A Novel Algebraic Geometry Compiling Framework for

Algebraic geometry in optimizationAlgebraic geometry for Graph Minor Theory

Applications

A Novel Algebraic Geometry CompilingFramework for Adiabatic Quantum

Computation

Raouf Dridi1 Hedayat Alghassi1 Sridhar Tayur1

1Quantum Computing GroupTepper School of BusinessCarnegie Mellon University

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Applications

Plan1 Algebraic geometry in optimization

What is algebraic geometry?Solving optimization pbs with Groebner bases : S. Tayur’smethodGroebner bases : QuadratizationToric ideals : Conti and Traverso

2 Algebraic geometry for Graph Minor TheoryToric ideals again ! Reduction to QUBOFrom embeddings to fiber-bundlesCounting embeddings without solving equationsInvariant coordinates : A first step towards scaling

3 ApplicationsTranslator API - DemoAnalytical dependance of the spectral gap on the pointsof V(B)Ising architecture designForbidden minor characterizations

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Applications

What is algebraic geometry?Solving optimization pbs with Groebner bases : S. Tayur’s methodGroebner bases : QuadratizationToric ideals : Conti and Traverso

What is algebraic geometry?

Algebraic geometry is the study of geometric objects defined bypolynomial equations, using algebraic means. Its roots go backto Descartes’ introduction of coordinates to describe points inEuclidean space and his idea of describing curves andsurfaces 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 modx2 + y2 � 1

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Applications


What is algebraic geometry?

Introducing some terminology :Let S be a set of polynomials f 2 Q[x0, . . . , xn�1].V(S) is the affine variety defined by the polynomials f 2 S,that is, the set of common zeros of the equationsf = 0, f 2 S.The system S generates an ideal I by taking all linearcombinations over Q[x0, . . . , xn�1] of all polynomials in S ;we have V(S) = V(I). The ideal I reveals the hiddenpolynomials that are the consequence of the generatingpolynomials in S. For instance, if one of the hiddenpolynomials is the constant polynomial 1 (i.e., 1 2 I), thenthe system S is inconsistent (because 1 6= 0).

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Applications


Strictly speaking, the set of all hidden polynomials is givenby the so-called radical ideal

pI, which is defined byp

I = {g 2 Q[x0, . . . , xn�1]| 9r 2 N : gr 2 I}.In practice, the ideal

pI is infinite, so we represent such

an ideal using a Groebner basis B, which one might take tobe a triangularization of the ideal

pI.

In fact, the computation of Groebner bases generalizesGaussian elimination in linear systems.We also have

V(S) = V(I) = V(pI) = V(B).

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Applications


What is algebraic geometry? Solving system ofpolynomial equations

ExampleConsider the system

S = {x2 + y2 + z2 � 4, x2 + 2y2 � 5, xz � 1}.

We want to solve S. We need to compute a Groebner basisfor S !Notebook 1.

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Applications


Algebraic geometry in optimization

Given a binary optimization problem

(P) : argmin(y0,··· ,ym�1)2Bm f (y0, · · · , ym�1), (2)

where B = {0, 1} and f 2 Q[y0, · · · , ym�1].Algebraic geometry appears naturally !

First appearance :The objective function f defines an ideal

I = {z � f (y0, · · · , ym�1), y2i � yi},

subset of the larger ring

Q[z, y0, · · · , ym�1].

The variety V(I) is the graph of the objective function f (we willsolve (P) later with Groebner bases). 7 / 51


Applications



Given the binary optimization problem


where B = {0, 1}.

Second appearance : The variety of local minima

Define

f := f +nX

i=1

↵2i yi(yi � 1).

The gradient ideal of (P) is

I :=< @yi f , · · · , @↵2if > .

Its variety is the set of local minima of (P).8 / 51


Applications



Given a binary optimization problem


where B = {0, 1}.

Third connection : Solving (P) as an eigenvalue problem!

Consider again the gradient ideal (f := f +Pn

i=1 ↵2i yi(yi � 1))

I :=< @yi f , · · · , @↵2if > .

Its coordinate ring is the residue algebraA := Q[y0, . . . , ym�1,↵1, . . . ,↵n]/

˜I. Define the linear map

mf : A ! A (5)

g 7! f g9 / 51


Applications


Solving (P) as ev problem : ContinuedSince the number of local minima is finite, the algebra A isalways finite-dimensional. Additionally, we have :

The values of f , on the set of critical points V(I), aregiven by the eigenvalues of the matrix mf .Eigenvalues of myi and m↵i give the coordinates of thepoints of V(I).If v is an eigenvector for mf , then it is also an eigenvectorfor myi and m↵i for 1 i m.

Refs :- D. Cox’s Using algebraic geometry.- RD and H. Alghassi, Prime factorization using QA andalgebraic geometry, nature srep 2017.

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Applications


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

Consider the ideal

I = {z � f (y0, · · · , ym�1), y2i � yi} ⇢ Q[z, y0, · · · , ym�1].

associated to the binary optimization :


We would like to solve (P) using the ideal I.

ExampleSolve the IP

⇢argminyi2{0,1} y1 + 2y2 + 3y3 + 3y4,

y1 + y2 + 2y3 + y4 = 3 (7)

Notebook 2.11 / 51


Applications


Reduction to QUBOs without slack variablesConsider the quadratic polynomial

Hij := QiPj + Si,j + Zi,j � Si+1,j�1 � 2 Zi,j+1,

with the binary variables Pj ,Qi ,Si,j ,Si+1,j�1,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,j�1,Zi,j ,Zi,j+1}},

and look for a positive quadratic polynomialHij

+ =P

t2B| deg(t)2 at t . Note that global minima of Hij+

are the zeros of Hij .12 / 51


Applications


The Groebner basis B ist1 := 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+ =

Pt2B| deg(t)2 at t , and solve for the at . We can

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

ij , with respect to the variablesPj ,Qi ,Si,j ,Si+1,j�1,Zi,j , and Zi,j+1, be within [1� ✏, 1 + ✏]).Ref : RD and HA srep 2017.

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Applications


Solving IPs using Groebner bases of toric ideals

These are ideals generated by differences of monomials. TheirGroebner bases enjoy a clear structure given by kernels ofinteger matrices. Specifically, let A = (a1, · · · , an) be anyinteger m ⇥ n-matrix. Each column ai = (a1i , · · · , ani)

T isidentified with a Laurent monomial yai = ya1i

1 · · · yanim . The toric

ideal JA is the kernel of the algebra homomorphism

Q[x ]! Q[y ] (15)xi 7! yai . (16)

PropositionThe toric ideal JA is generated by the binomials xu+ � xu� ,where the vector u = u+ � u� 2 Z+n � Z+n runs over allinteger vectors in KerZA, the kernel of the matrix A.

Notebook 3 14 / 51


Applications


Part 2 : Algebraic geometry for Graph Minor Theory

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Applications

Toric ideals again ! Reduction to QUBOFrom embeddings to fiber-bundlesCounting embeddings without solving equationsInvariant 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, · · · , ym�1). (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 ofsuch pairs (with n0 + m n).

PropositionThe minimal reduction of the polynomial function f into aquadratic function is given by the toric ideal JA = KA \Q[x ].

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Applications


From embeddings to fiber-bundles

Consider the QUBO

argmin(y0,··· ,ym�1)2Bm

X

(yi1 ,yi2 )2Edges(Y )

Ji1i2yi1yi2 +m�1X

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 inEdges(X ) connecting the two subtrees �(y1) and �(y2). 17 / 51


Applications


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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Applications


A direct corollary of this representation, is that the map ⇡ hasthe form :

⇡(xi) =X

ij

↵ij yj (21)

withX

ij

↵ij = �i , ↵ij1↵ij2 = 0, ↵ij(↵ij � 1) = 0,

where the binary number �i is 1 if the physical qubits xi is usedand 0 otherwise. We write domain(⇡) = Vertices(X ) andsupport(⇡) = 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’tintersect (i.e., ⇡ is well defined map).

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Applications


Example :

Let X and Y be the two graphs depicted in Figure 1. Anexample 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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Applications


We this new definition, we systematically answer the followingquestions :

Existence (or non existence) of embeddings.Calculating all embeddings in a compact form given by aGroebner basis.Counting all embeddings without solving any equations.

We do so for any fixed size of the chains.

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Applications


Consider the surjection ⇡ : X ! Y

⇡(xi) =X

ij

↵ij yj (23)

withX

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 ofalgebraic constraints on the parameters ↵ij and �j .

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Applications


Size constraint

The number of usable physical qubits can be constrained : fixthe maximum size of the fibers ⇡�1(yj) to a certainsize k card(Edges(X )). This can be enforced using :

8j :X

xi2Vertices(X)

↵ij k or equivalently (25)

⇧k=1

0

@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 thesize of the shortest chain connecting xi1 and xi2 .

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Applications


Fiber condition

Fiber ConditionEach 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 . Ourconvention here is to define a chain as an ordered list ofvertices 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 xi1and xi2 .

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Applications


Fiber condition

We impose :

↵i1j↵i2j ⇥

0

@X

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

⇧x`2ck (xi1 ,xi2 )\{xi1 ,xi2}↵`j � 1

1

A = 0.

(28)For each pair of vertices in ⇡�1(yj), condition (28) implies theexistence of a unique chain connecting the pair and that iscompletely contained in the fiber ⇡�1(yj). Note that, theexistence of chains implies that ⇡�1(yj) is connected.

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Applications


Fiber condition

In case we wish the fiber ⇡�1(yj) to be a chain, a preferredminimal structure for the logical qubits, we constrain the degreeof 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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Applications


Pullback conditionEach edge (yj1 , yi2) in Y there exists at least one edgeconnecting the fibers ⇡�1(yj1) and ⇡�1(yi2).

We need a few more constructions. The map ⇡ given by theequations (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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Applications


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

)2Edges(X)

⇡(xi1)⇡(xi2

)

=X

(xi1,xi2

)2Edges(X)

0

B@X

0j1<j2m�1

⇣↵i1 j1

↵i2 j2+ ↵i1 j2

↵i2 j1

⌘yj1

yj2+

m�1X

j=0↵i1,j

↵i2,jyj

2

1

CA

=X

0j1<j2m�1

0

BB@X

(xi1,xi2

)2Edges(X)

⇣↵i1 j1

↵i2 j2+ ↵i1 j2

↵i2 j1

⌘1

CCA yj1yj2

+m�1X

j=0

0

BB@X

(xi1,xi2

)2Edges(X)

↵i1 j↵i2 j

1

CCA yj2. (33)

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Applications


Fiber condition

The sum 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 factthat this number is strictly non zero if the pair {yj1 , yj2} is anedges of Y .

The Pullback Condition is equivalent to the followingstatement : for each {yj1 , yj2} in Edges(Y ) we have

X


�↵i1j1↵i2j2 + ↵i1j2↵i2j1

�= 1 + �2

j1j2 , (34)

for some integer �j1j2 2 Z.29 / 51


Applications


Equations (23), in addition to the conditions in the previousFiber and Pullback conditions define an algebraic idealI ⇢ Q[↵, �, �]. The variety V(I) gives all embeddings of Y (ofsize k ) inside the hardware graph X . In fact, one has :

PropositionLet B be a reduced Groebner basis for the ideal I. Thefollowing statements are true :

A Y minor exists if and only if 1 /2 B.If B is computed using the elimination order ↵ � � � � and1 /2 B, then the intersection B \Q[�, �] gives all subgraphsX� of X that are minors for Y . The remainder of thereduced Groebner basis gives the correspondingembedding ⇡� : X� ! Y .

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Applications


Example

Consider the two graphs in Figure 2.

FIGURE – The set of all fiber bundles ⇡ : X ! Y defines an algebraicvariety. This variety is given by the Groebner basis (40).

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Applications


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.

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Applications


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.

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Applications


A part of the reduced Groebner basis of the resulted system isgiven by

B =n�1 � 1,�2 � 1,�3 � 1,�4 � 1,�2

i � �i , ↵2ij � ↵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,↵4,2�5 � ↵5,2,

...�↵2,2↵5,3 � ↵3,2↵5,3 + ↵1,2�5 + ↵2,2�5 + ↵3,2�5 + ↵3,3�5 + ↵5,2 + ↵5,3 � �5 } .

In particular, the intersectionB \Q[�] = (�1 � 1,�2 � 1,�3 � 1,�4 � 1,�5

2 � �5) gives thetwo Y minors (i.e., subgraphs X�) inside X . The remainder of Bgives the explicit expressions of the corresponding mappings.

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Applications


Counting embeddings without solving equations

The number of zeros of an ideal I ⇢ Q[x0, · · · , xn�1] can bedetermined without solving any equation in I. This is doneusing staircase diagrams, as follows. To each polynomial in Iwe assign a point in the Euclidean space En given by theexponents of its leading term (with respect to the givenmonomial order). Figure 3 depicts three staircase diagrams.

FIGURE – Staircase diagrams of three ideals in Q[x , y ]. The numberof zeros of the three ideals (left to right) are 8, 1 and 4 respectively.

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Applications


The application of this construction to the problem of countingall embeddings ⇡ : X ! Y is obvious. The ideal I is given bythe different requirements on the coefficients ↵ij of the map ⇡ asdiscussed previousely. Note that the dimension of Q[↵,�, �]/Icannot be infinite because there is (if any) only finite number ofpossible 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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Applications


Getting rid of redundancies

1. When determining the surjections ⇡ (or equivalently, theembeddings �), many of the solutions are redundant : they areof the form ⇡ � � with � 2 Aut(X ). This is not desirable becauseit affects the efficiency of the computations.

2. Instead of applying our method directly, we fold the hardwaregraph along it symmetries and proceed as before.

3. This amounts to re-expressing the quadratic form of thehardware graph in terms of the invariants !

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Applications


ExampleConsider the two graphs X and Y of the figure below. Thequadratic form of X is :

QX (x) = x1x2 + x2x3 + x3x4 + x1x4 + x4x5. (40)

Exchanging the two nodes x1 and x3 is a symmetry for X , andthe quantities K = x1 + x3, x2, x4, and x5 are invariants of thissymmetry.

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Applications


Example-ContinuedIn terms of these invariants, the quadratic function QX (x), takesthe simplified form :

QX (x ,K ) = Kx2 + Kx4 + x4x5, (41)

which shows (as expected) that graph X can be folded into achain (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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Applications


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

original coords invt coordsTime for computing a GB (in secs) 0.122 0.039Number of defining equations 58 30Maximum degree in the defining eqns 3 2Number of variables in the defining eqns 20 12Number 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 logicalgraph Y .

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Applications


Part 3 : Applications

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Applications

Translator API - DemoAnalytical dependance of the spectral gap on the points of V(B)Ising architecture designForbidden 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 theobjective function of (P).

2 Computation of a reduced Groebner basis for JA ; returnthe 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 AQCprocessor.

User gets the answer.

Notebook 442 / 51


Applications


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

Consider a hardware graph X and a problem graph Y . Let Bdenote the reduced Groebner basis that gives the set ofembeddings ⇡ : X ! Y . An important problem is to understandthe dependence of the computational complexity of AQC on thepoints of the variety V(B). That is, the dependence of thespectrum of the adiabatic Hamiltonian

H(t) = ↵(t)Hinitial + �(t)H(P) (44)

on the different choices of embedding given by B. One way toproceed is to obtain the most general expression of the(quadratic form of the) minor �(QY )(x) in terms of theparameters ↵ij , and �i determined by B.Ref : V. Choi, Minor-embedding in adiabatic quantumcomputation II, 2011.

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Applications


PropositionGiven a hardware graph X and a problem graph Y . Let Bdenote the reduced Groebner basis that gives the set ofembeddings ⇡ : X ! Y . The general form of the quadratic formof the Y minor is given by

�(QY )(x) =X

xi1 xi22Edges(X)

NFB

8<

:

0

@X

j

↵i1j

1

A

0

@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 thatmaintains the chain.

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Applications


ExampleConsider the two graphs given by the quadratic functionsQX (x) = x1x2 + x2x3 and QY (y) = y1y2. In this case, thereduced Groebner basis is given by

�2 � 1, (�1 � 1)(�3 � 1), �32 � �3, �1

2 � �1,

↵1,2↵3,2, ↵2,1 + ↵2,2 � 1, ↵3,1 + ↵3,2 � �3, ↵1,1 + ↵1,2 � �1, ↵1,2�1 � ↵1,2, ↵3,2�3 � ↵3,2,

↵1,2�3 + 1 + ↵2,2�3 � ↵1,2 � ↵2,2 � �3, ↵3,2�1 � ↵2,2�3 + ↵1,2 + ↵2,2 � �1,

↵2,2�1 + 1 + ↵2,2�3 � ↵1,2 � 2 ↵2,2 � ↵3,2, ↵1,2↵2,2 + 1 + ↵2,2↵3,2 � ↵1,2 � ↵2,2 � ↵3,2,

↵1,22 � ↵1,2, ↵3,2

2 � ↵3,2, ↵2,22 � ↵2,2. (45)

The first four polynomials give the reduced Groebner basisB \Q[�1,�2,�3], which gives the different domains for theprojection ⇡.

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Applications


Example - ContinuedThe 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 + ↵2,2�3 � 2↵2,2↵3,2.

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Applications


Ising architecture design

An important milestone in the development of AQC is thedesign 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 Yrepresents a class of problems of interest.Each Y minor is explicitly computable.

This problem as described was posed in [V. Choi 2011], wherethe following nomenclature was introduced :

Definition (V. Choi 2011)Let Y be a family of graphs. A graph X is called Y�minoruniversal if for any graph Y 2 Y, there exists a minorembedding of Y in X .

Ref : V. Choi, Minor-embedding in AQC II, 2011. 47 / 51


Applications


The first requirement translates into the conditionP

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

⇡µ(xi) =X

yi2V (Yi )

↵µij yj , (46)

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

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Applications


More precisely, we have

PropositionLet B be a reduced Groebner basis for the system L withrespect to the elimination order {↵µ

ij } � {qij}. The followingstatements are true :

the family of graphs Y = {Yµ} admits a Y�minor universalgraph of size n if and only if 1 /2 B (the choice of theordering used is not relevant for this statement).if 1 /2 B, the set of all Y�minor universal graphs of size n isgiven by the intersection B \Q[q].if 1 /2 B, the embeddings ⇡µ (i.e., the coefficients ↵µ

ij ) arealso given by B (as functions of the qij ).

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Applications


This approach can be applied to forbidden minorcharacterizations. Consider for instance the followingstatement : A graph X is a forest if and only if it does not containthe triangle K3 as a minor. This yields the following procedure :

(i) Generate the system of equations that gives allembeddings of K3 inside X .

(ii) Compute a reduced Groebner basis B : 1 2 B if and only ifX is a forest.

More generally, Robertson - Seymour theorem states that everyfamily of graphs that is closed under minors can be defined bya finite set of forbidden minors. The membership to this classcan be expressed as a Groebner basis computation using thisfinite set of forbidden minors.

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Applications


Thank you !

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