TQFTS AND QUANTUM COMPUTING MAHMUD AZAM AND STEVEN RAYAN Abstract. Quantum computing is captured in the formalism of the monoidal subcate-

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TQFTS AND QUANTUM COMPUTING

MAHMUD AZAM AND STEVEN RAYAN

Abstract. Quantum computing is captured in the formalism of the monoidal subcate-

gory of VectCgenerated by C2— in particular, quantum circuits are diagrams in VectC

— while topological quantum ﬁeld theories, in the sense of Atiyah, are diagrams in VectC

indexed by cobordisms. We initiate a program that formalizes this connection. In doing

so, we equip cobordisms with machinery for producing linear maps by parallel trans-

port along curves under a connection and then assemble these structures into a double

category. Finite-dimensional complex vector spaces and linear maps between them are

given a suitable double categorical structure which we call FVectC. We realize quantum

circuits as images of cobordisms under monoidal double functors from these modiﬁed

cobordisms to FVectC, which are computed by taking parallel transports of vectors and

then combining the results in a pattern encoded in the domain double category.

Contents

1. Introduction 2

2. A Double Categorical Approach 3

3. Connections on Cobordisms 6

3.1. Gauge Transformations 6

3.2. Bundle Cobordisms 10

3.3. Monoidal Double Category of Connections 12

4. Paths for Parallel Transport 16

4.1. Graphs Encoding Algebraic Expressions 16

4.2. Morphisms of Expression Graphs 21

4.3. Constructs on Expression Graphs 23

4.4. Geometric Realization 27

4.5. Single Manifold TQFT 30

5. Parallel Transport Calculus 32

5.1. Double Category of Finite-Dimensional Vector Spaces 32

5.2. Thick Tangles and Transport Graphs 34

5.3. Parallel Transport Calculus 37

5.4. Quantum Computing with Parallel Transport 38

5.5. Addition of Cobordisms 40

5.6. Quantum Computing Revisited 42

5.7. Connections to Operads and PROPs 43

6. Further Directions: Graphs, Categoriﬁcation, and Hyperbolic Matter 45

References 46

Date: October 10, 2022.

arXiv:2210.03556v1 [quant-ph] 7 Oct 2022

2 MAHMUD AZAM AND STEVEN RAYAN

1. Introduction

Perhaps unsurprisingly, quantum ﬁeld theories and quantum information enjoy natural

points of intersection as two sides of modern quantum theory. However, the essential pur-

poses and formalisms inherent to these subjects are rather diﬀerent, and many of these

observed intersections are coincidental or speculative in nature. Here, we capitalize on

shared aspects of the categorical frameworks associated to the two theories in order make

eﬀorts to close the gaps between them. To be precise, we consider topological quantum ﬁeld

theories (TQFTs) of thick tangled type. On the quantum information side, we impose no

restrictions.

The monoidal category 2Thick of thick tangles is the monoidal category freely generated

by the composition of the following morphisms [10]:

Pair-of-pants Cap Cylinder Cup Co-pair-of-pants

Consider a planar open string topological quantum ﬁeld theory F: 2Thick −! VectCas

deﬁned in [10]. Fis determined by the images of the above generating structures under

F. One possible to way to connect quantum information to topological quantum ﬁeld

theories is to assume that the image of the interval Iunder Fis some ﬁnite dimensional

C∗–algebra of operators on some Hilbert space of states. Coecke, Heunen, and Kissinger

[2] have shown that every such algebra is, in fact, a dagger Frobenius algebra, with some

additional structure, so that this assumption on Fis valid. In fact, every ﬁnite dimensional

C∗–algebra arises as the image of the interval under such a planar open string ﬁeld theory.

It is well-known that ﬁnite dimensional C∗–algebras, up to ∗–isomorphism, can be realized

as ﬁnite direct sums of square matrix algebras LiMniwhere Mnis the set of n×nmatrices

with complex entries equipped with the usual multiplication. For simplicity, we ﬁrst assume

that F(I) is the C∗–algebra Mn. Then, we can consider quantum gates to be elements of

Mnand circuits to be composites (products) of these elements. While Mnhas all gates

necessary for quantum computing for a ﬁnite quantum register, there is a major constraint

on the elements of Mnthat are in the image of F, as we describe below.

Elements a∈Mncan be seen as maps C−! Mn:z7−! za. The elements that

are accounted for by Fare images of thick tangles ∅−! I. However, these “element”

thick tangles are determined by their genus, since in 2Thick we identify morphisms up to

diﬀeomorphism, so that all element thick tangles must decompose as follows:

where we take the domain of the thick tangle to be on the left and the codomain, on the

right. Call this thick tangle R. Let m:Mn⊗Mn−! Mnbe the multiplication of Mn.

Then the multiplicative unit is e:C−! Mn:z7−! zInand, by the deﬁnition of dagger

Frobenius algebra, the comultiplication of Mnis m†. Hence, under F, the element thick

tangle above yields a map (mm†)ke, where the superscript kdenotes a k–fold composite for

TQFTS AND QUANTUM COMPUTING 3

some non-negative integer k. Now, mm†is the map a7−! na [2, p. 5189 — 5190] so that

F(R)(z) = z·F(R)(1) = z·(mm†)ke(1) = znkIn

This shows that the only elements, in the usual sense of elements of Mn, that are accessible

through Fare multiples of the identity matrix by powers of n. It is then easy to see that

for direct sums such as Lq

i=1 Mni, the only accessible elements are

(nk

1In1, . . . , nk

qInq)

so that up to ∗–isomorphism, there are some major constraints on the quantum gates that are

accessible through planar open string ﬁeld theories. We note that this situation is brought

about by the identiﬁcation of thick tangles up to diﬀeomorphisms. This motivates us to

look for methods in a setting where we drop this identiﬁcation — higher categories of thick

tangles (for instance, PT T as deﬁned in [6]) (or cobordisms) where gluing is associative and

unital up to higher isomorphisms. We will see that it suﬃces to consider double categories for

obtaining a reasonable method for formulating quantum information in terms of topological

ﬁeld theories.

Recall that a monoidal double category consists of a 1–category of objects and a 1–

category of morphisms with source, target and unit functors, a notion of horizontal compo-

sition of morphisms, and a monoidal structure on the object and morphism categories. In

addition, horizontal composition and monoidal products need to be associative and unital

up to isomorphism with several coherence and compatibility properties [16]. In this work,

however, by “monoidal double category” we will mean only the data of such a structure.

Nevertheless, wherever possible, we have commented on how the data of our constructions

inherit most of the necessary properties from the usual categories of sets, manifolds, vec-

tor spaces, and so on. Thus, we will construct several monoidal double categories in this

work but they should be seen as monoidal double categories in a somewhat relaxed sense

— they consist of all the required data but satisfy the required axioms with a few possible

exceptions. We will treat monoidal double functors in the same loose sense.

Acknowledgements. The second-named author is partially supported by a Natural Sciences

and Engineering Research Council of Canada (NSERC) Discovery Grant, a Canadian Tri-

Agency New Frontiers in Research (Exploration Stream) Grant, and a Paciﬁc Institute

for the Mathematical Sciences (PIMS) Collaborative Research Group (CRG) Award. The

ﬁrst-named author was supported by an NSERC Undergraduate Student Research (USRA)

Award and the funding of the second-named author. The code for ﬁgures 4.31 and 5.5 were

generated with the help of the Mathcha editor (mathcha.io).

2. A Double Categorical Approach

It is well known that taking d–dimensional manifolds without boundary as objects, diﬀeo-

morphisms between them as vertical 1–morphisms, (d+1)–dimensional cobordisms between

them as horizontal 1–morphisms and boundary preserving diﬀeomorphisms between cobor-

disms as 2–morphisms yields a ﬁbrant monoidal double category Cobd+1 under the disjoint

union of manifolds [16]. Shulman gives a trifunctor Hfrom the tricategory of ﬁbrant double

categories to the tricategory of bicategories that takes Cobd+1 to a monoidal bicategory —

in fact, Shulman proves that Htakes any ﬁbrant monoidal double category to a bicategory.

On the other hand, the monoidal category of thick tangles 2Thick as deﬁned in [10] is

a decategoriﬁcation of a monoidal bicategory of thick tangles PT T deﬁned in [6]. Taking

inspiration from this situation, we assume that there is a ﬁbrant monoidal double category

4 MAHMUD AZAM AND STEVEN RAYAN

2Thick which, under H, yields PT T . The structures deﬁning 2Thick are the ones analogous

to Cob2:

•objects are diﬀeomorphism classes of disjoint unions of the interval I= [0,1]1

•vertical 1–morphisms are only the identity morphisms

•horizontal 1–morphisms Iqn−! Iqmare surfaces with boundary IqnqIqmalong

with an embedding dinto R×Isuch that d−1(R× {0}) = Iqnand d−1(R× {1}) =

Iqm2

•2–morphisms are diﬀeomorphisms between cobordisms (horizontal 1–morphisms)

that preserve the boundary

We then attempt to use this notion to concretely deﬁne the data of a monoidal double

functor from 2Thick to a suitable monoidal double category of complex vector spaces that

yields enough unitary linear transformations in the image to facilitate quantum computing.

We deﬁne the object function F0of such a functor by assigning to each disjoint union

X=Iqnthe n–th tensor power F0(X) = A⊗nof some ﬁxed algebra A— we may be

speciﬁc enough to pick a consistent bracketing pattern for A⊗n. It is easy to see that this

assignment is well-deﬁned. Since the vertical 1–morphisms are only identities, the object

category of 2Thick is discrete and, hence, the vertical 1–morphism function is the unique,

obvious one: F0(idX) = idF0(X).

Next, we consider the horizontal 1–morphisms or the cobordisms Z:Iqn−! Iqm,

which are determined up to diﬀeomorphism by their genus. For positive mand n, we ﬁrst

consider some “canonical” genus kcobordism Z:I−! Iwhere the holes are circles with

centers along a straight line from one boundary interval to another. This cobordism then

decomposes into a k–fold composition of M∗W:I−! Iwith itself, where Mis the

“canonical” pair-of-pants and Wis the “canonical” co-pair-of-pants. An example is shown

below:

In associating a linear map “functorially” to Z, it suﬃces to associate linear maps to M

— we can then associate a linear map to Wby duality and get a linear map for Zby

composition.

For associating a linear map to M, we take a complex bundle π:E−! Mwith ﬁbre

Aalong with a connection ∇. Then, we choose two paths: one from the mid-point of each

in-boundary interval to the mid-point of the out-boudnary interval. By parallel transport

along each curve, we get two linear maps l1, l2:A−! A. We then have a linear map

l:A⊗A−! Agiven by l(x⊗y) = l1(x)l2(y) where the product in the right is the algebra

product in A. Then, the linear map associated to Wis the conjugate transpose l†. We then

obtain a linear map associated to Z: the k–fold composite (ll†)k. We set F1(Z) := (ll†)k.

1The justiﬁcation for equating disjoint unions of the interval up to diﬀeomorphism is that the monoidal

product on the objects of PT T (and 2Thick) is strict given that the objects are taken to be the integers n

as opposed to n–fold disjoint unions Iqnwith diﬀerent bracketings.

2There are ﬁner details here which will be unimportant for our purposes.

TQFTS AND QUANTUM COMPUTING 5

Now, consider a cobordism Y:Iqn−! Iqmof genus ksuch that Y=RZQ, where Q

is a cobordism Iqn−! Iformed in some “canonical” way by a gluing of pairs of pants

and cylinders (“identity” cobordisms) and Ris a cobordism I−! Iqmformed again in a

“canonical” way from co-pairs of pants and cylinders. Of course, we associate a linear map

to the cylinder by another parallel transport along a curve between the mid-points of its

two boundaries. Then, again by composition, we get a linear map F1(Y) : A⊗n−! A⊗m.

This gives an assignment F1(Y) for a representative Yof each diﬀeomorphism class of

cobordisms in 2Thick. For the assignment of a linear map to every cobordism in 2Thick,

we take the following approach. Let Y0be an arbitrary cobordism in the class of some Yfor

which F1(Y) has been deﬁned as above. Then we pick a boundary preserving diﬀeomorphism

f∈AutMan(Y) such that f(Y) = Y03. Let {γi}be the family of curves which along which

parallel transport gave us the linear maps F1(Y). Then, {f◦γi}is a family of curves in Y0

which yield linear maps by parallel transport under a connection f∇, to be made precise

later. Combining these using the same “pattern” or “expression” of algebra multiplications,

tensor products and compositions as we had for Y, we can obtain a linear map F1(Y0).

Continuing the example, we have:

We now turn our attention to the case when mor nis zero. We can treat the case

n= 0 and obtain the other case by duality. Let Y:∅−! Iqmwith genus k. Then Y

decomposes as R∗Z∗Z0where Zand Rare as before and Z0is a genus zero cobordism

∅−! I. We call cobordisms ∅−! Ielements and we call elements of genus zero, atomic

elements because they will not decompose into any simpler structures. Z0has a boundary

preserving diﬀeomorphism f:Z00 −! Z0for some atomic element Z00 deemed “canonical”.

We associate a linear map a:C−! Ato Z00 as follows. Take a loop γin Z00 on the mid-

point of its only boundary interval and obtain an element a∈A, or equivalently a linear

map a:C−! Aby parallel transport of some ﬁxed element a0∈A. We set F1(Z00) = a

and get F1(Z0) by parallel transport of a0along fγ.F1(Y) is then obtained by composition.

This completes the deﬁnition of an object function F1for the morphism category of

2Thick. Now, we turn our attention to 2–morphisms — boundary preserving diﬀeomor-

phisms between cobordisms. Let f:Y−! Y0be one such diﬀeomorphism. We must assign

to fsome object that functions as roughly a “morphism of morphisms of vector spaces”.

One natural choice is homotopy classes of paths between linear functions in spaces of linear

functions under some suitable norm. In this case, we can take the following approach. There

exists a path ψin the diﬀeomorphism group Diﬀ(Y) of Yfrom idYto fsuch that we can

“move” the parallel transport machinery “along” ψ— that is, taking connections ψ(t)∇and

paths {ψ(t)γi}, for t∈[0,1] — to get linear maps for each ψ(t)(Y), which constitute a path

in the space Hom(dom F1(Y),codom F1(Y)). Note that, implicit in this is the assumption

that ψ(1)∇=f∇and {ψ(1)γi}={fγi}yield the linear map F1(Y0) by parallel transport,

which need not hold in general.

3Of course, we take the underlying topological space for each manifold in a diﬀeomorphism class to be

the same.

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TQFTSANDQUANTUMCOMPUTINGMAHMUDAZAMANDSTEVENRAYANAbstract.Quantumcomputingiscapturedintheformalismofthemonoidalsubcate-goryofVectCgeneratedbyC2|inparticular,quantumcircuitsarediagramsinVectC|whiletopologicalquantumeldtheories,inthesenseofAtiyah,arediagramsinVectCindexedbycobordisms.Weinitiateaprogra...

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