Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications A Novel Algebraic Geometry Compiling Framework for Adiabatic Quantum Computation Raouf Dridi 1 Hedayat Alghassi 1 Sridhar Tayur 1 1 Quantum Computing Group Tepper School of Business Carnegie Mellon University 1 / 51
Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Plan Algebraic geometry in optimization 1 What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso Algebraic geometry for Graph Minor Theory 2 Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling Applications 3 Translator API - Demo Analytical dependance of the spectral gap on the points of V ( B ) Ising architecture design 2 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso What is algebraic geometry? Algebraic geometry is the study of geometric objects defined by polynomial equations, using algebraic means. Its roots go back to Descartes’ introduction of coordinates to describe points in Euclidean space and his idea of describing curves and surfaces by algebraic equations. The basic correspondence in algebraic geometry Algebraic varieties ' Polynomial rings (1) (Equivalence of categories!) Example : Circle The variety : V := { ( x , y ) 2 R 2 : x 2 + y 2 � 1 = 0 } . The ring : Q [ x , y ] / < x 2 + y 2 � 1 > = polynomials mod x 2 + y 2 � 1 3 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso What is algebraic geometry? Introducing some terminology : Let S be a set of polynomials f 2 Q [ x 0 , . . . , x n � 1 ] . V ( S ) is the affine variety defined by the polynomials f 2 S , that is, the set of common zeros of the equations f = 0 , f 2 S . The system S generates an ideal I by taking all linear combinations over Q [ x 0 , . . . , x n � 1 ] of all polynomials in S ; we have V ( S ) = V ( I ) . The ideal I reveals the hidden polynomials that are the consequence of the generating polynomials in S . For instance, if one of the hidden polynomials is the constant polynomial 1 (i.e., 1 2 I ), then the system S is inconsistent (because 1 6 = 0). 4 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso Strictly speaking, the set of all hidden polynomials is given p by the so-called radical ideal I , which is defined by p I = { g 2 Q [ x 0 , . . . , x n � 1 ] | 9 r 2 N : g r 2 I} . p In practice, the ideal I is infinite, so we represent such an ideal using a Groebner basis B , which one might take to p be a triangularization of the ideal I . In fact, the computation of Groebner bases generalizes Gaussian elimination in linear systems. We also have p V ( S ) = V ( I ) = V ( I ) = V ( B ) . 5 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso What is algebraic geometry? Solving system of polynomial equations Example Consider the system S = { x 2 + y 2 + z 2 � 4 , x 2 + 2 y 2 � 5 , xz � 1 } . We want to solve S . We need to compute a Groebner basis for S ! Notebook 1. 6 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso Algebraic geometry in optimization Given a binary optimization problem ( P ) : argmin ( y 0 , ··· , y m − 1 ) 2 B m f ( y 0 , · · · , y m � 1 ) , (2) where B = { 0 , 1 } and f 2 Q [ y 0 , · · · , y m � 1 ] . Algebraic geometry appears naturally! First appearance : The objective function f defines an ideal I = { z � f ( y 0 , · · · , y m � 1 ) , y 2 i � y i } , subset of the larger ring Q [ z , y 0 , · · · , y m � 1 ] . The variety V ( I ) is the graph of the objective function f (we will solve ( P ) later with Groebner bases). 7 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso Algebraic geometry in optimization Given the binary optimization problem ( P ) : argmin ( y 0 , ··· , y m − 1 ) 2 B m f ( y 0 , · · · , y m � 1 ) , (3) where B = { 0 , 1 } . Second appearance : The variety of local minima Define n ˜ X ↵ 2 f := f + i y i ( y i � 1 ) . i = 1 The gradient ideal of ( P ) is ˜ I := < @ y i ˜ ˜ f , · · · , @ ↵ 2 f > . i Its variety is the set of local minima of ( P ) . 8 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso Algebraic geometry in optimization Given a binary optimization problem ( P ) : argmin ( y 0 , ··· , y m − 1 ) 2 B m f ( y 0 , · · · , y m � 1 ) , (4) where B = { 0 , 1 } . Third connection : Solving ( P ) as an eigenvalue problem! Consider again the gradient ideal ( ˜ f := f + P n i = 1 ↵ 2 i y i ( y i � 1 ) ) I := < @ y i ˜ ˜ f , · · · , @ ↵ 2 i f > . Its coordinate ring is the residue algebra A := Q [ y 0 , . . . , y m � 1 , ↵ 1 , . . . , ↵ n ] / ˜ ˜ I . Define the linear map m ˜ f : A ! A (5) 7! ˜ g fg 9 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso Solving ( P ) as ev problem : Continued Since the number of local minima is finite, the algebra A is always finite-dimensional. Additionally, we have : The values of ˜ f , on the set of critical points V (˜ I ) , are given by the eigenvalues of the matrix m ˜ f . Eigenvalues of m y i and m ↵ i give the coordinates of the points of V (˜ I ) . If v is an eigenvector for m ˜ f , then it is also an eigenvector for m y i and m ↵ i for 1 i m . Refs : - D. Cox’s Using algebraic geometry. - RD and H. Alghassi, Prime factorization using QA and algebraic geometry, nature srep 2017. 10 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso Solving optimization pbs with Groebner bases : S. Tayur’s method Consider the ideal I = { z � f ( y 0 , · · · , y m � 1 ) , y 2 i � y i } ⇢ Q [ z , y 0 , · · · , y m � 1 ] . associated to the binary optimization : ( P ) : argmin ( y 0 , ··· , y m − 1 ) 2 B m f ( y 0 , · · · , y m � 1 ) , (6) We would like to solve ( P ) using the ideal I . Example Solve the IP ⇢ argmin y i 2 { 0 , 1 } y 1 + 2 y 2 + 3 y 3 + 3 y 4 , (7) y 1 + y 2 + 2 y 3 + y 4 = 3 Notebook 2. 11 / 51
What is algebraic geometry? Algebraic geometry in optimization Solving optimization pbs with Groebner bases : S. Tayur’s method Algebraic geometry for Graph Minor Theory Groebner bases : Quadratization Applications Toric ideals : Conti and Traverso Reduction to QUBOs without slack variables Consider the quadratic polynomial H ij := Q i P j + S i , j + Z i , j � S i + 1 , j � 1 � 2 Z i , j + 1 , with the binary variables P j , Q i , S i , j , S i + 1 , j � 1 , Z i , j , Z i , j + 1 . The goal is solve H ij (obtain its zeros) as a QUBO (eg., using DWave) We can square H ij and reduce using slack variables! Or, instead, we compute a Groebner basis B of the system S = { H ij } [ { x 2 � x , x 2 { P j , Q i , S i , j , S i + 1 , j � 1 , Z i , j , Z i , j + 1 }} , and look for a positive quadratic polynomial + = P + H ij t 2 B| deg ( t ) 2 a t t . Note that global minima of H ij are the zeros of H ij . 12 / 51
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