Frederic Green

Frederic Green
Clark University · Department of Mathematics and Computer Science

Ph.D.

About

87
Publications
6,682
Reads
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1,211
Citations
Citations since 2017
17 Research Items
183 Citations
2017201820192020202120222023010203040
2017201820192020202120222023010203040
2017201820192020202120222023010203040
2017201820192020202120222023010203040
Additional affiliations
September 1986 - present
Clark University
Position
  • Professor (Full)
September 1984 - May 1986
Worcester Polytechnic Institute
Position
  • Instructor
September 1981 - May 1984
Northeastern University
Position
  • Research Associate
Education
September 1973 - May 1979
Yale University
Field of study
  • Physics
September 1969 - May 1973

Publications

Publications (87)
Article
In this column we review two books, both mathematical, the second containing more of an emphasis on applications:
Article
There are probably at least as many different approaches to number theory as there are books written about it. Some broad distinctions include those taking an historical versus (say) a purely modern approach, with many gradations in between, or those that are algebraically oriented (e.g., with an emphasis on reciprocity laws, or questions that rela...
Article
In this column we look at four books, ranging from the theoretical to the applied, in these three reviews:
Article
In this column we review these three books: 1. Compact Data Structures - A Practical Approach, by Gonzalo Navarro. An unusual approach focussing on data structures that use as little space as is theoretically possible, while maintaining practicality. Review by Laszlo Kozma. 2. Power Up: Unlocking the Hidden Mathematics in Video Games, by Matthew La...
Article
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For a number of years we have been investigating the geometric and algebraic properties of a family of discrete sets of points in Euclidean space generated by a simple binary operation: pairwise affine combination by a xed parameter, which we call xed-parameter extrapolation. By varying the parameter and the set of initial points, a large variety o...
Article
While the approaches are all different, there is at least one thread running through the three books reviewed in this column: the graph, and/or its more complex relative as manifested in the real world, the network: 1. Words and Graphs, by Sergey Kitaev and Vadim Lozin. The mathematics connecting two ideas that are more closely related than might a...
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We show that degree-d block-symmetric polynomials in n variables modulo any odd p correlate with parity exponentially better than degree-d symmetric polynomials, if ncd2logd and d[0995pt−1pt) for some t1. For these infinitely many degrees, our result solves an open problem raised by a number of researchers including Alon and Beigel in 2001 \cite{Al...
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http://cjtcs.cs.uchicago.edu/articles/2016/10/cj16-10.pdf
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We show that if a language is recognized within certain error bounds by constant-depth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, for 0 < ∈ ≤ δ ≤ 1, we define BQNC∈,δ0 to be the class of languages recognized by constant depth, polynomial-size quantum circuits with acceptance...
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An affine-invariant property over a finite field is a property of functions over Fn/p that is closed under all affine transformations of the domain. This class of properties includes such well-known beasts as low-degree polynomials, polynomials that ...
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Fix any $\lambda\in\complexes$. We say that a set $S\subseteq\complexes$ is $\lambda$-convex if, whenever $a$ and $b$ are in $S$, the point $(1-\lambda)a+\lambda b$ is also in $S$. If $S$ is also (topologically) closed, then we say that $S$ is $\lambda$-clonvex. We investigate the properties of $\lambda$-convex and $\lambda$-clonvex sets and prove...
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We report on some initial results of a brute-force search for determining the maximum correlation between degree-d polynomials modulo p and the n-bit mod q function. For various settings of the parameters n, d, p, and q, our results indicate that symmetric polynomials yield the maximum correlation. This contrasts with the previouslyanalyzed setting...
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TextIn this paper, we completely characterize the quadratic polynomials modulo 3 with the largest (hence “optimal”) correlation with parity. This result is obtained by analysis of the exponential sumS(t,k,n)=12n∑xi∈{1,−1}1⩽i⩽n(∏i=1nxi)ωt(x1,x2,…,xn)+k(x1,x2,…,xn) where t(x1,…,xn) and k(x1,…,xn) are quadratic and linear forms respectively, over Z3[x...
Article
Universal circuits can be viewed as general-purpose simulators for central classes of circuits and can be used to capture the computational power of the circuit class being simulated. We define and construct quantum universal circuits which are efficient and has very little overhead in simulation. For depth we construct universal circuits whose dep...
Conference Paper
Full-text available
Universal circuits can be viewed as general-purpose simulators for central classes of circuits and can be used to capture the computational power of the circuit class being simulated. We define and construct quantum universal circuits which are efficient and have very little overhead in simulation. For depth we construct universal circuits whose de...
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We dene and construct ecient depth-universal and almost- size-universal quantum circuits. Such circuits can be viewed as general- purpose simulators for central classes of quantum circuits and can be used to capture the computational power of the circuit class being simulated. For depth we construct universal circuits whose depth is the same order...
Article
In this column we review the following books. 1. The Access Principle by John Willinsky. Review by Scott Aaronson. The book is about how journals should be more available on-line and other issues about journals. I invited Scott Aaronson to do ...
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Small depth quantum circuits have proved to be unexpectedly powerful in comparison to their classical counterparts. We survey some of the recent work on this and present some open problems.
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We prove that the quadratic polynomials modulo $3$ with the largest correlation with parity are unique up to permutation of variables and constant factors. As a consequence of our result, we completely characterize the smallest MAJ~$circ mbox{MOD}_3 circ { m AND}_2$ circuits that compute parity, where a MAJ~$circ mbox{MOD}_3 circ { m AND}_2$ circui...
Article
We study exponential sums of the form S = 2(-n) Sigma(x{0,1}n) e(m)(h(x))e(q) (p(x)), where m, q is an element of Z(+) are relatively prime, p is a polynomial with coefficients in Z(q), and h(x) = a(x(1) + ... + x(n)) for some 1 <= a < m. We prove an upper bound of the form 2(-Omega(n)) on vertical bar S vertical bar. This generalizes a result of J...
Article
We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2 . This is the first result of this type for...
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We prove several new lower bounds for constant depth quantum circuits. The main result is that parity (and hence fanout) requires log depth circuits, when the circuits are composed of single qubit and arbitrary size Toffoli gates, and when they use only constantly many ancill\ae. Under this constraint, this bound is close to optimal. In the case of...
Conference Paper
Full-text available
We show that if a language is recognized within certain error bounds by constant-depth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, for 0<ε≤δ≤ 1, we define BQNC \(^{0}_{\epsilon ,\delta}\) to be the class of languages recognized by constant depth, polynomial-size quantum circui...
Conference Paper
Full-text available
We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2<sup>Ω(n)</sup>. This is the first result of...
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q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes...
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An oracle is constructed relative to which quantum polynomial time () is not polynomial-time Turing reducible to . That is, there is an A such that . This generalizes and simplifies previous separations of from and , due to Berthiaume and Brassard. A key element of the proof is the use of a special property of Grover's algorithm for database search...
Article
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For any q?1, let MOD q be a quantum gate that determines if the number of 1's in the input is divisible by q.Weshow that for any q# t ? 1, MOD q is equivalent to MOD t (up to constant depth). Based on the case q = 2, Moore [8] has shown that quantum analogs of AC ,ACC[q], and ACC, denoted QAC wf , QACC[2], QACC respectively, define the same class o...
Article
It is shown that determining whether a quantum computation has a non-zero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition, or whether a given quantum bit has positive expect...
Article
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An oracle is constructed relative to which quantum polynomialtime (EQP) is not polynomial-time Turing reducible to NP. That is, there is an A such that EQP A 6 P NP A . This generalizes and simplies previous separations of EQP from NP and ZPP, due to Berthiaume and Brassard. A key element of the proof is the use of a special property of Grover's al...
Article
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We say an integer polynomial p, on Boolean inputs, weakly m-represents a Boolean function f if p is nonconstant and is zero (mod m), whenever f is zero. In this paper we prove that if a polynomial weakly m-represents the Modq function on n inputs, where q and m are relatively prime and m is otherwise arbitrary, then the degree of the polynomial is...
Article
It is shown that determining whether a quantum computation has a nonzero probability of accepting is at least as hard as the polynomialtime hierarchy. This hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition, or whether a given quantum bit has positive expectat...
Conference Paper
Full-text available
For any $q > 1$, let $\MOD_q$ be a quantum gate that determines if the number of 1's in the input is divisible by $q$. We show that for any $q,t > 1$, $\MOD_q$ is equivalent to $\MOD_t$ (up to constant depth). Based on the case $q=2$, Moore \cite{moore99} has shown that quantum analogs of AC$^{(0)}$, ACC$[q]$, and ACC, denoted QAC$^{(0)}_{wf}$, QAC...
Article
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Consider circuits consisting of a threshold gate at the top, Modm gates at the next level (for a fixed m ), and polylog fan-in AND gates at the lowest level. It is a difficult and important open problem to obtain exponential lower bounds for such circuits. Here we prove exponential lower bounds for restricted versions of this model, in which each M...
Article
Full-text available
It is shown that determining whether a quantum computation has a non-zero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness result also applies to determining in general whether a given quantum basis state appears with nonzero amplitude in a superposition, or whether a given quantum bit has positive expect...
Article
Full-text available
We study the class coNPMV of complements of NPMV functions. Though defined symmetrically to NPMV this class exhibits very different properties. We clarify the complexity of coNPMV by showing that it is essentially the same as that of NPMV NP . Complete functions for coNPMV are exhibited and central complexity-theoretic properties of this class are...
Article
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this paper we take a further step in this direction by proving that testing for non-zero acceptance probability of a quantum machine is classically an extremely hard problem. In fact, we prove that this problem -- which we call QAP ("quantum acceptance possibility") and which is complete for NQP (a quantum analog of NP) -- is hard for the polynomia...
Article
We say an integer polynomial p, on Boolean inputs, weakly m-represents a Boolean function f if p is non-constant and is zero (mod m), whenever f is zero. In this paper we prove that if a polynomial weakly m-represents the Mod q function on n inputs, where q and m are relatively prime and m is otherwise arbitrary, then the degree of the polynomial i...
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The counting class C= P, which captures the notion of "exact counting", while extremely powerful under various nondeterministic reductions, is quite weak under polynomial-time deterministic reductions. We discuss the analogies between NP and co-C = P, which allow us to derive many interesting results for such deterministic reductions to co-C = P. W...
Conference Paper
Full-text available
We study the class coNPMV of complements of NPMV functions. Though defined symmetrically to NPMV this class exhibits very different properties. We clarify the complexity of coNPMV by showing that it is essentially the same as that of NPMV<sup>NP</sup> complete functions for coNPMV are exhibited and central complexity-theoretic properties of this cl...
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Thecorrelation between two Boolean functions ofn inputs is defined as the number of times the functions agree minus the number of times they disagree, all divided by 2 n . In this paper we compute, in closed form, the correlation between any twosymmetric Boolean functions. As a consequence of our main result, we get that every symmetric Boolean fun...
Conference Paper
Full-text available
We say an integer polynomial p, on Boolean inputs, weakly m-represents a Boolean function f if p is non-constant and is zero (mod m) whenever f is zero. In this paper we prove that if a polynomial weakly m-represents the Modq function on n inputs, where q and m are relatively prime and m is otherwise arbitrary, then the degree of the polynomial is...
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It is proved that there is a monotone function in AC 4 0 which requires exponential-size monotone perceptrons of depth 3. This solves the monotone version of a problem which, in the general case, would imply an oracle separation of PPPH.
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This paper studies the class MP of languages which can be solved in polynomial time with the additional information of one bit from a #P function ƒ. The middle bit of ƒ(x) is shown to be as powerful as any other bit, whereas the O(log n) bits at either end are apparently weaker. The polynomial hierarchy and the classes Modk P, k ≥ 2, are shown to b...
Conference Paper
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The class of languages that can be recognized in polynomial time with the additional information of one bit from a P function is studied. In particular, it is shown that every Mod<sub>k</sub>P class and every class contained in PH are low for this class. These results are translated to the area of circuit complexity using MidBit (middle bit) gates....
Article
We prove the existence of an oracle A such that ⊕P A is not contained in PP PH A . This separation follows in a straightforward manner from a circuit complexity result, which is also proved here: To compute the parity of n inputs, any constant depth circuit consisting of a single threshold gate on top of and’s and or’s requires exponential size in...
Conference Paper
The existence of an oracle A such that ⊕P<sup>A </sup> is not contained in PP<sup>PHA</sup> is proved. This separation follows in a straightforward manner from a circuit complexity result, which is also proved: to compute the parity of n inputs, any constant depth circuit consisting of a single threshold gate on top of ANDs and ORs requires exponen...
Article
Summary Is is pointed out that a recent proof of the NP-completeness of determining the ground state of spin-glasses is missing an important step. We fill in the step omitted by the author and in the process find that a stronger theorem can be proved.
Article
We present a Monte Carlo analysis of the SU(2) finite-temperature deconfinement transition on a body-centered hypercubic lattice. The behavior of the transition temperature T/sub c/ as a function of the lattice coupling ..beta.. appears to deviate from the perturbative prediction of asymptotic scaling.
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Full-text available
The deconfining transitions of SU(N) lattice gauge theories, both with and without quarks, are studied using strong coupling techniques combined with a mean field analysis. In the pure gauge sector our analysis suggests first-order transitions for N ⩾ 3 and a second-order transition only for N = 2. Quarks are incorporated via an effective external...
Article
The transition to a chirally symmetric phase in lattice QCD is investigatedwith Susskind fermions. In the limit of large coupling and small temperature, the partition function is found to be that of an antiferromagnetic Ising model. The phase transition of the Ising model corresponds to chiral symmetry restoration. It occurs near zero temperature,...
Article
It is shown that Schwinger-Dyson equations for the infinite volume U(N) chiral model and the reduced chiral model, quenched according ot Parisi's prescription, agree to leading order in N for all values of the coupling and in any number of dimensions d. The conditions under which these equations can determined a theory uniquely are discussed. It is...
Article
It is shown how, in the large-N limit of SU(N) gauge theory, a local symmetry can be truly broken. This is illustrated by use of the exact solution of the two-dimensional theory in the presence of a vanishingly small symmetry-breaking source term. The model is also solved at finite N in order to study the approach to N=∞. It is found that any finit...
Article
It is shown, for any non-zero temperature and finite coupling, that there are large transverse fluctuations in the electric flux tubes connecting quark-antiquark pairs. For large separation L, the average distance of the flux tube from its equilibrium position diverges like . Hence the flux tube is rough. The consequent absence of a roughening sing...
Article
The SU(2) euclidean lattice gauge theory is formulated at finite temperature. Previous hamiltonian derivations of the deconfining phase transition are reproduced in the euclidean formalism. Corrections to infinite coupling are calculable, and expansions for the string tension are obtained. It is shown that the string tension is always in a rough ph...
Article
We analyze the large-N phase transition phenomenon in lattice gauge and chiral models. In particular, we prove the existence of a large-N phase transition in 4d lattice QCD without fermions and present a physical understanding of why it takes place. The phase transition occurs at g2N = 2.53.
Article
Two-dimensional chiral models are investigated from a strong coupling viewpoint, with the ultimate objective of understanding the phase structure of gauge theories. Predictions for the average link and beta function are obtained. The abelian case agrees well with previous work. Strong evidence for a phase transition in the U(N) cases is found, in c...
Article
We present an outline for a proof (the precise details of which will be presented in a follow-up paper) of a large-N phase transition in dimensions greater than two. The critical couplings are calculated in d=3 and d=4 and are found to be beta=0.44 and beta=0.40, respectively.
Article
We prove the existence of a large-N phase transition in the d = 2 chiral model and calculate via strong-coupling methods the phase-transition point. The critical coupling constant is 0.324. We also treat the chiral model chains (equivalent to the gauge model on polyhedrons) and our approximate calculations come very close to the exact results for t...
Article
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We present an approximation for the integration over a link in the Feynman path integral for U(N) lattice quantum chromodynamics (QCD). The approximation is valid when N→∞ and g2N is fixed to a large value. The result is such that subsequent link integrations have the same form, allowing a complete evaluation of the path integral in any dimension o...
Article
We establish the gauge invariance property of recently proposed variables that lead to flux tubes in the strong coupling limit of quantum chromodynamics. We present a method for deriving the SU(N)1 ⊗ SU(N)2 ⊗ SU(N)3 algebra satisfied by these variables, which was previously guessed in an axial gauge formalism. We make use of Dirac brackets and appl...
Article
Quantum chromodynamics is investigated in the axial gauge with particular attention to boundary terms. The Poincaré algebra has anomalies and closes only in the gauge-invariant physical sector. A simple method is proposed for dealing with the boundary condition E3a --> 0 as x3 --> ∞ in a Hamiltonian formalism. It is found that the gauge potentials...
Article
Total neutron cross sections on deuterium in the energy range 1 to 1000 keV have been measured, and basically agree with our previously reported results. [NUCLEAR REACTIONS 2H(n, n), E=1.0-1000 keV; measured sigmaT. Enriched gaseous target. Iron-filtered neutron beam.]
Article
Total neutron cross sections of deuterium have been measured from less than 1 keV to 1000 keV. We also performed a three-body calculation of $n$-${}\mathrm{D}$ total scattering cross sections, using separable potentials. The theoretical and experimental results show good agreement. A marked increase in cross section is observed for decreasing energ...
Article
The effective theory describing SU(4) lattice gauge theory at finite temperature and strong coupling is investigated via Monte Carlo methods. A first-order deconfining transition is found. Introducing dynamical quarks leads to the disappearance of the transition at a large value of the quark mass, in agreement with recent mean-field calculations.

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