A Novel Algebraic Geometry Compiling Framework for Adiabatic Quantum - - PDF document

Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications

A Novel Algebraic Geometry Compiling Framework for Adiabatic Quantum Computation

Raouf Dridi1 Hedayat Alghassi1 Sridhar Tayur1

1Quantum Computing Group

Tepper School of Business Carnegie Mellon University

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications

Plan

1

Algebraic geometry in optimization What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

2

Algebraic geometry for Graph Minor Theory Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

3

Applications Translator API - Demo Analytical dependance of the spectral gap on the points

• f V(B)

Ising architecture design

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization 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 R2 : x2 + y2 1 = 0}. The ring : Q[x, y]/ < x2 + y2 1 > = polynomials mod x2 + y2 1

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

What is algebraic geometry?

Introducing some terminology : Let S be a set of polynomials f 2 Q[x0, . . . , xn1]. 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[x0, . . . , xn1] 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).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

Strictly speaking, the set of all hidden polynomials is given by the so-called radical ideal p I, which is defined by p I = {g 2 Q[x0, . . . , xn1]| 9r 2 N : gr 2 I}. In practice, the ideal p I is infinite, so we represent such an ideal using a Groebner basis B, which one might take to be a triangularization of the ideal p I. In fact, the computation of Groebner bases generalizes Gaussian elimination in linear systems. We also have V(S) = V(I) = V( p I) = V(B).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

What is algebraic geometry? Solving system of polynomial equations

Example Consider the system S = {x2 + y2 + z2 4, x2 + 2y2 5, xz 1}. We want to solve S. We need to compute a Groebner basis for S ! Notebook 1.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

Algebraic geometry in optimization

Given a binary optimization problem (P) : argmin(y0,··· ,ym−1)2Bm f(y0, · · · , ym1), (2) where B = {0, 1} and f 2 Q[y0, · · · , ym1]. Algebraic geometry appears naturally! First appearance : The objective function f defines an ideal I = {z f(y0, · · · , ym1), y2

i yi},

subset of the larger ring Q[z, y0, · · · , ym1]. The variety V(I) is the graph of the objective function f (we will solve (P) later with Groebner bases).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

Algebraic geometry in optimization

Given the binary optimization problem (P) : argmin(y0,··· ,ym−1)2Bm f(y0, · · · , ym1), (3) where B = {0, 1}. Second appearance : The variety of local minima Define ˜ f := f +

n

X

i=1

↵2

i yi(yi 1).

The gradient ideal of (P) is ˜ I :=< @yi˜ f, · · · , @↵2

i

˜ f > . Its variety is the set of local minima of (P).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

Algebraic geometry in optimization

Given a binary optimization problem (P) : argmin(y0,··· ,ym−1)2Bm f(y0, · · · , ym1), (4) where B = {0, 1}. Third connection : Solving (P) as an eigenvalue problem! Consider again the gradient ideal (˜ f := f + Pn

i=1 ↵2 i yi(yi 1))

˜ I :=< @yi˜ f, · · · , @↵2

i f > .

Its coordinate ring is the residue algebra A := Q[y0, . . . , ym1, ↵1, . . . , ↵n]/˜ ˜

• I. Define the linear map

m˜

f :

A ! A (5) g 7! ˜ fg

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization 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 myi 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 myi 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.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

Solving optimization pbs with Groebner bases : S. Tayur’s method

Consider the ideal I = {z f(y0, · · · , ym1), y2

i yi} ⇢ Q[z, y0, · · · , ym1].

associated to the binary optimization : (P) : argmin(y0,··· ,ym−1)2Bm f(y0, · · · , ym1), (6) We would like to solve (P) using the ideal I. Example Solve the IP ⇢ argminyi2{0,1} y1 + 2y2 + 3y3 + 3y4, y1 + y2 + 2y3 + y4 = 3 (7) Notebook 2.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

Reduction to QUBOs without slack variables Consider the quadratic polynomial Hij := QiPj + Si,j + Zi,j Si+1,j1 2 Zi,j+1, with the binary variables Pj, Qi, Si,j, Si+1,j1, Zi,j, Zi,j+1. The goal is solve Hij (obtain its zeros) as a QUBO (eg., using DWave) We can square Hij and reduce using slack variables! Or, instead, we compute a Groebner basis B of the system S = {Hij} [ {x2 x, x 2 {Pj, Qi, Si,j, Si+1,j1, Zi,j, Zi,j+1}}, and look for a positive quadratic polynomial Hij

+ = P t2B| deg(t)2 att. Note that global minima of Hij +

are the zeros of Hij.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

The Groebner basis B is

t1 := Qi Pj + Si,j + Zi,j − Si+1,j−1 − 2 Zi,j+1, (8) t2 := ⇣ −Zi,j+1 + Zi,j ⌘ Si+1,j−1 + ⇣ Zi,j+1 − 1 ⌘ Zi,j , (9) t3 := ⇣ −Zi,j+1 + Zi,j ⌘ Si,j + Zi,j+1 − Zi,j+1Zi,j , (10) t4 := ⇣ Si+1,j−1 + Zi,j+1 − 1 ⌘ Si,j − Si+1,j−1Zi,j+1, (11) t5 := ⇣ −Si+1,j−1 − 2 Zi,j+1 + Zi,j + Si,j ⌘ Qi − Si,j − Zi,j + Si+1,j−1 + 2 Zi,j+1, (12) t6 := ⇣ −Si+1,j−1 − 2 Zi,j+1 + Zi,j + Si,j ⌘ Pj − Si,j − Zi,j + Si+1,j−1 + 2 Zi,j+1, (13) in addition to 3 more cubic polynomials, (14)

We take Hij

+ = P t2B| deg(t)2 att, and solve for the at. We can

require that the coefficients at are subject to the dynamic range allowed by the quantum processor (eg., the absolute values of the coefficients of H+

ij , with respect to the variables

Pj, Qi, Si,j, Si+1,j1, Zi,j, and Zi,j+1, be within [1 ✏, 1 + ✏]). Ref : RD and HA srep 2017.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

Solving IPs using Groebner bases of toric ideals

These are ideals generated by differences of monomials. Their Groebner bases enjoy a clear structure given by kernels of integer matrices. Specifically, let A = (a1, · · · , an) be any integer m ⇥ n-matrix. Each column ai = (a1i, · · · , ani)T is identified with a Laurent monomial yai = ya1i

1

· · · yani

m . The toric

ideal JA is the kernel of the algebra homomorphism Q[x] ! Q[y] (15) xi 7! yai. (16) Proposition The toric ideal JA is generated by the binomials xu+ xu−, where the vector u = u+ u 2 Z+n Z+n runs over all integer vectors in KerZA, the kernel of the matrix A. Notebook 3

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications What is algebraic geometry? Solving optimization pbs with Groebner bases : S. Tayur’s method Groebner bases : Quadratization Toric ideals : Conti and Traverso

Part 2 : Algebraic geometry for Graph Minor Theory

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Toric ideals again! Reduction to QUBO

Consider the binary optimization problem : (P) : argmin(y0,··· ,ym−1)2Bm f(y0, · · · , ym1). (17) Define the ideal KA = ⌦ x1 y1, x2 y2, x3 y3, · · · , xm ym, (18) xk yi1yi2, for each pair (yi1, yi2) contained in f ↵ , where k runs from 1 to m + n0, where n0 is the total number of such pairs (with n0 + m  n). Proposition The minimal reduction of the polynomial function f into a quadratic function is given by the toric ideal JA = KA \ Q[x].

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

From embeddings to fiber-bundles

Consider the QUBO argmin(y0,··· ,ym−1)2Bm X

(yi1,yi2)2Edges(Y)

Ji1i2yi1yi2 +

m1

X

j=0

hjyj. (19) We recall the following definition Definition (Embedding) Let X be a fixed hardware graph. A minor-embedding (embedding for short) of the graph Y is a map : Vertices(Y) ! Subtrees(X) (20) that satisfies the following condition for each : (y1, y2) 2 Edges(Y), there exists at least one edge in Edges(X) connecting the two subtrees (y1) and (y2).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

An embedding is a mapping : Logical graph ! Hardware graph. We flip the direction and define the surjection ⇡ : Hardware graph ! Logical graph such that for each logical qubit y ⇡1(y) = (y). The chain (y) is projected into the logical qubit y. The triplet : (X, Y, ⇡) is a fiber bundle.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

A direct corollary of this representation, is that the map ⇡ has the form : ⇡(xi) = X

ij

↵ijyj (21) with X

ij

↵ij = i, ↵ij1↵ij2 = 0, ↵ij(↵ij 1) = 0, where the binary number i is 1 if the physical qubits xi is used and 0 otherwise. We write domain(⇡) = Vertices(X) and support(⇡) = Vertices(X ) with X ⇢subgraph X. The fiber of the map ⇡ at yj 2 Vertices(Y) is given by ⇡1(yj) = (yj) = {xi 2 Vertices(X)| ↵ij = 1}. (22) The conditions on the parameters ↵ij guarantee that fibers don’t intersect (i.e., ⇡ is well defined map).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Example :

Let X and Y be the two graphs depicted in Figure 1. An example of the map ⇡ is defined by ⇡(x1) = ⇡(x4) = y1 and ⇡(x2) = y2 and ⇡(x3) = y3.

FIGURE – An example of a fiber bundle.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

We this new definition, we systematically answer the following questions : Existence (or non existence) of embeddings. Calculating all embeddings in a compact form given by a Groebner basis. Counting all embeddings without solving any equations. We do so for any fixed size of the chains.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Consider the surjection ⇡ : X ! Y ⇡(xi) = X

ij

↵ijyj (23) with X

ij

↵ij = i, ↵ij1↵ij2 = 0, and ↵ij(↵ij 1) = 0, j(j 1) = 0, The fiber at y is ⇡1(yj) = (yj) = {xi 2 Vertices(X)| ↵ij = 1}. (24) Task : Translate the definition of embedding into a system of algebraic constraints on the parameters ↵ij and j.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Size constraint

The number of usable physical qubits can be constrained : fix the maximum size of the fibers ⇡1(yj) to a certain size k  card(Edges(X)). This can be enforced using : 8j : X

xi2Vertices(X)

↵ij  k

• r equivalently

(25) Πk

=1

@ X

xi2Vertices(X)

↵ij  1 A = 0. (26) Additionally, we have 8j : ↵i1j↵i2j = 0, (27) for all pairs (xi1, xi2) with d(xi1, xi2) > k, where d(xi1, xi2) is the size of the shortest chain connecting xi1 and xi2.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Fiber condition

Fiber Condition Each fiber ⇡1(y) of ⇡ is a connected subtree. We need the following notations : ck(xi1, xi2) is a chain of size  k connecting xi1 and xi2. Our convention here is to define a chain as an ordered list of vertices that includes the end points xi1 and xi2, thus, card(Ck(xi1, xi2))  k + 1. Ck(xi1, xi2) is the set of all chains of size  k connecting xi1 and xi2.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Fiber condition

We impose : ↵i1j↵i2j ⇥ @ X

ck(xi1,xi2)2Ck(xi1,xi2)

Πx`2ck(xi1,xi2)\{xi1,xi2}↵`j 1 1 A = 0. (28) For each pair of vertices in ⇡1(yj), condition (28) implies the existence of a unique chain connecting the pair and that is completely contained in the fiber ⇡1(yj). Note that, the existence of chains implies that ⇡1(yj) is connected.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Fiber condition

In case we wish the fiber ⇡1(yj) to be a chain, a preferred minimal structure for the logical qubits, we constrain the degree

• f each vertex xi1 to be in {1, 2}, which translates into

1 + X

i2: (xi1,xi2)2Edges(X)

↵i1j↵i2j (29) is binary for all xi1 2 ⇡1(yj).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Pullback condition Each edge (yj1, yi2) in Y there exists at least one edge connecting the fibers ⇡1(yj1) and ⇡1(yi2). We need a few more constructions. The map ⇡ given by the equations (23) extends to a linear and multiplicative map ⇡ : Q[Vertices(X)] ! Q[Vertices(Y)] (30) by ⇡(xi1xi2) = ⇡(xi1)⇡(xi2) and ⇡(ai1xi1+ai2xi2) = ai1⇡(xi1)+ai2⇡(xi2), (31) for all ai 2 Q. Additionally, the pullback of the polynomial P(x) by ⇡ is the polynomial ⇡⇤(P)(y) = P(⇡(x)) 2 Q[Vertices(Y)]. (32)

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Pullback condition

In particular, the pullback of the quadratic form QX(x) = X

(xi1,xi2)2Edges(X)

xi1xj2 by ⇡ is the quadratic form

⇡∗(QX )(y) = X

(xi1 ,xi2 )∈Edges(X)

⇡(xi1 )⇡(xi2 ) = X

(xi1 ,xi2 )∈Edges(X)

B @ X

0≤j1<j2≤m−1

⇣ ↵i1j1 ↵i2j2 + ↵i1j2 ↵i2j1 ⌘ yj1 yj2 +

m−1

X

j=0

↵i1,j ↵i2,j yj

2

1 C A = X

0≤j1<j2≤m−1

B B @ X

(xi1 ,xi2 )∈Edges(X)

⇣ ↵i1j1 ↵i2j2 + ↵i1j2 ↵i2j1 ⌘ 1 C C A yj1 yj2 +

m−1

X

j=0

B B @ X

(xi1 ,xi2 )∈Edges(X)

↵i1j ↵i2j 1 C C A yj

2.

(33) 28 / 51

Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Fiber condition

The sum X

(xi1,xi2)2Edges(X)

• ↵i1j1↵i2j2 + ↵i1j2↵i2j1
• gives the number of edges in Edges(X) that connect ⇡1(yj1)

and ⇡1(yj2). The Pullback Condition is equivalent to the fact that this number is strictly non zero if the pair {yj1, yj2} is an edges of Y. The Pullback Condition is equivalent to the following statement : for each {yj1, yj2} in Edges(Y) we have X

(xi1,xi2)2Edges(X)

• ↵i1j1↵i2j2 + ↵i1j2↵i2j1
• = 1 + 2

j1j2,

(34) for some integer j1j2 2 Z.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Equations (23), in addition to the conditions in the previous Fiber and Pullback conditions define an algebraic ideal I ⇢ Q[↵, , ]. The variety V(I) gives all embeddings of Y (of size  k) inside the hardware graph X. In fact, one has : Proposition Let B be a reduced Groebner basis for the ideal I. The following statements are true : A Y minor exists if and only if 1 / 2 B. If B is computed using the elimination order ↵ and 1 / 2 B, then the intersection B \ Q[, ] gives all subgraphs X of X that are minors for Y. The remainder of the reduced Groebner basis gives the corresponding embedding ⇡ : X ! Y.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Example

Consider the two graphs in Figure 2.

FIGURE – The set of all fiber bundles ⇡ : X ! Y defines an algebraic

• variety. This variety is given by the Groebner basis (40).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

In this case, equations (23) are given by ↵1,1↵1,2, ↵1,1↵1,3, ↵1,2↵1,3, (35) ↵2,1↵2,2, ↵2,1↵2,3, ↵2,2↵2,3, (36) ↵3,1↵3,2, ↵3,1↵3,3, ↵3,2↵3,3, (37) ↵4,1↵4,2, ↵4,1↵4,3, ↵4,2↵4,3, (38) ↵5,1↵5,2, ↵5,1↵5,3, ↵5,2↵5,3, (39) and

↵1,1 + ↵1,2 + ↵1,3 − 1, ↵2,1 + ↵2,2 + ↵2,3 − 2, ↵3,1 + ↵3,2 + ↵3,3 − 3, ↵4,1 + ↵4,2 + ↵4,3 − 4, ↵5,1 + ↵5,2 + ↵5,3 − 5. 32 / 51

Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

The Pullback Condition reads

−1 + ↵4,1↵5,2 + ↵3,1↵4,2 + ↵1,1↵2,2 + ↵3,2↵4,1 + ↵1,2↵2,1 + ↵1,2↵4,1 + ↵2,2↵3,1 +↵1,1↵4,2 + ↵2,1↵3,2 + ↵4,2↵5,1, −1 + ↵3,3↵4,1 + ↵1,3↵2,1 + ↵2,3↵3,1 + ↵4,1↵5,3 + ↵1,3↵4,1 + ↵1,1↵2,3 + ↵4,3↵5,1 +↵2,1↵3,3 + ↵3,1↵4,3 + ↵1,1↵4,3, −1 + ↵3,3↵4,2 + ↵1,2↵2,3 + ↵1,2↵4,3 + ↵1,3↵2,2 + ↵1,3↵4,2 + ↵2,3↵3,2 + ↵2,2↵3,3 +↵4,2↵5,3 + ↵3,2↵4,3 + ↵4,3↵5,2.

The Fiber Condition is given by

−↵1,1↵2,1↵5,1, −↵1,1↵3,1↵5,1, −↵1,2↵2,2↵5,2, −↵1,2↵3,2↵5,2, −↵1,3↵2,3↵5,3, −↵1,3↵3,3↵5,3 −↵2,1↵3,1↵5,1, −↵2,1↵4,1↵5,1, −↵2,2↵3,2↵5,2, −↵2,2↵4,2↵5,2, −↵2,3↵3,3↵5,3, −↵2,3↵4,3↵5,3, ↵2,1↵5,1, ↵2,2↵5,2, ↵2,3↵5,3. 33 / 51

Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

A part of the reduced Groebner basis of the resulted system is given by B = n 1 1, 2 1, 3 1, 4 1, 2

i i, ↵2 ij ↵ij,

↵1,2↵1,3, ↵1,2↵3,2, ↵1,3↵3,3, ↵2,2↵2,3, ↵2,2↵4,2, ↵2,2↵5,2, ↵4,2↵5,3, ↵4,3↵5,2, ↵5,2↵5,3, ↵4,2↵5,2 ↵5,2, ↵4,25 ↵5,2, . . . ↵2,2↵5,3 ↵3,2↵5,3 + ↵1,25 + ↵2,25 + ↵3,25 + ↵3,35 + ↵5,2 In particular, the intersection B \ Q[] = (1 1, 2 1, 3 1, 4 1, 52 5) gives the two Y minors (i.e., subgraphs X ) inside X. The remainder of B gives the explicit expressions of the corresponding mappings.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Counting embeddings without solving equations

The number of zeros of an ideal I ⇢ Q[x0, · · · , xn1] can be determined without solving any equation in I. This is done using staircase diagrams, as follows. To each polynomial in I we assign a point in the Euclidean space En given by the exponents of its leading term (with respect to the given monomial order). Figure 3 depicts three staircase diagrams.

FIGURE – Staircase diagrams of three ideals in Q[x, y]. The number

• f zeros of the three ideals (left to right) are 8, 1 and 4 respectively.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

The application of this construction to the problem of counting all embeddings ⇡ : X ! Y is obvious. The ideal I is given by the different requirements on the coefficients ↵ij of the map ⇡ as discussed previousely. Note that the dimension of Q[↵, , ]/I cannot be infinite because there is (if any) only finite number of possible embeddings. An example is depicted in Figure 4.

FIGURE – There are 360 embeddings, with chains of size at most 2, for the bottom graph into the upper graph.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Getting rid of redundancies

• 1. When determining the surjections ⇡ (or equivalently, the

embeddings ), many of the solutions are redundant : they are

• f the form ⇡ with 2 Aut(X). This is not desirable because

it affects the efficiency of the computations.

• 2. Instead of applying our method directly, we fold the hardware

graph along it symmetries and proceed as before.

• 3. This amounts to re-expressing the quadratic form of the

hardware graph in terms of the invariants!

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Example Consider the two graphs X and Y of the figure below. The quadratic form of X is : QX(x) = x1x2 + x2x3 + x3x4 + x1x4 + x4x5. (40) Exchanging the two nodes x1 and x3 is a symmetry for X, and the quantities K = x1 + x3, x2, x4, and x5 are invariants of this symmetry.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Example-Continued In terms of these invariants, the quadratic function QX(x), takes the simplified form : QX(x, K) = Kx2 + Kx4 + x4x5, (41) which shows (as expected) that graph X can be folded into a chain (given by [x2, K, x4, x5]). The surjective homomorphism ⇡ : X ! Y now takes the form K = ↵01y1 + ↵02y2 + ↵03y3. (42) xi = ↵i1y1 + ↵i2y2 + ↵i3y3 for i = 2, 4, 5. (43) The coefficients are constrained as usual.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Example-Continued The table below compares the computations of the surjections ⇡ with and without the use of invariants :

• riginal coords

invt coords Time for computing a GB (in secs) 0.122 0.039 Number of defining equations 58 30 Maximum degree in the defining eqns 3 2 Number of variables in the defining eqns 20 12 Number of solutions 48 24

In particular, the number of solutions is down to 24, that is, four (non symmetric) minors times the six symmetries of the logical graph Y.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Toric ideals again! Reduction to QUBO From embeddings to fiber-bundles Counting embeddings without solving equations Invariant coordinates : A first step towards scaling

Part 3 : Applications

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

Flowchart of the Translator : ! The user inputs the optimization problem (P). A Reduction to a quadratic form :

1 Generation of the toric ideal JA from the monomials of the

• bjective function of (P).

2 Computation of a reduced Groebner basis for JA ; return the quadratic function.

B Embedding inside the AQC processor graph :

3 Generation of the ideal I that gives the embeddings ⇡. 4 Computation of a reduced Groebner basis B of the ideal I.

C Solution using a selected embedding on the AQC processor. User gets the answer. Notebook 4

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

Analytical dependance of the spectral gap on the points of the variety V(B)

Consider a hardware graph X and a problem graph Y. Let B denote the reduced Groebner basis that gives the set of embeddings ⇡ : X ! Y. An important problem is to understand the dependence of the computational complexity of AQC on the points of the variety V(B). That is, the dependence of the spectrum of the adiabatic Hamiltonian H(t) = ↵(t)Hinitial + (t)H(P) (44)

• n the different choices of embedding given by B. One way to

proceed is to obtain the most general expression of the (quadratic form of the) minor ˜ (QY)(x) in terms of the parameters ↵ij, and i determined by B. Ref : V. Choi, Minor-embedding in adiabatic quantum computation II, 2011.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

Proposition Given a hardware graph X and a problem graph Y. Let B denote the reduced Groebner basis that gives the set of embeddings ⇡ : X ! Y. The general form of the quadratic form

• f the Y minor is given by

˜ (QY)(x) = X

xi1xi22Edges(X)

NFB 8 < : @X

j

↵i1j 1 A @X

j

↵i2j 1 A 9 = ; xi1xi2 + M ⇥ NFB 8 < : X

j

↵i1j↵i2j 9 = ; (2xi1 + 1)(2xi2 + 1), with M being one (or more) strong ferromagnetic coupling that maintains the chain.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

Example Consider the two graphs given by the quadratic functions QX(x) = x1x2 + x2x3 and QY(y) = y1y2. In this case, the reduced Groebner basis is given by

2 − 1, (1 − 1)(3 − 1), 3

2 − 3,

1

2 − 1,

↵1,2↵3,2, ↵2,1 + ↵2,2 − 1, ↵3,1 + ↵3,2 − 3, ↵1,1 + ↵1,2 − 1, ↵1,21 − ↵1,2, ↵3,23 − ↵3,2, ↵1,23 + 1 + ↵2,23 − ↵1,2 − ↵2,2 − 3, ↵3,21 − ↵2,23 + ↵1,2 + ↵2,2 − 1, ↵2,21 + 1 + ↵2,23 − ↵1,2 − 2 ↵2,2 − ↵3,2, ↵1,2↵2,2 + 1 + ↵2,2↵3,2 − ↵1,2 − ↵2,2 − ↵3,2, ↵1,2

2 − ↵1,2,

↵3,2

2 − ↵3,2,

↵2,2

2 − ↵2,2.

(45)

The first four polynomials give the reduced Groebner basis B \ Q[1, 2, 3], which gives the different domains for the projection ⇡.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

Example - Continued The general form of Y minor is given by ˜ (QY)(x) = 1x1x2 + 3x2x3 M (1 1 ) (2x1 1)(2x2 1) + M (3 ) (2x2 1)(2x3 1), with = ↵3,2 + ↵2,23 2↵2,2↵3,2.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

Ising architecture design

An important milestone in the development of AQC is the design of Ising architectures that satisfy the following : The degree of X cannot exceed a limited degree d (imposed by current manufacturing limitations). X contains a minor for each graph Y 2 Y, where Y represents a class of problems of interest. Each Y minor is explicitly computable. This problem as described was posed in [V. Choi 2011], where the following nomenclature was introduced : Definition (V. Choi 2011) Let Y be a family of graphs. A graph X is called Yminor universal if for any graph Y 2 Y, there exists a minor embedding of Y in X. Ref : V. Choi, Minor-embedding in AQC II, 2011.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

The first requirement translates into the condition P

j qij  d,

where (qij)1i,jn is the unknown adjacency matrix of X. Additionally, if the family Y is given by a finite number of graphs Yµ (where µ belongs to a finite range), then for each graph Yµ, we define the transformation ⇡µ(xi) = X

yi2V(Yi)

↵µ

ij yj,

(46) where the binary coefficients are subject to the conditions (23) for each index µ. These conditions, in addition to the pullback and chain conditions for all µ as well as the degree condition above, form a system of polynomials L ⇢ Q[↵µ, q] that has all information needed to determine the coefficients qij.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

More precisely, we have Proposition Let B be a reduced Groebner basis for the system L with respect to the elimination order {↵µ

ij } {qij}. The following

statements are true : the family of graphs Y = {Yµ} admits a Yminor universal graph of size n if and only if 1 / 2 B (the choice of the

• rdering used is not relevant for this statement).

if 1 / 2 B, the set of all Yminor universal graphs of size n is given by the intersection B \ Q[q]. if 1 / 2 B, the embeddings ⇡µ (i.e., the coefficients ↵µ

ij ) are

also given by B (as functions of the qij).

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

This approach can be applied to forbidden minor

• characterizations. Consider for instance the following

statement : A graph X is a forest if and only if it does not contain the triangle K3 as a minor. This yields the following procedure : (i) Generate the system of equations that gives all embeddings of K3 inside X. (ii) Compute a reduced Groebner basis B : 1 2 B if and only if X is a forest. More generally, Robertson - Seymour theorem states that every family of graphs that is closed under minors can be defined by a finite set of forbidden minors. The membership to this class can be expressed as a Groebner basis computation using this finite set of forbidden minors.

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Algebraic geometry in optimization Algebraic geometry for Graph Minor Theory Applications Translator API - Demo Analytical dependance of the spectral gap on the points of V(B) Ising architecture design Forbidden minor characterizations

Thank you!

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