aa r X i v : . [ qu a n t - ph ] F e b Categorical computation

Liang Kong a , b , Hao Zheng a , b , c a Shenzhen Institute for Quantum Science and Engineering,Southern University of Science and Technology, Shenzhen 518055, China b Guangdong Provincial Key Laboratory of Quantum Science and Engineering,Southern University of Science and Technology, Shenzhen 518055, China c Department of Mathematics, Peking UniversityBeijing 100871, China

Abstract

In quantum computation, the computation is achieved by linear operators in Hilbert spaces.In this work, we explain an idea of a new computation scheme, in which the linear operatorsare replaced by (higher) functors between two (higher) categories. The fundamental problemin realizing this idea is the physical realization of (higher) functors. We provide a theoreticalidea of realizing (higher) functors based on the physics of topological orders.

Classical computation or Turing computation can be summarized mathematically as the com-putation of functions between sets. More precisely, for any positive integers n , anda function f : { , } n → { , } n , we design a ﬁnite set of gates S , each of which is also a function between two sets, such that f can be realized by a circuit constructed from the gates in S . The classical computer is physicallypossible because we can realize or simulate a function between two sets physically through acircuit consisting of a family of more elementary physically realizable functions (i.e. gates).The quantum computation can be summarized mathematically as the computation of linearmaps (or linear operators). More precisely, for any positive integers n , anda linear map F : ( C ) ⊗ n → ( C ) ⊗ n , we design a ﬁnite set of gates, each of which is also a linear map between two ﬁnite dimensionalHilbert spaces, such that F can be physically realized by a quantum circuit of gates. A quantumcomputer is physically possible because we can physically realize linear operators in Hilbertspaces. A quantum computation is a physical way of realizing parallel computation via quantummechanics. The idea is to replace the bit { , } by a qubit C , which can be viewed as a linearspan of two states | i and | i . Di ﬀ erent from { , } , the states in a qubit are much more than | i and | i . A generic state | a i is a superposition of them, i.e. | a i = a | i + a | i = a a ! , ∀ a , a ∈ C . It means that | a i can be in either the state | i or the state | i with potentially non-trivial proba-bilities. Using | i and | i , a linear map F : C → C can be expressed as a matrix. F : a a ! F F F F ! a a ! = F a + F a F a + F a ! . Emails: [email protected], [email protected] F that realizes the parallel computation of both F ( | i ) and F ( | i ) at the sametime can be viewed as a probabilistic circuit (or quantum circuit), which can be simulated by afamily of more elementary linear operators called quantum gates. The idea of categoriﬁcation was originated from the second quantization in physics. Secondquantization has a long history. It is a way to obtain higher dimensional quantum ﬁeld theories(QFT) from quantum mechanics, which can also be viewed as a 0 +

1D QFT. In early 90’s,Luise Crain and Igor Frenkel introduced the idea of categoriﬁcation aiming at constructinghigher dimensional QFTs [CF94]. Roughly speaking, the categoriﬁcation can be viewed as amathematical formulation of the second quantization. The process of categoriﬁcation lifts lowercategorical structures to higher categorical structures. The idea of categoriﬁcation that we needin this work is rather simple and can be summarized by the following diagram:numbers c / / functions (cid:15) (cid:15) vector spaces c / / linear maps (cid:15) (cid:15) c / / − functors (cid:15) (cid:15) c / / − functors (cid:15) (cid:15) · · · higher functors (cid:15) (cid:15) numbers c / / vector spaces c / / c / / c / / · · · where we use “ c ” to represent the process of categoriﬁcation. Remark 2.1.

The reversed process is called de-categoriﬁcation and can be realized by computingthe K -group. For example, the K -group of a 2-category is a 1-category, that of a 1-category is avector space, and that of a vector space is its dimension (i.e. a number).Therefore, we see that if we categorify the usual quantum computation, we should obtainthe computation of n -functors between n -categories. This new computation can be called asecond quantized computation or a categoriﬁed quantum computation, or perhaps, even betteran n-categorical computation .Now we explain why a 1-categorical computation can be viewed as a parallel quantum com-putation. We recall the deﬁnition of a 1-category. For the convenience of physical applications,1-categories and 1-functors are all assumed to be C -linear. Deﬁnition 2.2.

A 1-category C consists of1. a set Ob( C ) of objects a , b , c , · · · ;2. a vector space hom C ( a , b ) over C for each orders pair ( a , b ) and a , b ∈ Ob( C ); (a vector f ∈ hom C ( a , b ) is called a morphism from a to b and is often denoted by a f −→ b or f : a → b ).3. a distinguished vector 1 a ∈ hom C ( a , a );4. a C -linear composition map ◦ for a , b , c ∈ Ob( C ):hom C ( b , c ) ⊗ C hom C ( a , b ) ◦ −→ hom C ( a , c ) g ⊗ C f g ◦ f ;satisfying the following conditions:1. ( h ◦ g ) ◦ f = h ◦ ( g ◦ f ) for all a f −→ b g −→ c h −→ d ;2. f ◦ a = f and 1 b ◦ f = f for all f : a → b . 2f you encounter the notion of a category for the ﬁrst time, it is useful to keep in mind thefollowing picture of the basic structures in a category. a hom C ( a , a ) (cid:4) (cid:4) hom C ( a , b ) + + b hom C ( b , b ) (cid:5) (cid:5) hom C ( b , a ) k k hom C ( b , c ) + + c hom C ( c , c ) (cid:4) (cid:4) hom C ( c , b ) k k (2.1) Example 2.3.

The most useful example of a C -linear category to keep in mind is the categoryof ﬁnite dimensional vector spaces over C , denoted by Vec. Its objects a , b , c , · · · are ﬁnitedimensional vector spaces, and hom Vec ( a , b ) are precisely the space of all the linear maps from a to b . The composition maps are the usual composition of linear maps, and the identity morphism1 a : a → a is the identity linear map. Deﬁnition 2.4.

A 1-functor F : C → D between two 1-categories consists of1. a map F : Ob( C ) → Ob( D ) (we slightly abuse the notation here);2. a linear map F a , b : hom C ( a , b ) → hom D ( F ( a ) , F ( b )) for all a , b ∈ Ob( C );satisfying the following conditions:1. F a , a (1 a ) = F ( a ) , for all a ∈ C ;2. F b , c ( g ) ◦ F a , b ( f ) = F a , c ( g ◦ f ) for all f ∈ hom C ( a , b ) , g ∈ hom C ( b , c ). Example 2.5.

For a given vector space V ∈ Vec, we can deﬁne a 1-functor − ⊗ V : Vec → Vec by a a ⊗ V for objects a ∈ Vec and by ( a f −→ b ) ( a ⊗ V f ⊗ V −−−→ b ⊗ V ) for morphisms.We can get an intuitive picture of such a 1-functor F by the following diagram: ∀ a , b , c ∈ Ob( C ), a hom C ( a , b ) / / ❴ (cid:15) (cid:15) b ❴ (cid:15) (cid:15) hom C ( b , c ) / / c ❴ (cid:15) (cid:15) ↓ F a , b ↓ F b , c F ( a ) hom C ( F ( a ) , F ( b )) / / F ( b ) hom C ( F ( b ) , F ( c )) / / F ( c ) . (2.2)Then one can see immediately that if we can physically realize the 1-functor F , it automaticallyrealizes all linear maps F a , b for all a , b ∈ Ob( C ) at the same time. In this sense, a 1-functor can beviewed as a parallel quantum computing.A natural categoriﬁcation of the one-dimensional vector space C is Vec and a natural categori-ﬁcation of the n -dimensional vector space C n is the product (or, equivalently, direct sum) Vec n of n copies of Vec. A 1-category in the form Vec n is referred to as a ﬁnite semisimple 1-category orseparable 1-category. Then the categoriﬁcation of tensor product ⊗ of vector spaces is Deligne’stensor product ⊠ of 1-categories, i.e. Vec m ⊠ Vec n = Vec mn . A is thereforedeﬁned to be a 1-functor F : (Vec ) ⊠ n → (Vec ) ⊠ n , where Vec is better referred to as a categorical bit . In this section, we explain that a 1-functor can be physically realized by quantum phases ofmatter with topological orders, or topological orders for simplicity (see [Wen19] for a review of3un( X , X )( X , x ) Fun( X , X ) F ( X , x ) fusing F with x −−−−−−−−−−−−−−−→ Fun( X , X )( X , F ( x ))Figure 1: The physical realization of a 1-functor F : X → X .topological orders and references therein). The ﬁrst topological orders that were discovered inphysics labs are in 2d (spatial dimension) fractional quantum Hall systems. Since this work onlydiscusses a theoretical idea, we assume that physical materials that realize topological ordersare abundant.Consider the physical conﬁguration of topological orders depicted in the ﬁrst picture inFigure 1. It depicts a (potentially unstable) anomaly-free 1d (spatial dimension) topologicalorder, together with a 0d boundary. By the mathematical theory of topological order (see forexample [KWZ15]), the 0d boundary can be mathematically described by a pair ( X , x ), where X is a ﬁnite semisimple 1-category and x is an object in X . Physically, this x is a particle-like defectlocated at the boundary.By the boundary-bulk relation [KWZ15, KWZ17], the 1d topological order in the ﬁrst picturein Figure 1 can be described by the 1-category of 1-functors from X to X , denoted by Fun( X , X ).Objects F , G in Fun( X , X ) are particle-like topological defects in this 1d topological order. Thefusion of two such defects corresponds to the composition of two 1-functors. Given such aparticle-like defect, say F , if we push it to the boundary, then it fuses with the boundary particle x and change it to F ( x ) (see Figure 1). In other words, a 1-functor F can be realized by a creatinga particle-like defect in the 1d topological order followed by a fusion process. Creating a defectin a topological order can be physically achieved by inserting an impurity. The whole processcan be simpliﬁed to inserting an impurity in the neighborhood of the boundary. Remark 3.1.

In this way, we can realize any 1-functors between two di ﬀ erent ﬁnite semisimple1-categories X and X because they are just special cases of above situation by taking X = X ⊕ X .All (anomaly-free) 1d topological orders can be mathematically described by Fun( X , X ) forsome ﬁnite semisimple 1-category X . When X , Vec, Fun( X , X ) is a multi-fusion 1-categoryinstead of a fusion 1-category (see a review [EGNO15]). Mathematically, it means that the identity1-functor id X is not a simple object in Fun( X , X ), or equivalently, hom Fun( X , X ) (id X , id X ) ; C .Physically, it means that the multi-fusion 1-category Fun( X , X ) describes an unstable phase,which can ﬂow to a stable one if we introduce certain perturbation to the phase [KWZ15]. Inthis case, this unstableness demands ﬁne tuning, which makes the physical realization not faulttolerant.This problem can be solved by replacing the potentially unstable physical conﬁgurationdepicted in the ﬁrst picture in Figure 2 by a stable higher dimensional physical conﬁguration asdepicted in the second picture in Figure 2. In this new conﬁguration, C labels an anomaly-free 2dtopological order, and P , Q label two 1d gapped boundaries of C , and the same pair ( X , x ) is nowrealized as a domain wall between P and Q . Mathematically, C is a modular tensor 1-category[T94], and P , Q are fusion 1-categories. All ﬁnite semisimple 1-category X can be realized asa corner of this conﬁguration by properly selecting P , C , Q . The second picture in Figure 2 canreproduce the ﬁrst picture by a dimensional reduction process (i.e. by closing the fan). After More precisely, X should be a unitary 1-category. We hide the unitarity requirement for convenience because theunitarity is not essential in the understanding of our ideas. X , X )( X , x ) dimensional reduction ←−−−−−−−−−−−−−− QP ( X , x ) C Q QP P R ( X , x ) ( N , n )( M , m ) CC Figure 2: These ﬁgures illustrate the idea of ﬁxing the unstable problem in Figure 1.the dimensional reduction, Fun( X , X ) is unstable as a 1d phase, but before the dimensionalreduction, everything is stable. Then a 1-functor F : X → X can be realized by introducinga 0d domain wall ( M , m ) in P and a 0d wall ( N , n ) in Q , and a 1d domain wall R in C , thenpushing ( M , m ) , R , ( N , n ) down to the corner (i.e. squeezing the triangle to a point) such thatthey fuse with ( X , x ) to give ( X , F ( x )) [AKZ17]. In reality, this process can be achieved by simplyinserting new materials in a highly controlled way in the neighborhood of the corner directly.All 1-functors F : X → X can be physically realized in this way by properly selecting M , R , N .We believe that this way of doing computation is fault tolerant because topological defects arestable under the perturbation of local operators. Above ideas automatically generalize to higher categories and higher functors but with impor-tant new features.Roughly speaking, an n -category C consists of a set of objects a , b , c , · · · , the hom spacehom C ( a , b ) is an ( n − n − n -functor is deﬁned similarly to a 1-functor. In particular, an n -functor F : C → D consistsof all ( n − F a , b : hom C ( a , b ) → hom D ( F ( a ) , F ( b )) at the same time. Therefore, an n -categorical computation can be viewed as a parallel ( n − X be a unitary n -category [DR18, GJF19, JF20, KZ20] and x an object in X . The pair ( X , x ) now describes a potentially anomalous ( n − n d bulk is again givenby Fun( X , X ), which is now a unitary multi-fusion n -category. An n -fucntor F : X → X labels atopological defect of codimension 1 in the n d bulk, and can be realized physically in the sameway as illustrated in Figure 1 and Figure 2.Unlike the n = ∗ -structure), a unitary n -category does not have such a decomposition when n ≥

2. Indeed, atpresent time, there is no idea about how to classify unitary 2-categories at all. It is known thata unitary n -category is the representation category of a unitary multi-fusion ( n − n ≥

2, a categorical bit no longer makes sense. An n-categorical computation is then should be deﬁned as an n -functor F : X → X , where X is a unitary n -category. Unitary higher categories are highly structural, in contrastto unitary 1-categories. This means that higher categorical computation di ﬀ ers from classical5nd quantum computation dramatically. There must be a lot of new features worthy of furtherexploration. Remark 4.1.

The data ( X , x ) is essentially equivalent to the unitary multi-fusion ( n − X ( x , x ) [JF20, KZ20]. The above physical realization of an n -functor F : X → X can also beviewed as a physical realization of a monoidal ( n − X ( x , x ) to hom X ( F ( x ) , F ( x )).When n =

2, this physical realization of monoidal 1-functor can be stated as a mathematicaltheorem [KZ18, Theorem 3.2.3].

Remark 4.2.

Although the physical realization of topological orders cannot go beyond 3d (spatialdimension), it is already very interesting and rich in 1d and 2d cases because anomalous 1dtopological orders and 2d topological orders are abundant.

Remark 4.3.

Topological orders was proposed long ago to provide the physical realization ofthe fault tolerant quantum computation [K03, F98] (see for example [Wan10] for a review andreferences therein). Our proposal says that they can also do n -categorical computations in a faulttolerant way.At this stage, it is too early to tell if the categorical computation is technologically possible orimpossible. It depends on the future development of the physics of topological orders, the dis-covery of new topological materials and the technology advances in controlling and engineeringtopological materials. We also do not know if it can be more e ﬃ cient than classical / quantumcomputation. However, it is likely to be useful in computing categories and functors. We be-lieve that it is worthwhile to make this naive idea available to experts so that more ideas anddiscussion can follow. Acknowledgement : We thank Ce Shen, Xiao-Ming Sun and Zhong Wang for helpful comments.We are supported by Guangdong Provincial Key Laboratory (Grant No.2019B121203002). LKis also supported by NSFC under Grant No. 11971219. HZ is also supported by NSFC underGrant No. 11871078.

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