withp = 1 (mod k). The evaluation of G, was first achieved by Gauss. The sums Gk for k = 3,4, 5, and 6 have also been studied. It is known that G, is a root of a certain irreducible cubic polynomial. Except for a sign ambiguity, the value of G4 is known. See Hasse’s text [24, pp. 478-4941 for a detailed treatment of G, and G, , and a brief account of G, . For an account of G, , see a paper of E. Lehmer [29]. In Section 3, we shall determine G, (up to two sign ambiguities). Using our formula for G, , the second author [18] has recently evaluated G,, (up to four sign ambiguities). We shall also evaluate G, , G,, , and Gz4 in terms of G, . For completeness, we include in Sections 3.1 and 3.2 short proofs of known results on G, and G 4 ; these results will be used frequently in the sequel. (We do not discuss G, , since elaborate computations are involved, and G, is not needed in the sequel.) While evaluations of G, are of interest in number theory, they also have * This paper was originally accepted for publication in the Rocky Mountain Journal of Mathematics. -
Reciprocity Laws Through Formal Groups
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 5, May 2013, Pages 1591–1596 S 0002-9939(2012)11632-6 Article electronically published on November 8, 2012 RECIPROCITY LAWS THROUGH FORMAL GROUPS OLEG DEMCHENKO AND ALEXANDER GUREVICH (Communicated by Matthew A. Papanikolas) Abstract. A relation between formal groups and reciprocity laws is studied following the approach initiated by Honda. Let ξ denote an mth primitive root of unity. For a character χ of order m, we define two one-dimensional formal groups over Z[ξ] and prove the existence of an integral homomorphism between them with linear coefficient equal to the Gauss sum of χ. This allows us to deduce a reciprocity formula for the mth residue symbol which, in particular, implies the cubic reciprocity law. Introduction In the pioneering work [H2], Honda related the quadratic reciprocity law to an isomorphism between certain formal groups. More precisely, he showed that the multiplicative formal group twisted by the Gauss sum of a quadratic character is strongly isomorphic to a formal group corresponding to the L-series attached to this character (the so-called L-series of Hecke type). From this result, Honda deduced a reciprocity formula which implies the quadratic reciprocity law. Moreover he explained that the idea of this proof comes from the fact that the Gauss sum generates a quadratic extension of Q, and hence, the twist of the multiplicative formal group corresponds to the L-series of this quadratic extension (the so-called L-series of Artin type). Proving the existence of the strong isomorphism, Honda, in fact, shows that these two L-series coincide, which gives the reciprocity law. -
Arxiv:0804.2233V4 [Math.NT]
October 26, 2018 MEAN VALUES WITH CUBIC CHARACTERS STEPHAN BAIER AND MATTHEW P. YOUNG Abstract. We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters. 1. Introduction and Main results Dirichlet characters of a given order appear naturally in many applications in number theory. The quadratic characters have seen a lot of attention due to attractive questions to ranks of elliptic curves, class numbers, etc., yet the cubic characters have been relatively neglected. In this article we are interested in mean values of L-functions twisted by characters of order three, and also large sieve-type inequalities for these characters. Our first result on such L-functions is the following Theorem 1.1. Let w : (0, ) R be a smooth, compactly supported function. Then ∞ → q (1) ∗ L( 1 , χ)w = cQw(0) + O(Q37/38+ε), 2 Q (q,3)=1 χ (mod q) X χX3=χ 0 b where c> 0 is a constant that can be given explicitly in terms of an Euler product (see (23) below), and w is the Fourier transform of w. Here the on the sum over χ restricts the sum ∗ to primitive characters, and χ0 denotes the principal character. This resultb is most similar (in terms of method of proof) to the main result of [L], who considered the analogous mean value but for the case of cubic Hecke L-functions on Q(ω), ω = e2πi/3. -
Quadratic Reciprocity Via Linear Algebra
QUADRATIC RECIPROCITY VIA LINEAR ALGEBRA M. RAM MURTY Abstract. We adapt a method of Schur to determine the sign in the quadratic Gauss sum and derive from this, the law of quadratic reciprocity. 1. Introduction Let p and q be odd primes, with p = q. We define the Legendre symbol (p/q) to be 1 if p is a square modulo q and 16 otherwise. The law of quadratic reciprocity, first proved by Gauss in 1801, states− that (p/q)(q/p) = ( 1)(p−1)(q−1)/4. − It reveals the amazing fact that the solvability of the congruence x2 p(mod q) is equivalent to the solvability of x2 q(mod p). ≡ There are many proofs of this law≡ in the literature. Gauss himself published five in his lifetime. But all of them hinge on properties of certain trigonometric sums, now called Gauss sums, and this usually requires substantial background on the part of the reader. We present below a proof, essentially due to Schur [2] that uses only basic notions from linear algebra. We say “essentially” because Schur’s goal was to deduce the sign in the quadratic Gauss sum by studying the matrix A = (ζrs) 0 r n 1, 0 s n 1 ≤ ≤ − ≤ ≤ − where ζ = e2πi/n. For the case of n = p a prime, Schur’s proof is reproduced in [1] and a ‘slicker’ proof was later supplied by Waterhouse [3]. Our point is to show that Schur’s proof can be modified to determine tr A when n is an odd number and this allows us to deduce the law of quadratic reciprocity. -
The Fourth Moment of Dirichlet L-Functions
Annals of Mathematics 173 (2011), 1{50 doi: 10.4007/annals.2011.173.1.1 The fourth moment of Dirichlet L-functions By Matthew P. Young Abstract We compute the fourth moment of Dirichlet L-functions at the central point for prime moduli, with a power savings in the error term. 1. Introduction Estimating moments of families of L-functions is a central problem in number theory due in large part to extensive applications. Yet, these moments are seen to be natural objects to study in their own right as they illuminate structure of the family and display beautiful symmetries. The Riemann zeta function has by far garnered the most attention from researchers. Ingham [Ing26] proved the asymptotic formula Z T 1 1 4 4 3 jζ( 2 + it)j dt = a4(log T ) + O((log T ) ) T 0 2 −1 where a4 = (2π ) . Heath-Brown [HB79] improved this result by obtaining Z T 4 1 1 4 X i − 1 +" (1.1) jζ( + it)j dt = a (log T ) + O(T 8 ) T 2 i 0 i=0 for certain explicitly computable constants ai (see (5.1.4) of [CFK+05]). Ob- taining a power savings in the error term was a significant challenge because it requires a difficult analysis of off-diagonal terms which contribute to the lower-order terms in the asymptotic formula. The fourth moment problem is related to the problem of estimating (1.2) X d(n)d(n + f) n≤x uniformly for f as large as possible with respect to x. Extensive discussion of this binary divisor problem can be found in [Mot94]. -
Arxiv:1512.06480V1 [Math.NT] 21 Dec 2015 N Definition
CHARACTERIZATIONS OF QUADRATIC, CUBIC, AND QUARTIC RESIDUE MATRICES David S. Dummit, Evan P. Dummit, Hershy Kisilevsky Abstract. We construct a collection of matrices defined by quadratic residue symbols, termed “quadratic residue matrices”, associated to the splitting behavior of prime ideals in a composite of quadratic extensions of Q, and prove a simple criterion characterizing such matrices. We also study the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of Q(√ 3) and Q(i), respectively. − 1. Introduction Let n> 1 be an integer and p1, p2, ... , pn be a set of distinct odd primes. ∗ The possible splitting behavior of the pi in the composite of the quadratic extensions Q( pj ), where p∗ = ( 1)(p−1)/2p (a minimally tamely ramified multiquadratic extension, cf. [K-S]), is determinedp by the quadratic− residue symbols pi . Quadratic reciprocity imposes a relation on the splitting of p in Q( p∗) pj i j ∗ and pj in Q( pi ) and this leads to the definition of a “quadratic residue matrix”. The main purposep of this article isp to give a simple criterion that characterizes such matrices. These matrices seem to be natural elementary objects for study, but to the authors’ knowledge have not previously appeared in the literature. We then consider higher-degree variants of this question arising from cubic and quartic residue symbols for primes of Q(√ 3) and Q(i), respectively. − 2. Quadratic Residue Matrices We begin with two elementary definitions. Definition. A “sign matrix” is an n n matrix whose diagonal entries are all 0 and whose off-diagonal entries are all 1. -
As an Entire Function; Gauss Sums We First Give, As Promised, the Anal
Math 259: Introduction to Analytic Number Theory L(s; χ) as an entire function; Gauss sums We first give, as promised, the analytic proof of the nonvanishing of L(1; χ) for a Dirichlet character χ mod q; this will complete our proof of Dirichlet's theorem that there are infinitely primes in the arithmetic progression mq+a : m Z>0 whenever (a; q) = 1, and that the logarithmic density of suchf primes is 12='(q).g1 We follow [Serre 1973, Ch. VI 2]. Functions such as ζ(s), L(s; χ) and their products are special cases of whatx Serre calls \Dirichlet series": functions 1 f(s) := a e λns (1) X n − n=1 with an C and 0 λn < λn+1 . [For instance, L(s; χ) is of this form with 2 ≤ !1 λn = log n and an = χ(n).] We assume that the an are small enough that the sum in (1) converges for some s C. We are particularly interested in series such as 2 ζ (s) := L(s; χ) q Y χ mod q whose coefficients an are nonnegative. Then if (1) converges at some real σ0, it converges uniformly on σ σ0, and f(s) is analytic on σ > σ0. Thus a series (1) has a maximal open half-plane≥ of convergence (if we agree to regard C itself as an open half-plane for this purpose), namely σ > σ0 where σ0 is the infimum of the real parts of s C at which (1) converges. This σ0 is then called the \abscissa of convergence"2 2 of (1). -
Sign Ambiguities of Gaussian Sums Heon Kim Louisiana State University and Agricultural and Mechanical College, [email protected]
Louisiana State University LSU Digital Commons LSU Doctoral Dissertations Graduate School 2007 Sign Ambiguities of Gaussian Sums Heon Kim Louisiana State University and Agricultural and Mechanical College, [email protected] Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_dissertations Part of the Applied Mathematics Commons Recommended Citation Kim, Heon, "Sign Ambiguities of Gaussian Sums" (2007). LSU Doctoral Dissertations. 633. https://digitalcommons.lsu.edu/gradschool_dissertations/633 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Doctoral Dissertations by an authorized graduate school editor of LSU Digital Commons. For more information, please [email protected]. SIGN AMBIGUITIES OF GAUSSIAN SUMS A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Department of Mathematics by Heon Kim B.S. in Math., Chonbuk National University, 1995 M.S. in Math., Chonbuk National University, 1997 M.A. in Math., University of Georgia, 2002 December 2007 Acknowledgments This dissertation would not be possible without several contributions. The love of family and friends provided my inspiration and was my driving force. It has been a long journey and completing this work is definitely a high point in my academic career. I could not have come this far without the assistance of many individuals and I want to express my deepest appreciation to them. It is a pleasure to thank my dissertation advisor, Dr. Helena Verrill, and coad- visor Dr. -
The Voronoi Formula and Double Dirichlet Series
The Voronoi formula and double Dirichlet series Eren Mehmet Kıral and Fan Zhou May 10, 2016 Abstract We prove a Voronoi formula for coefficients of a large class of L-functions including Maass cusp forms, Rankin-Selberg convolutions, and certain non-cuspidal forms. Our proof is based on the functional equations of L-functions twisted by Dirichlet characters and does not directly depend on automorphy. Hence it has wider application than previous proofs. The key ingredient is the construction of a double Dirichlet series. MSC: 11F30 (Primary), 11F68, 11L05 Keywords: Voronoi formula, automorphic form, Maass form, multiple Dirichlet series, Gauss sum, Kloosterman sum, Rankin-Selberg L-function 1 Introduction A Voronoi formula is an identity involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. A history of the Voronoi formula can be found in [MS04]. Since its introduction in [MS06], the Voronoi formula on GL(3) of Miller and Schmid has become a standard tool in the study of L-functions arising from GL(3), and has found important applications such as [BB], [BKY13], [Kha12], [Li09], [Li11], [LY12], [Mil06], [Mun13] and [Mun15]. As of yet the general GL(N) formula has had fewer applications, a notable one being [KR14]. The first proof of a Voronoi formula on GL(3) was found by Miller and Schmid in [MS06] using the theory of automorphic distributions. Later, a Voronoi formula was established for GL(N) with N ≥ 4 in [GL06], [GL08], and [MS11], with [MS11] being more general and earlier than [GL08] (see the addendum, loc.