Recall that two posts ago, I mentioned that Landsberg et al. observed that the matchgate identities are merely the defining equations of the spinor varieties. In this post, I will make explore what this really means in the hopes of describing to the reader their vision of the beautiful geometric reasons why holographic algorithms work. The big picture is that holographic algorithms are all about the efficient computation of some custom-made Holant that captures the answer to the combinatorial problem at hand. We come up with this exponentially large vector pairing (i.e. tensor contraction) and magically manage to compute it in polynomial time basically by computing a determinant. When do such efficiently computable pairings occur in geometry? Well if you take vectors in the cones of and (informally, vectors close enough to a -plane that they can be regarded as decomposable in or respectively) they turn out to be vectors consisting of determinants of minors of some matrix, and their pairing is merely the determinant of a related matrix. As it will turn out, if you take points in the variety of pure of spinors, they can likewise be characterized as vectors of “sub-Pfaffians” of some matrix, and a pairing between two such vectors is merely the Pfaffian of a related matrix. These are the kinds of efficiently computable pairings holographic algorithms take advantage of.
But it’s not enough to know that there exist special vectors whose pairings are easy to compute. After all, as a stupid example, the pairing between two -vectors supported in only dimensions is also computable in time. What distinguishes the vectors that feature in holographic algorithms from these vectors is the surprising availability of the former in complexity theory problems. Whereas there are way too many vectors in not supported on only polynomially many dimensions, the codimensions of for of low dimension and the spinor variety over low dimensions are quite small, and just as we’ve seen with matchgates, the vector spaces involved can often be decomposed into small enough “local” components (this local decomposition is key, because the abovementioned codimensions blow up very quickly).
In what follows, I hope to give a rigorous treatment of these ideas. My main references Chevalley’s The Algebraic Theory of Spinors and Clifford Algebras and the fifth chapter of Varadarajan’s lecture notes on supersymmetry for the development of the basic theory of Clifford algebras and spinors, and Landsberg’s Tensors: Geometry and Applications and Manivel’s On Spinor Varieties and Their Secants for establishing the connection between the matchgate identities and the variety of pure spinors. As always, if there are any mistakes in presentation below, they are entirely my fault.
1. Clifford Algebras and Spinors
“No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.” -Michael Atiyah
As a remark, I make no attempt to cover technical material related to spinors and Clifford algebras beyond the scope of their relevance to holographic algorithms, so I apologize for failing to include certain key ideas (e.g. anything more than a basic mention of the graded structure of the Clifford algebra).
The notion of a spinor, while not named at the time, first came up in Cartan’s famous classification of the simple representations of all simple Lie algebras when he discovered an “exotic” representation of not obtainable from tensors over the defining representation (the reason being that the weights of representations obtained from the usual tensor constructions are integral while those of the spin representation are odd multiples of 1/2). We see the spin representation come up immediately, for example, from the fact that (the belt trick): the spin representation is merely the faithful representation of degree 2 of . Spinors are just the vectors acted on by the spin representation.
In any case, the cleanest route to developing these notions will be from the point of view of Clifford algebras, which can be thought of in one way as providing a generalization of the multiplicative structure of the vector spaces of complex numbers, quaternions, and octonions to higher dimensions. In our discussion below, we will for convenience restrict ourselves to the field , though the results are valid over all fields of characteristic not equal to 2. Let denote a real vector space of dimension , let be some quadratic form, and let denote the symmetric bilinear form associated with (namely, ).
The Clifford algebra of associated to is defined to be the -algebra , where is the ideal in generated by all elements of the form .
is graded, so denote by and the direct sums of the respectively even and odd components of , and denote by and the spaces .
As a first observation, note that there is a natural map , and it’s easy to see that in fact this natural map is an injection; so we’ll regard as sitting inside . Now let’s prove some results on the structure of which will allow us to define spin representations.
As a second observation, we can characterize in terms of the exterior algebra of :
Proof: There is a natural filtration corresponding to , and projecting this filtration down to gives a filtration on . But the graded ring associated to this filtration is isomorphic to . Indeed, , and for any , , from which we conclude that we have a natural isomorphism of graded algebras . But as vector spaces, and are isomorphic, so we’re done.
As a third observation, real Clifford algebras are parametrized quite nicely. By Sylvester’s law of intertia, quadratic forms can be diagonalized into the form for some which are invariants of , so the real Clifford algebras are parametrized by these invariants. We will thus denote them by when we wish, in which case if , we can choose generators of satisfying and .
If is even, then is a central simple algebra.
Proof: Say that for , and pick the generators described above, but assume that we’ve diagonalized so that . Then note that the elements and satisfy and and thus generate , which is the real matrix algebra , while generate . The conclusion is that , and it remains to show that is central simple. For those familiar with the proof of Bott periodicity with Clifford algebras, it is easy to check using this same , trick that while the and don’t consistently generate the same algebra , the intermediate algebras generated at each step tensor together nicely, so that for all , we get , and we’re done.
While we cannot make the same claim for odd, we can make this claim about :
If is odd, then is a central simple algebra.
Proof: is generated by , which satisfy and so that is isomorphic to just or , which we know from the previous discussion to be central simple.
Now let denote the orthogonal group of transformations preserving .
The Clifford group of is the set of for which conjugation of any gives something inside . Let be the linear representation of sending to this conjugation map; call this the vector representation of .
In fact, sends not just into but into : . Moreover, it is known that if the rank of is even (resp. odd), is all of (resp. the even-graded elements of with determinant one), though we’ll omit this proof.
Moving onto the spin representation, assume first that is even. We know is simple, and simple representations of simple algebras are equivalent, so pick any such representation, and this is the spin representation. Vectors in the space over which it acts are called spinors.
On the other hand, assume that is odd. We can no longer say that is simple, but we do know that is, so again, all simple representations of this are equivalent. By Lemma 1 below, there are two ways to extend this unique spin representation of to a representation, and these are the spin representations of , with their corresponding spinors defined as above.
Proof: Before we jump to the proof of this, let’s prove a basic but key fact about the center of :
Claim 1: If is of odd dimension, then for some odd element for which .
Proof: For orthogonal basis , we will denote . If , then ; otherwise . Pick in the center, and write it as . Then one can check that for all implies that for all except , and is the odd element that we’re looking for, up to scaling.
Now write any uniquely as for . The maps and are homomorphisms . Now simply compose the spin representations with these and to obtain two distinct representations and .
Now say that we had another representation of extending . Call it , and say that . The space of spinors decomposes into , those for which , and , those for which . But preserves the action of both of these subspaces, and by the fact that is simple, we conclude that must be either or .
2. Pure Spinors
The particular kinds of spinors which form the bridge between geometry and holographic algebras are the so-called “pure spinors.” We begin by exploring these in the classical sense. The intuition is that pure spinors corresponding to a certain kind of subspace of are the analogue of “decomposable vectors” associated to a subspace of . They’re particularly nice because they correspond to maximal linear subvarieties of the quadric .
Before we can define pure spinors, here is the key preliminary result. For simplicity, we will assume that is of even dimension. Pick a splitting into maximal isotropic spaces . Denote the uniquely defined spin representation of by .
The intersection between a minimal left ideal and a minimal right ideal of is a 1-dimensional vector space.
Proof: A theorem due to Brauer tells us that and for some idempotents and .
If is the kernel of and is the image of under the projection , then we see first off that for all , maps to zero. On the other hand, for any that maps to zero, ( acts as the identity on ), from which we conclude that is the set of elements in that kill . But the set of elements in that kill any bigger than likewise corresponds to a left ideal smaller than , contradicting minimality. We conclude that is a hyperplane.
The argument for is analogous. If is the image of under the projection , then we see first off that for all , sends to itself. On the other hand, for any that does this, ( acts as the identity on ). The set of elements of that project into a smaller than corresponds to a right ideal smaller than , contradicting minimality. We conclude that is a line.
is thus the set of which kill and project into . For , lies in a line. If we pick a basis element for , is determined exactly by the for which , so we conclude that is of dimension 1.
Now take to be a maximal isotropic subspace of . and say generates as a subalgebra of . Pick a basis of and define to be . Also define to be for basis elements of . By Lemma 1, we note that both and is defined up to a multiplicative factor, independent of the choice of bases. But one can show that and are minimal left and right ideals respectively.
Now because and intersect on a line, say that their intersection is some for one-dimensional subspace . is called the line of representative spinors of .
A pure spinor is any spinors representative of some maximal isotropic .
These spinors are called “representative” for the following reason:
If is a representative spinor of , then is precisely the set of all which kill .
Proof: Recall that for of even dimension, , meaning there must exist some by which can be conjugated to obtain . As such, we’ll assume that , in which case the corresponding line of pure spinors contains . But for any is multiplication by the component of inside , and this map is zero iff this component is zero, i.e iff .
There is thus a correspondence between maximal isotropic subspaces of and lines of pure spinors. As such, we’ll define the so-called spinor variety as the variety of all maximal isotropic subspaces of .
This variety consists of two connected components and such that two maximal isotropic subspaces lie in the same component iff their intersection has dimension of the same parity as .
As an aside, recall that one of the ingredients for holographic algorithms to work is that “efficient pairings can be computed quite often.” Once we establish the connection between standard signatures and points on the spinor variety, this statement translates to the fact that “the spinor varieties over low dimensions have low codimension.” Indeed, the spinor varieties over dimensions 2, 3, and 4 are , , and .
3. Sub-Pfaffians and Matchgates Revisited
Having developed the foundations for the theory of spinors, we will now show how they relate to the matchgate identities. In this section, we will eventually characterize points on the spinor variety in terms of sub-Pfaffians.
Before we do this however, let’s look at the analogous correspondence in the case of the Grassmannian manifold .
The Plucker embedding sends to the projectivization of via so that the cone of the Grassmannian is the set of decomposable vectors inside .
Write as for -plane . Say that has basis elements and has basis elements . The neighbors of , can be locally parametrized by matrices of coordinates of (i.e. the tangent space at the point of ) as . Then one can check that for and of the same size, the coordinate of the th basis element where for and otherwise equals the determinant of with rows indexed by and columns indexed by .
The relationship between the spinor variety and sub-Pfaffians of some coordinate matrix is completely analogous. But first, we give a slightly different, cleaner formulation of spinors as sub-Pfaffians using an exponential map. Pick a basis of . If , define . This map has all the properties one would expect of an exponential map: 1) , 2) it is approximated by in the sense that if is the multiple of any basis element or in fact anything decomposable, then , and . Furthermore, one can check that conjugation by any individual factor preserves , so is always in , and furthermore is just multiplication by . The reason we introduce this map is that the coefficients of for all are precisely the Pfaffian of the submatrix of with rows and columns indexed by the .
The main result of this section is the following characterization due to Chevalley of all pure spinors:
Proof: To prove this, we’ll need two auxiliary results.
Let’s first see how these two lemmas prove Theorem 3. We know that represents a space whose -component has basis . By our characterization of the space that a pure spinor represents as the space of all things killing it, the space that represents is conjugated by . But because the elements of are fixed by conjugation by , so we also get the second part of the theorem.
In the other direction, start with any maximal isotropic . Take , , and in the second lemma to be , , and as defined above. As we’ve already noted, represents , and by the second lemma we have some for which sends to , so the line of representative spinors consists of multiples of , as desired.
It remains to prove our lemmas. For the first lemma, and clearly have dimensions which sum to that of , and the bilinear form associated to is nondegerate, so , which is clearly isotropic, is maximal. represents it because is the space of all elements killing it.
For the second lemma, we will show that any map in the orthogonal group fixing the points of is of the form for some . After that, it suffices to construct a map sending to while fixing .
Start with any fixing points in . It turns out that it’s possible to construct bases for and , call them and , such that , for all for some , and for all , where the depends on the parity of . The we are looking for is then simply .
Finally, to construct the desired map sending to , define to be the subspace of spanned by , and to be . It suffices to show that and can both be sent to while fixing points of . We will omit the details, but the map from to sends via the identity and via taking the -component.
The big conclusion is that the the matchgate identities cut out the variety of pure spinors!
Anyways, for the sake of concreteness and completeness, let’s continue the analogy we started with. Instead of a point on , consider a point on the spinor variety of . In other words, rather than a -plane , let’s take a maximal isotropic subspace , and let as before. Whereas the Plucker map embeds into the projectivization of , our discussion of the spinor variety tells us we have an embedding of the projectivization of (where denotes the sum of the even graded components of and denotes the sum of the odd ones). Whereas the neighbors of a -plane in were parametrized by any matrix of coordinates inside , as one can check, the neighbors of a maximal isotropic subspace in the spinor variety are parametrized by skew-symmetric matrices :
The space generated by is maximal isotropic iff is skew-symmetric.
Whereas the decomposable vector in had coordinates the determinants of submatrices of , the pure spinor corresponding to , by Chevalley’s result, has coordinates the Pfaffians of certain submatrices of . Specifically, a bit of computation gives us the following observation.
For even, the th coordinate of this pure spinor is the Pfaffian of the submatrix of with rows and columns indexed by .
In this post, we began with a classical presentation of the matchgate identities and proceeded to develop the appropriate theory for understanding where these identities really come from. To reiterate, this latter discussion in some sense “demystifies” holographic algorithms by uncovering the underlying geometric reason why holographic algorithms work: realizable signatures of arity are parametrized by points on the variety of pure spinors over dimension , tensor contractions are often easy when there is this kind of additional structure because, and for low such varieties have low codimension so that this happens nontrivially often.
Of course, the bigger question is still whether holographic algorithms could ever be used to give a proof that (in fact, such a proof would actually imply the more astounding result that ), and if not, how geometry could somehow “know” that certain problems are intractable.
I don’t know if anyone has a convincing answer yet. To further explore the full potential of holographic algorithms, I will take a detour from these more theoretical considerations in the next post and cover some more of Cai’s work, namely his recent work on holographic algorithms over domains of larger size than just the Boolean domain, asking whether larger domains, unlike larger bases, can add power to holographic algorithms.