Algebra and Coalgebra on Categorical Tensor Network States

Jacob D Biamonte1, ∗

1Oxford University Computing Laboratory

We present a set of new tools which extend the problem solving techniques and range of appli-cability of network theory as currently applied to quantum many-body physics. We use this newframework to give a solution to the quantum decomposition problem. Specifically, given a quantumstate S, we are now able to directly construct a tensor network that describes the state S. This solu-tion became possible by synthesizing and tailoring several powerful modern techniques from highermathematics: category theory, algebra and coalgebra and applicable results from classical networktheory and graphical calculus. We present several examples (such as categorical MERA networksetc.) which illustrate how the established methods surrounding tensor network states arise as aspecial instance of this more general framework, which we call Categorical Tensor Network States.

∗jake@qubit.org

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I. MOTIVATION SUMMARY: A NEW QUANTUM NETWORK THEORY

We report the development of a new tool set and corresponding framework which is significantly differentand outside the range of methods used to address problems in network descriptions of many-body physicsand related disciplines. In the categorical network model of quantum states we present, each of the internalcomponents that form our network building blocks are completely defined in terms of their mathematicalproperties, and these properties are given in terms of equations which have a purely graphical interpretation:category theory [1] replaces ad hoc graphical methods in network descriptions of many-body physics ande.g. enables rigorous proofs to now be done graphically. We went out of our way to write this article for ageneral reader with a background in tensor network states and/or quantum circuit theory: no backgroundin category theory or higher algebra is assumed.To explain the key motivation behind developing this new machinery, let us recall the success of established

numerical simulation methods, such as density matrix renormalization group (DMRG) and quantum MonteCarlo (QMC) which have become key to studying strongly correlated systems in regimes and at scales wherequantum mechanical effects are crucial [2, 3]. However, substantial limitations have remained in the size,dimensionality, and classes of Hamiltonians these methods can be used to simulate.Tensor network states arose recently from the field of Quantum Information Science as the backbone of

applicable and general methods to simulate quantum systems using classical computers (for the currentcapabilities of the existing graphical language see [4, 5]). As a matter of necessity, and one of opportunity,by utilizing concepts from quantum information science several novel and powerful computer algorithms (allbased on tensor network states) have recently been proposed which have overcome many existing numericallimitations. Specifically: the Multi-scale Entanglement Renormalization Ansatz (MERA) [6] and ProjectedEntangled Pairs (PEPS) [7] — see also [8–10]. In addition, tensor network based numerical algorithmshave recently been successfully adapted to the simulation of stochastic classical systems [11]. These andother related methods have been used to perform highly accurate calculations on a broad class of stronglycorrelated systems. This has attracted significant interest from several research communities concerned withcomputer simulations of physical systems.Computer Science techniques such as semantics and logic emerged out of increasingly general methods

to depict intuitive and descriptive models of systems and processes. Such conceptual methods rely on theunifying language of category theory [12, 13]: for both its expressive power and as a unification tool touniformly reason over wide classes of a priori seemingly different scenarios. Many otherwise obscure aspectsof mathematical models can be made vivid at the level of categories, and the associated differences can bepinpointed on-the-nose in terms of clear, definable structure. This continues to set the stage for the formalanalysis of a range of generally applicable scientific concepts.The expressiveness and range of applicability of tensor network based algorithmic techniques is fostered by

an intuitive graphical language describing the tensor networks which represent physical states and processes.This graphical language can now take a broader direction by being connected to the long existing rigorouslanguage of category theory. This will immediately open the door to apply many established techniques:category theory provides the exact arena of mathematics concerned with such diagrammatic reasoning.The diagrammatic language of tensor network states in current use is a special case of this long existingrich framework. Again, as stated, we present the category theory underpinning our approach with a wideaudience in mind — this work is largely self contained and assumes little if any knowledge of higher algebraor category theory. We will explore several category theoretic results that arose from this study in futurepublications.Categorical models of tensor network states allow us to both “zoom out” and expose high-level structure,

but also to “zoom in” and expose hosts of “hidden” algebraic structures that are not currently being con-sidered in the graphical language used throughout the tensor network simulation community. Enhancingthe graphical language component of these numerical methods should lead to the discovery new theoreticalmodels and numerical algorithms which challenge and shape our understanding of many-body physics.Category theory is often used as a unifying language for mathematics [1] and in more recent times to

formulate physical theories [12, 13]. To accomplish our goals, we will build on ideas across several fields.This includes the work by Lafont [14] which was aimed at providing an algebraic theory for classical Booleancircuits (this is a different direction than the work [15]). We represent this algebraic theory [14] on tensorsand use these to express quantum states. Using more category theory tightens this approach and removessome redundancy in his graphical lemmas [14]. Lafont’s work is related to the more recent work on prooftheory by Guiraud [16]. One of the strong points of categorical modeling is that it comes equipped with

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many flavors of intuitive graphical calculi. We consider a so-called Penrose-Joyal-Street calculus (and actuallythe Kelly-Laplaza-Selinger coherence result [1, 17, 18]) and as a matter of convenience, we make use of †-compactness already present in categorical quantum theory [19]. The graphical calculus formally extends toa rigorous tool. See for instance, Selinger’s survey of graphical languages for monoidal categories (these arethe categories which describe Hilbert spaces and quantum theory [18]).

II. RESULTS OVERVIEW: II A, II B AND IIC

We will now quickly introduce the key concepts of this paper before going into full detail in the remainingsections. The main idea (representing quantum states in terms of networks) is reviewed next in IIA with thecorresponding algebraic definitions of these network components reviewed in II B and the concept of a statedefining an algebra in II C. We summarize this new theory in IID and outline the structure of the remainderof the manuscript in III. These first three review sections (II A, II B and IIC) where made accessible for therange of readers working in the area of quantum information science with a background in quantum circuits,and/or the existing tensor network theory.

A. A New Representation of Quantum States

We give algebraic operations a representation on quantum states. This allows us to do the converse,that is, give tensor network states a representation in terms of algebraic operations! The starting point ofclassical network theory was seminal work resulting in Shannon and Davio decompositions of functions intonetworks. These powerful methods formed the backbone and enabled the last century of methods surroundingclassical network theory. The present paper presents a solution to the related quantum problem: that is, thedecomposition of a quantum state into a categorical tensor network.To get an idea of what we’re about to do, let’s consider a key network building block: the so-called

“quantum AND-state” which we define in Section IVD and was given in [20]. This is a representation of thefamiliar Boolean operation in the bit pattern of a tri-qubit quantum state as

ψAND

def=

∑

x1,x2,x3=0/1

|x1〉 ⊗ |x2〉 ⊗ |x1 ∧ x2〉 = |000〉+ |010〉+ |100〉+ |111〉 (1)

and hence, the truth table of a function is encoded in the bit pattern of the superposition. This gives riseto a linear representation of Boolean gates (represented on quantum states) as opposed to the typical directsum representation common in Boolean algebra.

Remark 1 (Physical Realisation). Such a state is in fact deterministically realisable from a Toffoli gate(see 13). It allows us to create some interesting states experimentally, for instance, post-selection of theoutput to |0〉 would yield the state |00〉 + |01〉+ |10〉. (See the course notes [10] and Corollary 43 for moreon how these techniques can be used as an experimental prescription to generate quantum states.)

= =

(a) (b)

time

time

FIG. 1. Example of the Boolean quantum AND-state. In (a) the network is run backwards (post-selected) to |1〉resulting in the product state |11〉. In (b) the gate is post-selected to |0〉 resulting in the entangled state |00〉+ |01〉+|10〉.

In this work however, the physical interpretation of such states is of less interest: we are concerned withnetwork constructions as this allows us to study many-body states in new ways and elevate the existingtensor network theory by creating a theory of categorical tensor network states.

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For starters, we can compose AND-states (by connecting wires etc.) — together with NOT-gates, thisenables one to create the class of Boolean states (3). That is, we can realise a network that outputs logical-one any time the input qubits represent a desired term in a quantum state (e.g. create a function thatoutputs say logical-one on designated inputs |00〉, |01〉 and |10〉 and zero otherwise). We then insert a |1〉 atthe network output (physically you might call this post-selection, but again a physical interpretation is notneeded for our purposes). This recovers the desired Boolean state

1

2n/2

∑

x1,x2,...,xn=0/1

〈1|f(x1, x2, ..., xn)〉|x1, x2, ..., xn〉 (2)

where in terms of a network, we read the network backwards from output to input (a related idea arose inmy work on adiabatic circuits [21]). This full class of Boolean states is defined as:

Definition 2 (Boolean Many-Body Quantum States). We define the class of Boolean states as those stateswhich can be expressed up to a scalar in the form

∑

x1,x2,...,xn=0/1

|x1, x2, ..., xn〉|f(x1, x2, ..., xn)〉 (3)

where f is a switching function and the abusive notation in the sum is over all variables taking 0 and 1.

Examples of Boolean states include the familiar GHZ -state |00...0〉+|11...1〉 and W-state |00...1〉+|01...0〉+...+ |10...0〉. Our full method to decompose any quantum state into a tensor network subsumes the importantclass of Boolean states as a subclass. In fact, we’re able to translate any quantum state directly into acategorical tensor network. This appears to open a door: a new and different research direction in quantumnetwork theory by providing a new handle on quantum states. This is captured by the following result (seeTheorem 35).

Result 3 (Translating Quantum States into Categorical Tensor Networks). Given quantum state |S〉, Theo-rem 35 asserts a constructive method to represent |S〉 in a categorical tensor network with rank-3 and rank-2tensors.

An attractive aspect of our approach is that we’re able to place the network components into clearlydefined building blocks. Indeed, these building blocks are defined in terms of a rich graphical language —we utilize the theory of monoidal categories [1] and related ideas in computer science for this.

B. Network components fully defined by diagrammatic laws

The theory of categories provides a framework to elevate diagrammatic reasoning to a rigorous tool —e.g. proofs can be done graphically! We will in addition, use this framework to fully define the algebraicoperations appearing in this work, and this definition will be done graphically. This picture calculus can beused whenever working in the the dagger-category (that is †-category) of von Neumann quantum mechanics(for details see [18, 19]).To get an idea of how this will work, consider Figure 2, which forms a presentation of the linear fragment

of the Boolean calculus [14, 22]: that is, the calculus of Boolean algebra we represent on quantum states,restricted to the building blocks that can be used to generate linear Boolean functions.To recover the full Boolean-calculus, we must consider a non-linear Boolean gate: we use AND-gates and

Figure 2 together with Figure 3 to form a full presentation of the calculus [14] (the present work seems tobe the first to apply the full Boolean calculus to quantum physics, where the linear fragment was appliedin [22]). As stated, in this work, we will give the Boolean-calculus a representation (on quantum states) andmake use of the categorical generalisation of map-state duality found first in [19, 23], and which we studiedin the setting of quantum circuits and called cup/cap induced duality in [24].

Remark 4 (Full Set of Defining Equations). We note that the presentations in Figure 2 together with Figure 3are not just a set of relations and identities on circuit components, but instead represent a complete set ofdefining equations [25]. To our knowledge, the representation of the calculus in 3 on tensors and quantumstates is new to the physics communities concerned with diagrammatic reasoning (e.g. those working oncategorical quantum theory as well as those working on tensor network states).

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(a) (b) (c)

==

(d)

=

=

= =

=

=

=

=

=

(e)

(f)

(g)

FIG. 2. Read top to bottom. A presentation of the linear fragment of the Boolean calculus. The plus (⊕) dots areXOR and the black (•) dots represent COPY. The details of (a)-(g) will be given in Sections IV and V. For instance,(d) represents the bialgebra law and (g) the Hopf-law (in this case true as x⊕ x = 0).

(a) (b) (c) (d)

=

=

=

=

(f) (h)

==

=

=

= =

(g)(e)

FIG. 3. Read top to bottom. A presentation of the Boolean-calculus with Figure 2. The details of (a)-(g) will begiven in Sections IV and V. For instance, (h) represents distributivity of AND(∧) over XOR (⊕), and (d) shows thatx ∧ x = x.

We need to add a bit more to the presentation of the Boolean calculus to represent quantum states.Proceeding systematically by adding just a bit more structure we’re able to do a whole lot more. One wayforward is to add what are called compact structures in category theory [19, 22, 23] (see Section VC fordetails). These compact structures are given diagrammatically as

(a) (b)time

and this allows us to define the transpose of a linear map/state. We understand (a) as a cup, given as∑

i |ii〉and (b) as the so-called cap which is

∑

i 〈ii|.A second way forward is to consider what’s called a Frobenius form [26] on either of the structures in the

linear fragment (COPY or XOR). This is simply a functional that turns a product/coproduct into a cup/cap.This allows one to recover the above compact structures (that is, the cups and caps given above). We willuse these cups and caps to bend wires in tensor network’s which in turn is thought of as reshape of a matrix.

=

+

=0

(a) (b)

Remark 5 (Bending Wires is Transpose). We note here (see Definition 29) that care must be taken, asflipping a ket |ψ〉 to a bra 〈ψ| is conjugate transpose, and bending a wire is simply transposition, so theconjugate must be taken: e.g. acting on |ψ〉 with a cap given as

∑

i 〈ii| results in 〈ψ|. See Section VC.

Compact structures allow us to bend wires — indeed, we can now connect a diagram, bend all the wirestowards the same direction and it then can be thought of as representing a state, bend them the otherway and it then can be thought of as representing a measurement outcome, that is, an effect. One canalso connect inputs to outputs, creating larger and larger networks. What’s more is that these compact

6

structures have a vivid physical meaning in terms of an algebra on quantum states. A less user friendlyvariant of this algebra on quantum states is already present in categorical quantum theory [22] — it wascleaned up and made more user friendly in [10]. We will clarify this algebraic structure on quantum statesand explain its physical meaning next. We however again mention that most of what we consider here issimply an abstract network theory, and the physical interpretations of these operations, and their interactionsin terms of quantum mechanics, is not really necessary for applications in numerical algorithms involvingtensor network simulation.

C. A new type of algebra on quantum states

We are concerned with a network theory of quantum states. This on the one hand can be used as a toolto solve problems about states and operators in quantum theory, but does have a physical interpretation onthe other. This is largely based on what you might call an operational interpretation of quantum states andprocesses. An algebra is a pairing on a vector space, taking two vectors and producing a third (you mightcall it a monoid if there is a unit). Let’s now see how every tripartite quantum state forms an algebra.Consider a tripartite quantum state (subsystems labeled 1,2 and 3), and ask the simple question, how

would the state of the third system change after we measure systems one and two? Enter Algebras: asstated, an algebra on a vector space, or on a Hilbert space is formed by a product taking two elements fromthe vector space to produce a third element in the vector space. Algebra on states can then be studied byconsidering duality of the state, that is considering the adjunction between the maps of type

1 → H⊗H⊗H and H⊗H → H (4)

This duality is made evident by using the †-compact structure of the category (e.g. the cups and caps). Itis given vivid physical meaning by considering the effect measuring (that is two events) two components ofa state has on the third component.

Remark 6 (Overbar notation on Spaces). Given a Hilbert space H, we can consider the Hilbert space Hwhich can be simply thought of as the Hilbert space H will all basis vectors complex conjugates (overbar).That is, H is a vector space whose elements are in one-to-one correspondence with the elements of H:

H = v | v ∈ H, (5)

with the following rules for addition and scalar multiplication:

v + w = v + w and αv = αv . (6)

Remark 7 (Definition of Algebra). We consider an algebra as a vector space A endowed with a product,taking a pair of elements (e.g. from A ⊗ A) and producing an element in A. So the product is a mapA⊗A → A, which may not be associative or have a unit (that is, a multiplicative identity — see 18 for anexample of an algebra on a quantum state without a unit).

Observation 8 (Every tripartite Quantum State Forms an Algebra). Let ψ ∈ H ⊗ H ⊗ H be a quantumstate and let Mi, Mj be complete sets of measurement operators. Then (ψ,Mi,Mj) forms an algebra.

= := =

time

The quantum state is drawn as a triangle, with the identity operator on each subsystem acting as timegoes to the right on the page. Projective measurements with respect to Mi and Mj are made. We define

7

these complete measurement operators as

M1 =

N∑

i=1

i · |ψi〉〈ψi| (7)

M2 =N∑

j=1

j · |φj〉〈φj | (8)

such that we recover the identity operator on the N -level subsystem viz

N∑

j=1

|φj〉〈φj | =N∑

i=1

|ψi〉〈ψi| = 1N (9)

The measurements result in eigenvalues i, j leaving the state of the unmeasured system in

|ω〉 =∑

x1,x2,x3

〈ψi|x1〉〈φj |x2〉|x3〉 (10)

where 〈Q| def= |Q〉⊤ that is, the transpose is factored into: (i) taking the dagger (diagrammatically this

mirrors states across the page) and (ii) taking the complex conjugate. Hence,

〈Q|† = |Q〉⊤ = 〈Q| = 〈Q|† (11)

and if we pick a real valued basis for x1, x2, x3 we recover

|ω〉 =∑

x1,x2,x3

〈x1|ψi〉〈x2|φj〉|x3〉 (12)

As stated, this physical interpretation is not our main interest. It’s a nice feature, but even in its absence,we’re able to write down and represent a quantum state purely in terms of a connected network, where eachcomponent is fully defined in terms of algebraic laws.

D. Putting it all together: connecting the dots

This new formalism allows us to express a range of new a priori hidden tensor network structure. Indeed, aswe mentioned categorical tensor network states allow us to both “zoom out” and expose high-level structure,but also to “zoom in” and expose hosts of algebraic structures that are not currently being considered inthe tensor network simulation community. As will be shown, by formally defining these network buildingblocks, we’re able to see a lot more of what’s going on inside these networks. Importantly, we’re able to dothings that are not possible using the current approach to tensor network states: translate a given quantumstate directly into a representative network. This provides a quantum network analog of classical networkdecomposition methods.We hope that presenting a solution to the quantum decomposition problem and that by enhancing the

graphical language component of these numerical methods, that our work will lead to new theoretical modelsand numerical algorithms which will challenge and shape our understanding of many-body physics.

III. REMAINING MANUSCRIPT STRUCTURE

We have organized this manuscript in the following way: We continue next in IV by defining our networkbuilding blocks including rank-3 tensors such as defining the quantum AND-state in Equation 18. We thenconsider how these components interact in Section V. This is done in terms of algebraic laws, such asBialgebra (VB) and Hopf-algebras (VB1). With these definitions in place, we’re able to continue onto

8

Section VI: we applying this framework to create a new type of tensor network theory. We zoom in andexpose internal structure of an MPS and particularly consider categorical tensor networks for many-bodyW-states in VI. An example categorical MERA network along with reduced two- and four-point correlationfunctions (graphically in our language) are given in Section VID. In conclusion, we mention some futuredirections in VII and importantly, how this work opens the door to apply tensor network simulation methodsto NP-complete problems. We have included Appendix A on the Boolean XOR-algebra we represent onquantum states.a. Assumed Background. We assume readers are familiar with the basics of tensor network states (see

the reviews in [7, 27]). We have gone out of our way to make the category theory necessary in this work asuser friendly as possible. For general background see [1] and for more related work see [10, 24]. In a furtherattempt to make this document readable across the range of people working on these topics, we assume onlyminimal knowledge of Boolean algebra, discrete set functions and circuit theory (see a quick review of XOR-algebra in Appendix A and more generally see [28] and [29] for background on pseudo Boolean functions).We assume readers have experience with quantum circuits and basic quantum computing concepts (e.g. atthe level of [10, 30, 31]).

Remark 9 (Normalisation factors omitted). As one would expect in quantum theory, where rays describe thestate space, without loss of generality we will often omit global scale factors mainly for ease of presentation.We note that for Hilbert space H there is a truly natural isomorphism

C⊗H ∼= H ∼= H⊗ C (13)

so the isomorphism of the ⊗ of a scalar M and a vector S is given by multiplication.

Remark 10 (Diagrammatic conventions: top to bottom and left to right). Diagrams will typically be drawnwith ‘time’ going down the page. However, in certain instances we will draw them from left to right across thepage to aid in presentation. We note that in general open legs can be attached to other open legs (contracted)and that nodes, maps, etc. all have evident meaning, which should be clear from context.

IV. CONSTITUENT NETWORK COMPONENTS

Any vector space V has a dual V∗: this is the space of linear functions f from V to the ground field C,that is f : V → C. This defines the dual uniquely. We must however fix a basis to identify the vector spaceV with its dual. Given a basis, any basis vector ei in V gives a basis vector f j in V∗ defined by f j(ei) = δji(Kronecker’s delta). This defines an isomorphism V → V∗ sending ei to f i and allowing us to identify Vwith V∗. In what follows, we will fix a particular arbitrarily chosen basis (called the computational basis inquantum information science). We will proceed to give only the necessary building blocks that are neededin our construction.

A. COPY: the “diagonal”

The copy operation arises in digital circuits and more generally, in the context of category theory andAlgebra, where it is called a diagonal in cartesian categories. (although not directly relevant for the presentwork, see [32] for details on using COPY to define a basis). The operation is defined as

def=

∑

i

|ii〉〈i| (14)

where the sum is over ∀i which could be, e.g. iterating a complete Boolean basis: for qubits, that is i = 0, 1.As |0〉 and |1〉 are eigenstates of σz, we might give the alternative name of Z-copy (this was done in [22, 32]when considering COPY as a quantum observable) — which in the case of qubits is succinctly presented byconsidering the map that copies σz-eigenstates:

: C2 → C2 ⊗ C

2 ::

|0〉 7→ |00〉|1〉 7→ |11〉

9

This map can be written as : |00〉〈0| + |11〉〈1| and under cup/cap induced duality (on the right bra)this state becomes a GHZ -state as ψGHZ = |000〉 + |111〉. The standard properties of COPY are givendiagrammatically in Figure 4 and a list of its relevant mathematical properties are found in Figure 5.

(a) (b)

=

(c)

==

(d)

=

time

time

tim

e

=

FIG. 4. The COPY-dot. (a) Full-symmetry. (b) Copy points, e.g. |x〉 7→ |xx〉 for x = 0, 1. (c) The unit — in this

case the unit corresponds to deletion, or a map to the terminal object |+〉 def= |0〉 + |1〉 (the bi-direction of time is

explained in by considering co-diagonals in IVE). (d) Co-interaction with the unit creates a Bell state. This is thecompact structure of the †-category of quantum theory.

Remark 11 (The COPY-gate from CNOT). The CNOT-gate is defined as |0〉〈0|1⊗12+ |1〉〈1|1⊗σx2 . We will

set the input that the target acts on to |0〉 we then calculate CNOT(11 ⊗ |0〉2) = |0〉〈0|1⊗ |0〉2+ |1〉〈1|1 ⊗ |1〉2.We have hence defined the desired map (COPY) from the Hilbert space with label 1 (subscript) to the jointHilbert space labeled 1 and 2.

Gate Type Co-copy point(s) Unit Co-unit InteractionCOPY |0〉,|1〉 (b) |+〉 (c) Bell state: |00〉 + |11〉 (d)

Symmetry Associative Commutative Frobenius AlgebraFull (a) Yes Yes Yes (Spider Law)

FIG. 5. Summary of the COPY-gate.

B. XOR: the “addition”

The XOR-gate logic gate that implements exclusive disjunction or addition (mod 2) — written with symbol⊕. By what could be called “dot-duality”, the XOR-gate is simply a Hadamard transform of the COPY-gate,applied to all of the dots legs. This can be captured diagrammatically in the slightly different form

=

which clarifies several examples. To define the gate on the computational basis, we consider f(x1, x2) =x1 ⊕ x2 then f = 0 corresponds to (x1, x2) = (0, 0), (1, 1) and f = 1 corresponds to (x1, x2) = (1, 0), (0, 1),where the truth table for XOR follows

x1 x2 f(x1, x2) = x1 ⊕ x20 0 00 1 11 0 11 1 0

Under cap/cap induced duality, the state defined by XOR is given as

ψ⊕def=

∑

x1,x2

|x1〉|x2〉|f(x1, x2)〉 = |000〉+ |110〉+ |011〉+ |101〉 (15)

10

which is in the GHZ -class — by LOCC equivalence viz. ψ⊕ = H⊗ H⊗ H(|000〉+ |111〉). The operation ofXOR is summarized in the table appearing in Figure 6. Since the XOR-gate is related to the COPY-gate by achange of basis its diagrammatic laws have the same structure as those already appearing in Figure 4. Thegate acting backwards (co-XOR) is defined on a basis as follows:

⊕ : C2 → C2 ⊗ C

2 ::

|0〉 7→ |00〉+ |11〉|1〉 7→ |10〉+ |01〉 or equivalently

|+〉 7→ |++〉|−〉 7→ | − −〉

Gate Type Co-copy point(s) Unit Co-unit InteractionXOR |+〉,|−〉 (b) |0〉 (c) Bell state: |00〉 + |11〉 (d)

Symmetry Associative Commutative Frobenius AlgebraFull (a) Yes Yes Yes (Spider Law)

FIG. 6. Summary of the XOR-gate.

C. The constant 1: negation

Linear Boolean functions, are functions which have uncomplimented variables that appear individually(e.g. variable couplings are not allowed such as x1x2 etc. see A). Linear functions take the general form

f(x1, x2, ..., xn) = c1x1 ⊕ c2x2 ⊕ ...⊕ cnxn (16)

where the vector (c1, c2, ..., cn) determines the function. The affine boolean functions are linear functionsthat allow variables to appear in both complimented and uncomplimented form. Affine functions take thegeneral form

f(x1, x2, ..., xn) = c0 ⊕ c1x1 ⊕ c2x2 ⊕ ...⊕ cnxn (17)

where c0 = 1 gives functions outside the linear class. Together, XOR and COPY are not universal for classicalcircuits. However, When used in conjunction, XOR- and COPY-gates compose to create the class of linearcircuits. The affine circuits are generated by considering constant |1〉. This point (|1〉) is indeed copied bythe black dot. However, an axomitisation can proceed through only considering the XOR- and COPY-gatestogether with |+〉, the unit for COPY and |0〉 the unit for XOR. It is by appending the constant |1〉 into thesystem that the affine class of circuits can be realised.

Remark 12 (Affine functions correspond to a basis). Each affine function is labeled by a correspondingbit pattern. This forms a function basis for the space of Boolean polynomials and can also be thought of aslabeling the computational basis (see A).

D. Quantum AND-states: enter Boolean non-linearity

AND (that is, ∧) implements logical conjunction. By what could be called “dot-duality”, the AND-gaterelates to the OR-gate via De Morgan’s law. This can be captured diagrammatically as

=

To define the gate on the computational basis, we consider f(x1, x2) = x1 ∧ x2 which we write as x1x2.Here f = 0 corresponds to (x1, x2) = (0, 0), (0, 1), (1, 0) and f = 1 corresponds to (x1, x2) = (1, 1), wherethe truth table for AND follows

x1 x2 f(x1, x2) = x1 ∧ x20 0 00 1 01 0 01 1 1

11

Under cap/cap induced duality, the state defined by AND is given as

ψ∧def=

∑

x1,x2

|x1〉|x2〉|f(x1, x2)〉 = |000〉+ |010〉+ |010〉+ |111〉 (18)

The operation of AND is summarized in the table appearing in Figure 6. The key diagrammatic propertiesare presented in Figure 7.

∧ : C2 → C2 ⊗ C

2 ::

|0〉 7→ |00〉+ |01〉+ |10〉|1〉 7→ |11〉 or

|+〉 7→ |++〉|−〉 7→ |00〉+ |01〉+ |10〉 − |11〉

(a) (b)

=

(c)

==

(d)

time

tim

e=

tim

e

FIG. 7. The AND-gate. (a) Input-symmetry. (b) Existence of a zero or fixed-point. (c) The unit |1〉. (d) Co-interaction with the unit creates a product-state. Note that the gate forms a valid quantum operation when runbackwards as in (d).

Example 13 (AND-states from Toffoli-gates). The AND-state is readily constructed from the ‘ as illustratedin Figure 8.

FIG. 8. Illustrates the use of compact structures for black and plus dots to prepare the state ψAND = |000〉+ |010〉+|100〉+ |111〉. Using only single qubit NOT-gates, one can use this method to construct any of the states representingthe non-linear Boolean functions in Figure 9. We note that the box around the Toffoli gate (left) is meant to illustratethat those two connected dots do not satisfy the spider-law 22.

Gate Type Co-copy point(s) Unit Co-unit InteractionAND |1〉 (b) |1〉 (c) Product state: |11〉 (d)

Symmetry Associative Commutative Bialgebra LawInputs (a) Yes Yes Yes (with GHZ )

Remark 14 (Universal States). We should note that quantum universal diagrams are possible by consideringsimple Hadamard states (e.g. ψH = |00〉+ |01〉+ |10〉 − |11〉) and AND-states. This follows from the simpleproof that Hadamard and Toffoli are quantum universal [33].

E. co-COPY: the co-diagonal

It should be evident from the preceding discussions that our gates are what a physicist would call tensors(with the evident graphical interpretation apparently first pointed out in [34]) and that open legs on tensor

12

correspond to say spin degrees of freedom (and are hence either states or dual to states by bending wires).In this manner, we say that gates can be used both forwards in backwards in time.We already mentioned in the results summary that we utilize the †-compact structure from categorical

quantum theory to take the adjoint of a linear map. This let’s us take the transpose (e.g. bend wires). Whathappens if we flip a copy operation upside down, that is, instead of having a single leg split into two legs,have two legs merge into one. The first thing one might ask is if this is physical?Appending a physical interpretation to these operations in terms of a quantum process is possible, by

considering, e.g. post-selection, but not necessary for our purposes. Indeed, this is not our goal here aswe’re concerned with representing states in terms of categorical tensor networks — we expose an elegant,user friendly language to accomplish just that. So the co-COPY is simply thought of as a being a dual(transpose) to the familiar COPY.This is common in algebra: to consider the dual notation to algebra, that is co-algebra. In general, while

a product is a joining or paring (e.g. taking two vectors and producing a third) a co-product is a co-pairingtaking a single vector in say A and producing a vector in A⊗A.

Remark 15 (Coalgebras). Coalgebras are structures that are dual (in the sense of reversing arrows) tounital associative algebras such as COPY and AND the axioms of which we formulated in terms of picturecalculi (IVA and IVD). Every coalgebra, by (vector space) duality, gives rise to an algebra, and in finitedimensions, this duality goes in both directions.

Co-COPY can be thought of as applying a delta function in the transition from input to output. That is,given a copy point x

(|x〉) = |x〉 ⊗ |x〉 (19)

we have that

(|i〉, |j〉) = δij |i〉 (20)

that is, the diagram get’s mapped to zero (or empty) if the inputs don’t agree. This is succiently written interms of a Delta-function dependent on inputs i, j.

Example 16 (Simple co-pairing). Measurement effects on tri-state quantum systems can be thought of asa coproducts. This is given as a map from one system (measuring the first) into two systems (the effect thishas on the other two). GHZ -states are prototypical examples of co-pairings: an example left to the readerto explore.

F. The remaining Boolean states: NAND-states etc.

We have represented a complete logical system on quantum states — this enables us to represent anyBoolean function quantum mechanically and hence any Boolean state. We chose as our generators, constant|1〉, COPY, XOR, AND. Other generators could have also been chosen such as NAND-states. Our choicehowever, was made as a matter of convenience, as the definitions work well, and elegantly fit together (e.g.representing the XOR-algebra). If we would have considered other generators, we could have ended upconsidering the following cases: weak-units (17) and fixed point pairs (19). Note that NAND-states wereused in [35] for fault-tolerant quantum computation — see also [36].

Definition 17 (Weak Units). An algebra (or product) on a tri-party state ψ has a unit (equivalently thestate is unital) if there exists an effect φ which the product acts on to produce an invertible map B, whereB = 1 (see Example 18). If no such φ exists to make B = 1, and B has an inverse, we call φ a weak unit,and say the state ψ is weak unital and if B 6= 1 and B2 = 1 we call the algebra on ψ unital-involutive. Thisscenario is given diagrammatically as:

= =

13

Example 18 (NAND and NOR). NAND and NOR have weak units, respectively given by |1〉 and |0〉. Theseweak units are unital-involutive.

ψNAND = |001〉+ |011〉+ |011〉+ |110〉 (21)

ψNOR = |001〉+ |010〉+ |100〉+ |110〉 (22)

For ψNAND to have a unit, there must exist a |φ〉 such that

〈φ|0〉|01〉+ 〈φ|0〉|11〉+ 〈φ|0〉|11〉+ 〈φ|1〉|10〉 (23)

is equal to the Bell-state |00〉+ |11〉 and hence dual to |1〉〈1|+ |0〉〈0|. No choice of |φ〉 makes this possible.

Definition 19 (Fixed Point Pair). An algebra on a tri-party state ψ has a Fixed Point if there exists aneffect φ making the output constant. If the constant is output and φ are both |0〉, we say φ has a zero. Afixed point pair consists of two algebras with fixed points, such that the fixed point of one algebra is the unitof the other, and vise versa (see Example 20). Diagrammatically this is expressed in the following:

==

Example 20 (AND, OR form a Fixed Point Pair). AND and OR form a fixed point pair. That is, the unitfor AND (|1〉 see a) is the zero for OR (c) and vise versa: the unit of OR (|0〉 see a) is the zero for AND (b).

(a) (b) (c)

== = =

G. Summarizing: Network Composition of Quantum Logic States

We have considered set’s of universal classical structures in the categorical tensor network model. Inclassical computer science, a universal set of gates, is able to express any n-bit Boolean function

f : Bn → B :: (x1, ..., xn) 7→ f(x1, ..., xn) (24)

Universal sets include COPY, NAND, COPY, AND, NOT, COPY, AND, XOR,1, OR, XNOR,1 andothers. One can also consider the states ψ formed by the bit patterns of these functions f(a, b) as

ψf =∑

a,b∈0,1|a〉|b〉|f(a, b)〉 (25)

This allows a wide class of states to be constructed effectively. In the following Table (9) we illustrate thestates representing the classical function of two-inputs.

Remark 21 (Induced compact structure). The Boolean states in Table (9) represent true tri-state entan-glement. For each state, there exists an effect (a measurement outcome) on one of the states that leavesthe other two parties in an entangled state. Mathematically, this entangled state defines what’s called anon-degenerate pairing.

14

non-linear linear (Frobenius Algebras)ψAND = |000〉 + |010〉 + |100〉 + |111〉ψOR = |001〉 + |011〉 + |101〉 + |111〉 ψXOR = |000〉 + |011〉 + |101〉 + |110〉ψNAND = |001〉 + |011〉 + |101〉 + |110〉 ψXNOR = |001〉 + |010〉 + |100〉 + |111〉ψNOR = |001〉 + |010〉 + |100〉 + |110〉

FIG. 9. The bit pattern of these quantum states represents a Boolean function (given by the subscript) such thatthe right most bit is the Boolean functions output, and the two left bits are the functions inputs, and the non-linearBoolean functions are on the left side of the table and the linear functions on the right. Consider the state ψAND, andBoolean variables x1 and x2, then the superposition ψAND encodes the function |x1, x2, x1 ∧ x2〉 in each term in thesuperposition, and ψAND =

∑x1,x2∈0,1 |x1, x2, x1 ∧ x2〉. As outlined in the text, cup/cap induced-duality allows us

(for instance) to express this state as the operator |0〉〈00| + |0〉〈01| + |0〉〈01| + |1〉〈11| :: |x1, x2〉 7→ |x1 ∧ x2〉 whichprojects qubit states to the AND of their bit value.

V. INTERACTION OF THE NETWORK COMPONENTS

1. Merging Dots: Spider Law

Copy dots are readily generalized to an arbitrary number of input and output legs. As one would rightlysuspect, a copy dot with n inputs and m outputs corresponds to an n+m-partite GHZ state. Neighboringdots of the same color can be merged into a single dot: just like in digital circuits. These COPY-dotsrepresent Frobenius algebras [26, 37].

Theorem 22 (Spider Law [22]). Given a connected graph with m inputs and n outputs comprised solely ofFrobenius dots of equal dimension, this map can be equivalently expressed as a single m-to-n dot, as shownin Figure 10.

Example 23 (Two-site reduced density operator of n-party GHZ -states). GHZ -states on n-parties have awell known matrix product expression given as

GHZn = Tr

(

|0〉 00 |1〉

)n

= |00...0〉+ |11...1〉 (26)

where the internal matrix product is given by⊗. These MPS networks are known to be efficiently contactable.We note that the networks in Figure 10 are not a priori in a contractible form due to the number of of openlegs. What makes them contractible (in their present from) is the spider law. The reduced density matrixof an n-party GHZ -state then becomes (a) in Figure 11 and the expectation value of an observable is shownin (b). where we include the normalisation constant.

FIG. 10. Spider law: connected black-dots (•) as well as connected plus-dots (⊕) can be merged.

A. Associativity, Distributivity and Commutativity

The products we have considered are all associative and commutative. As algebras, AND, XOR and COPY

are associative, unital commutative algebras. This was already expressed diagrammatically in Figures 1 (a)

15

=

(a) (b)

=

FIG. 11. Reduced density operator. Left (a) reduced density operator ρ′GHZ found from applying the spider law toa n-qubit GHZ -state. Right (b) the expectation value of observable O1 ⊗ O2 found from connecting the observableand connecting the open legs (e.g. taking the trace).

and 3 (c). These diagrammatic laws represent the following Equations:

(x1 ∧ x2) ∧ x3 = x1 ∧ (x2 ∧ x3) (27)

(x1 ⊕ x2)⊕ x3 = x1 ⊕ (x2 ⊕ x3) (28)

Distributivity of AND over XOR then becomes (see (h) in Figure 3)

(x1 ⊕ x2) ∧ x3 = (x1 ∧ x2)⊕ (x1 ∧ x2) (29)

We of course have commutativity for any product symmetric in its inputs: this is the case for AND and XOR.

B. Bialgebras

There is a very powerful type of algebra that arises in our setting: a bialgebra (See Kassel, Chapter III [38],or [26]). Such an algebra is simultaneously an unital associative algebra (for the associativity condition see(b) in Figure 12)and coalgebra and are characterized by a compatibility condition. We consider the followingingredients:

(i): a product (black dot) with a unit (black triangle) see Figure 12 (a)

(ii): a coproduct (white dot) with a counit (white triangle)

precisely, the four compatibility conditions are satisfied if the following holds:

(i): The unit of the black dot is a copy-point of the white dot as in (e) from Figure 12.

(ii): The (co)unit of the white dot is a copy-point of the black dot as in (d) from Figure 12.

(iii): The bialgebra-law is satisfied given in (c) from Figure 12.

(iv): The inner product of the unit (black triangle) and the counit (white triangle) is non-zero (not shownin Figure 12).

= = ==

(a) (d)(c)

=

(b) (e)

= =

FIG. 12. Bialgebra axioms. (a) unit laws (these are of course left and right units); (b) associativity; (c) bialgebra;(d,e) co-copy points.

Example 24 (GHZ -AND form a bialgebra). We are in a position to study the interaction of GHZ -AND.This interaction satisfies the equations in the following diagrams: (a) the bialgebra law; (b) the co-copypoint of AND is |1〉; and (c) the co-interaction with the unit for GHZ creates a compact structure. Inaddition, (a,b) show the copy points for the black GHZ -dot; in (c) we have the unit and fixed point laws.

16

Even if two products don’t form bialgebras, they can still satisfy the bialgebra condition (and hence notsatisfy all of the axioms listed above). For this reason, so we define this law (examples of states that satisfythis law, but are not necessarily bialgebras are given in Example 26

Definition 25 (Bialgebra). A pair of quantum states (black, white below) satisfy the bialgebra law if thefollowing holds:

=

Example 26 (Boolean States from Bialgebras with COPY). The Boolean states, AND, OR, XOR, XNOR,NAND, NOR all satisfy the bialgebra law with COPY.

1. Hopf algebras

A particularly important class of bialgebras are known as Hopf-algebras [26]. This is characterized by theway in which algebras and coalgebras can interact. This is captured by the Hopf-law, where linear map A isknown as the antipode.

Definition 27 (Hopf-Law). A pair of quantum states satisfy the Hopf-Law if an A can be found such thatthe following equations hold:

= =

Example 28 (XOR and COPY are Hopf-algebras on Boolean States). It is well known (see e.g. [14]) thatthe Boolean state XOR, satisfies the Hopf-algebra law with trivial antipode with COPY.

C. Bending wires: Compact Structures

As mentioned in the preliminary section (II), we will introduce what’s called in category theory a compactstructure: this can be thought of as defining a non-degenerate pairing on a vector space, which allows us todefine transposition graphically. This problem was addressed in categorical quantum theory by consideringBell-states and their dual (this was key to axiomatizing the teleportation protocol [39, 40]). A secondapproach forward is by utilization of the induced compact structures contained in the linear fragment of theBoolean-calculus (e.g. the co-interaction of COPY with |+〉 results in a Bell-sate — see Section IVA).A compact structure on an object H consists of another object H∗ together with a pair of morphisms (note

that we use the equation H∗ = H in Hilbert space making objects self dual which simplifies what follows).

ηH : 1 −→ H⊗H ǫH : H⊗H −→ 1

where the canonical representation in Hilbert space with dimension N and basis |i〉 is given by

ηH =N∑

i=1

|i〉 ⊗ |i〉 ǫH =N∑

i=1

〈i| ⊗ 〈i|

and in string diagrams (read from the top to the bottom of the page) as

(a) (b)time

17

These cups and caps give rise to cup/cap-induced duality: this amounts to being able to create a linear mapthat “flips” a bra to a ket (and vise versa) and at the same time taking an (anti-linear) complex conjugate.Under cup/cap-induced duality, we flip the second ket on ηH and the first bra on ǫH to relate these mapsand the identity 1H of the Hilbert space: that is, we can fix a basis and construct invertible maps sendingηH ⋍ 1H ⋍ ǫH.More generally, the maps ηH and ǫH satisfy the following equations and their duals (under the dagger) in

the graphical language (b is known as the snake equation).

=

=

(a)

(b)f

fT

=

=

‒

= f

(c)

(d)

Definition 29 (Diagrammatic Adjoints). Cups and caps allow us to take the transpose of a linear map (b);and (a) following [19] we introduce the derived concept of adjoint.

=(a) (b)

=

VI. TRANSLATING ANY QUANTUM STATE INTO A CATEGORICAL TENSOR NETWORK

Typically only the converse is possible — that is, one determines a quantum state from a given tensornetwork or quantum circuit, or perhaps performs an optimization or renormalization procedure over a set ofnetwork parameters to find the network representing the state that best e.g. minimizes a given Hamiltonian.While tensor networks are in theory expressive enough to represent any quantum state, doing so will typicallynot expose additional internal structure (see the general from of a Matrix Product State in Figure 13). Onthe other hand, our new methods enable one to translate a quantum state directly into a new type of network:a so-called categorical tensor network. We have already presented the algebraic definitions and and definingproperties of these new components. Here we will illustrate their expressive power by considering a fewelementary examples before presenting our main theorem (35).

A. Extending the State of the Art

Tensor network states are in wide spread current use (see the reviews [7, 27]). The current approach doesnot expose much internal structure of the constituent tensors comprising a given network. Indeed, all MPS

states have essentially the same topological or network structure in the current incarnation (see Figure 13).There is however, ample internal structure to exploit. The current approach to write down a matrix productstate is ad hoc and via trial and error. For instance, the current approach shows little insight into why theW-state on n-qubits takes the form:

Wn = 〈0|(

|0〉 0|1〉 |0〉

)n

|1〉 = |10...0〉+ |01...0〉+ ...+ |00...1〉 (30)

or importantly, how to arrive at a tensor network for more complicated states. We will build on a specificexample, and show how our alternative approach reveals new found internal structure when representingquantum states in terms of a network.

18

FIG. 13. W-state on n-parties in the Matrix Product State formalism in wide spread current use (see (30)). Notethat the internal structure of the tensors themselves can not be exposed in the current formalism: all states in thisformalism have this same topological structure.

B. Example: W-states in the Categorical Tensor Network Formalism

W-states can arise in our framework in several ways. To help build a feeling for the general setting, considerthe following:

Example 30 (Functions on W- and GHZ -states). We consider the function fW which outputs logical-onegiven input bit string 001, 010 and 100 and logical-zero otherwise. Likewise the function fGHZ is defined tooutput logical-one on input bit strings 000 and 111 and logical-zero otherwise. See Examples (32) and (33)which consider representation of these functions as polynomials. We will of course continue to work with alinear representation of quantum states, where bit string 000 7→ |000〉 (etc.).

Remark 31 (Exact-value functions). The function fW takes value one on input vectors with k ones for afixed k. Such functions are known in the literature as a Exact-value symmetric Boolean functions.

Example 32 (Function Realisation of fW and fGHZ: the Boolean case). One can express

fW(x1, x2, x3) = x1x2x3 ⊕ x1x2x3 ⊕ x1x2x3 (31)

by noting that each term in the disjunctive normal form of fW are disjoint, and hence ∨ 7→ ⊕. The algebraicnormal form (see Appendix A) becomes

fW(x1, x2, x3) = x1 ⊕ x2 ⊕ x3 ⊕ x1x2x3 (32)

fGHZ(x1, x2, x3) = 1⊕ x1 ⊕ x2 ⊕ x3 ⊕ x1x2 ⊕ x1x3 ⊕ x2x3 (33)

Example 33 (Function Realisation of fW and fGHZ: the set function case). Set functions are mappings fromthe family of subsets of a finite ground set (e.g. Booleans) to the set of reals. In the Circuit Theory literature,functions from the Booleans to the reals are known as pseudo-Boolean functions and more commonly as multi-linear polynomials or forms (see [21] where these functions are used to embed logic gates in the ground stateenergy configuration of spin models). Their exists a unique multi-linear polynomial representation for eachpseudo-Boolean function found by mapping the negated Boolean variable as x 7→ (1 − x). For the GHZ -and W-functions defined in Example 30 we arrive at the unique polynomials (33) and (33).

fGHZ(x1, x2, x3) = 1− x1 − x2 + x1x2 − x3 + x1x3 + x2x3 (34)

fW(x1, x2, x3) = x1 + x2 + x3 − 2x1x2 − 2x1x3 − 2x2x3 + 3x1x2x3 (35)

These polynomials (33) and (33) are readily translated into categorical tensor networks.

Example 34 (Network realisation of W- and GHZ -states). A network realization of W- and GHZ -states inour framework then follows by post-selecting to |1〉 on the output bit — leaving the input qubits to representa W- or GHZ -state respectively. As example of this is shown in Figure 14.

19

=

tim

e

(a) (b)

=

time

FIG. 14. Left (a) the circuit realisation (internal to the triangle) of the function fW of e.g. (32) which outputslogical-one given input bit string |x1x2x3〉 = |001〉, |010〉 and |100〉 and logical-zero otherwise. Right (b) reversingtime and setting the output to |1〉 (e.g. post-selection) gives a network representing the W-state. See also Figure 15.

= =

(a) (b)

FIG. 15. W-class states in the categorical tensor network state formalism. (a) is the standard W-state. (b) is foundfrom applying De Morgan’s law (see Section IVD) to (a) and rearranging after inserting inverters on the output legs.See also Figure 16.

C. The General Case

A starting point of the classical network theory was seminal work resulting in Shannon and Davio decom-positions of functions into networks. These powerful methods formed the backbone and enabled the lastcentury of methods surrounding classical network theory, but no related methods to decompose a many-bodyquantum state into a tensor network had been found. We are now in a position to state the main theoremwhich provides a constructive method to realise any quantum state in terms of a categorical tensor network.In other words, we offer a solution to the quantum decomposition problem: translating a given state S intoa network representing the state S (see also Remark 38).

Theorem 35 (Network Representation of Quantum States). Fix a natural number n. Any quantum stateψ =

∑

i∈0,1n aie−iki |i〉 with ∀i, ai ∈ 0, 1 and 0 ≤ k < 2π can be represented as a network containing

tensors from the introduced quantum Boolean calculus together with states of the form |0〉 + α|1〉. Thisincludes all qubit states as an important subclass of representable states.

Network Representation of Quantum States 35. The proof is constructive and proceeds based on the contentof the main body of the text. The first step is to realise a function fS that outputs logical one on allinput bit strings corresponding to the desired state. Post selecting this network to |1〉 realises the desiredsuperposition of terms, but with all coefficients and hence relative phases equal. To adjust the phases andrelative amplitudes, we will construct diagonal operators. Given a term |k〉 in a state, with coefficient αk, weconstruct a function fd that outputs local zero for all inputs not equal to k, and logical one for input k. Thenetwork is then post selected to |0〉+ αk|1〉 and we transform fd into an operator by using COPY-dots fromSection IVE (see Figure 17 and Example 36). We note that the construction can be improved significantlyby considering several reductions. We of course group terms in the state with the same coefficients αi, butfurther reductions are also possible if say a given set of coffecients are given by products of other coffecients.This is illustrated by networks that take the form

20

FIG. 16. W-state (n-party) in the categorical tensor network state formalism. The feature of efficient networkcontraction remains, with the internal structure of the network components exposed in terms of well understoodstructures.

=

time

... ... ......

where we note that the fan-in present in the networks, can result in networks that are not thought tobe efficiently contactable. In addition, each of these networks gives a prescription to physically preparea state, however when fan-in is present, this prescription does not represent a deterministic process (seeCorollary 43).

Example 36 (Network realisation of S = |01〉+ |10〉+α|11〉). As a simple example, we will design a networkto realise the state |01〉+ |10〉+ α|11〉. We first write down a function fS such that

fS(0, 1) = fS(1, 0) = fS(1, 1) = 1 (36)

and fS(00) = 0 (in this case, fS is the logical OR-gate). We post select the network on |1〉, which results inthe state |01〉+ |10〉+ α|11〉, see Figure 17 (a). The next step is to realise a diagonal operator, that acts onidentity on all inputs, except |11〉 which gets sent to α|11〉. To do this, we design a function fd such that

fd(0, 1) = fd(1, 0) = fd(0, 0) = 0 (37)

and fd(1, 1) = 1 (in this case, fd is the logical AND-gate). This diagonal, takes the form in Figure 17 (b).The final state S = |01〉+ |10〉+ α|11〉 is realised by connecting both networks, leading to Figure 17 (c).

Remark 37 (Realisation of state (|0〉+ i|1〉)/√2). On a Blog post [41] I was asked for a network realisation

of the state (|0〉 + i|1〉)/√2. This proceeds from Example 36: except in this case we post select one of the

outputs (open legs in (c) from Figure 17) to (|0〉+ |1〉)/√2. We note that here we set fS to XOR, fd is again

AND with αdef= i. See Figure 18.

Remark 38 (Qubit States and Beyond). In Theorem 35 we considered the class of states of the formψ =

∑

i∈0,1n aie−iki |i〉. Using multivalued logic, it is possible to define a gate set similar to what was done

for the case of qubits, and construct a similar circuit.

21

(a) (b)=

=(c)

FIG. 17. Example of categorical tensor network representing state S = |01〉 + |10〉 + α|11〉. See Example 36 for fulldetails.

FIG. 18. Example of categorical tensor network representing state (|0〉 + i|1〉)/√2. See Remark 37 for details.

Remark 39 (Quantum Universality). The problem determining set’s of operations that can be used to defineuniversal quantum gate set’s has received a lot of research interest resulting in the surprising minimal gatesets appearing in [33, 42, 43]. In particular, it is even known that Toffoli and Hadamard are universal forquantum computation [33]. Toffoli can be generated by combining one AND-state and two COPY-states.Interestingly, the Hadamard gate can be generated from the AND-state together with |−〉 = 1√

2(|0〉 − |1〉) as

shown in Figure 19.

= H

FIG. 19. A Hadamard gate represented using the AND-state together with |−〉 def= 1√

2(|0〉 − |1〉).

D. Categorical MERA Networks and Solving SAT instances

In the previous sections we developed a powerful framework — we can use it to make seemingly dauntingcalculations elementary. As a token of the power, we will now consider examples of the presented calculusapplied to a categorical description of a MERA network and then in VIF explain how our approach enablesa rang of classical optimization problems (such as SAT) to be addressed by tensor contraction.

E. Categorical MERA Networks

The Multi-scale Entanglement Renormalization Ansatz (MERA) approach is a combination of the sem-inal ideas of Kadanoff’s spin-blocking, Wilson’s real-space renormalization and White’s DMRG procedure.Renormalization proceeds by coarse-graining lattice sites and truncating the description. DMRG’s success

22

is based on properly identifying the optimal truncation. The key feature of MERA is that it dramaticallyreduces information loss due to truncation by eliminating entanglement beforehand [6]. Repeating this en-tanglement renormalization procedure generates a hierarchical network, (shown below), where entanglementat different length scales is efficiently described. Within this structure the properties of quantum criticalsystems and emergent quantum phenomena are known to be efficiently computable.

In numerical algorithms, it is desirable to calculate correlation functions from the above network. We areinterested in comparing quantities such as < xixj > and < xi >< xj >. We can leverage our calculus tocomplete this task by noting that:

(⊕-dots): These dots are Hadamard transforms of the black COPY-dots and hence satisfy the evidentalgebraic properties: (i) the spider law (Theorem 22) and so can be merged into a single dot; (ii) theyform a bialgebra with COPY-dots (Section VB); (iii) they satisfy the Hopf-Law (with trivial antipode— Section VB); (iv) the unit of the ⊕-dot is |0〉 and its co-unit interaction leads to the familiar compactstructure.

(•-dots): These are the COPY-dots we have considered in Sections IVA and IVE which have all the sameproperties as above, with unit |+〉.

(AND-dots): AND-dots where defined in Section IVD. These dots correspond to quantum states thatare outside the stabilizer class and, as mentioned, have the following algebraic properties: (i) formbialgebras with COPY-dots; (ii) have unit |1〉; (iii) co-unit interaction that results in a copy-point; (iv)have a fixed point (|0〉); (v) as the dots form an associative algebra, there is no ambiguity in theirmerger.

Using the laws we have developed throughout this work, we immediately calculate these correlation functions.Correlator < xixj > is given in (a) and < xi >< xj > is from two copies of (b).

(a)(b)

We note that this drastic simplification corresponds to exact analytical expressions, and are independent oflength scales and hence the size of the original MERA-network. The quantum correlations in the (mixedstate) networks (a,b) can be studied by examining graph connections.

Remark 40 (Normalisation of the AND-state). We have defined the quantum AND-state as ψAND = |000〉+|010〉+ |100〉+ |111〉. We define the corresponding quantum logic tensor as |0〉〈00|+ |0〉〈01|+ |0〉〈10|+ |1〉〈11|and it’s dual is given by transposition |00〉〈0|+ |01〉〈0|+ |10〉〈0|+ |11〉〈1|. Care must then be taken to assurethat the appropriate composition of the original tensor and it’s dual map to the identity. This can be done byworking over the Booleans or by considering the state 1√

3(|000〉+ |010〉+ |100〉) + |111〉, it’s corresponding

quantum logic tensor and it’s dual under transposition.

23

F. SAT and read-once formula

In future work with Stephen Clark and Dieter Jaksch, we will study in detail how the presented calculusenables one to address Satisfiability and other related problems in terms of a network contraction. Indeed,the method leads immediately to a method to contract a function representing a SAT-instance: e.g. if thecontraction results in scalar zero, the function represents a NO instance. See Figure 20.

=tim

e(a) (b)

...

...

FIG. 20. Solving NP-complete problems by contracting a the categorical tensor network. (a) A SAT-formula realisedas a network. (b) contracting the network: if this contraction evaluates to one the SAT instance is satisfiable and ifit evaluates to zero it is not.

Remark 41 (MERA and read-once Boolean formula). The class of Boolean networks that only allow fan-in(e.g. no bit merging) are known as read-once formula or networks. MERA is a quantum version of this class(see also Table 21).

Corollary 42 (All read-once formula are SAT YES instances). Using the method described above for SAT(See Figure 20) it immediately follows that all read-once formula are satisfiable.

Corollary 43 (A prescription to realize any read-once quantum state deterministically). We call the class ofread-once Boolean quantum states as those states prepared by read-once binary networks as given in Figure 14where fW is a read-once formula and hence the network generates states encoding the constraint fW = 1 (seeExample 30). If fW is a read-once formula, it corresponds to a fanout only quantum network, and hence thisnetwork represents a deterministic process to realize the physical state corresponding to fW . This extendsto the evident way to quantum read-once states which are exactly the MERA class. See [20, 24] for moredetails.

VII. OUTLOOK AND CONCLUDING REMARKS

We have presented a solution to the quantum decomposition problem based on a representation of quantumstates in terms of categorical tensor networks. The expressiveness and power of this new method wasillustrated by considering several test cases: we unveiled hidden internal structure of MPS states (e.g. W-states) and illustrated the simplification power of these methods by considering our example applied toMERA-networks. We have opened up many future potential research directions. For instance, our methodsnow readily allow tensor network algorithms (which work by contracting tensors) to solve NP-completeproblems. We conclude by presenting a table (21) which summarizes some of the mathematical structuresthat are already present in the tensor networks community (and their corresponding categories) as well assome mathematical structures that arise as categorical tensor networks. I have plans to take this aspect ofthis work further in joint work with John Baez [25, 41].

ACKNOWLEDGMENTS

We thank John Baez, Stephen Clark, Dieter Jaksch, Martin Plenio and Mike Shulman. JDB receivedsupport from EPSRC grant EP/G003017/1 and completed large parts of this work visiting the Center forQuantum Technologies, at the National University of Singapore (these visits were hosted by Vlatko Vedral).

24

Categories Fan-in and Fan-out Only Fan-in

Many Types Symmetric Monoidal Category [17, 44] Symmetric MulticategoryOne Type PROP [45] Operad

Switching Networks Fan-in and Fan-out Only Fan-in

Many Types General Switching Networks General Read-once formulaOne Type General Boolean Networks [28] Read-once Boolean circuits

Tensor Networks Fan-in and Fan-out Only Fan-in

Many Types Categorical Tensor Networks Tree Tensor NetworksOne Type ? MERA Networks

FIG. 21. Table illustrating how the symmetric categories of interest fit together and their corresponding classicalnetwork, and tensor network. See also the related non-symmetric categories listed in Table 22.

Categories (no symmetry) Fan-in and Fan-out Only Fan-in

Many Types Monoidal Category MulticategoryOne Type PRO Planar Operad

FIG. 22. Table of the categories of interest without symmetry.

Appendix A: XOR-algebra

Here we review the concept of an algebraic normal form (ANF) on a boolean polynomial which is commonlyknown as PPRMs.

Definition 44. The XOR-algebra forms a commutative ring with presentation M = B,∧,⊕ where thefollowing product is called XOR

—⊕— : B× B 7→ B :: (a, b) → a+ b− ab mod 2 (A1)

and conjunction is given as

— ∧— : B× B 7→ B :: (a, b) → a · b. (A2)

One defines left negation ¬— in terms of ⊕ as ¬— ≡

(1⊕—) : B 7→ B :: a→ 1− a. (A3)

In the XOR-algebra, 1-5 hold. 1.) a ⊕ 0 = a, 2.) a ⊕ 1 = ¬a, 3.) a ⊕ a = 0, 4.) a ⊕ ¬a = 1 and 5.)a ∨ b = a ⊕ b ⊕ (a ∧ b). Hence, 0 is the unit of XOR and 1 is the unit of AND. The 5th rule reduces toa ∨ b = a⊕ b whenever a ∧ b = 0, which is the case for disjoint (mod 2) sums.

Definition 45. Any boolean equation may be uniquely expanded to the fixed polarity Reed-Muller formas:

f(x1, x2, ..., xk) = c0 ⊕ c1xσ1

1 ⊕ c2xσ2

2 ⊕ · · · ⊕ cnxσn

n ⊕cn+1x

σ1

1 xσn

n ⊕ · · · ⊕ c2k−1xσ1

1 xσ2

2 , ..., xσk

k , (A4)

where selection variable σi ∈ 0, 1, literal xσi

i represents a variable or its negation and any c term labeledc0 through cj is a binary constant 0 or 1. In Equation A4 only fixed polarity variables appear such that eachis in either un-complemented or complemented form.

Let us now consider derivation of the form from Definition 45. Because of the structure of the algebra,without loss of generality, one avoids keeping track of indices in the N node case, by considering the casewhere N ≡ 2n = 8.

Example 46. The vector c = (c0, c1, c2, c3, c4, c5, c6, c7, )⊺ represents all possible outputs of any function

f(x1, x2, x3) over the algebra formed from linear extension of Z2×Z2×Z2. We wish to construct a canonical

25

representation in terms of the vector c, where each ci ∈ 0, 1, and therefore c is a selection vector thatsimply represents the output of the function f : B × B × B → B :: (x1, x2, x3) 7→ f(x1, x2, x3). One mayexpand f as:

f(x1, x2, x3) = (c0 · ¬x1 · ¬x2 · ¬x3) ∨ (c1 · ¬x1 · ¬x2 · x3) ∨ (c2 · ¬x1 · x2 · ¬x3)∨(c3 · ¬x1 · x2 · x3) ∨ (c4 · x1 · ¬x2 · ¬x3) ∨ (c5 · x1 · ¬x2 · x3)∨(c6 · x1 · x2 · ¬x3) ∨ (c7 · x1 · x2 · x3) (A5)

Since each disjunctive term is disjoint the logical OR operation can be replaced with the logical XORoperation. By making the substitution ¬a = a⊕ 1 for all variables and rearranging terms one arrives at thefollowing canonical form:1

f(x1, x2, x3) = c0 ⊕ (c0 ⊕ c4) · x1 ⊕ (c0 ⊕ c2) · x2 ⊕ (c0 ⊕ c1) · x3 ⊕ (c0 ⊕ c2 ⊕ c4 ⊕ c6) · x1 · x2⊕(c0 ⊕ c1 ⊕ c4 ⊕ c5) · x1 · x3 ⊕ (c0 ⊕ c1 ⊕ c2 ⊕ c3) · x2 · x3⊕(c0 ⊕ c1 ⊕ c2 ⊕ c3 ⊕ c4 ⊕ c5 ⊕ c6 ⊕ c7) · x1 · x2 · x3 (A6)

The set of linearly independent vectors, x1, x2, x3, x1 · x2, x1 · x3, x2 · x3, x1 · x2 · x3 combined with a setof scalars from Equation A6 spans the eight dimensional space of the Hypercube representing the Algebra.A similar form holds for arbitrary N .

f(x1, x2, x3) = (a1) · x1 ⊕ (a2) · x2 ⊕ (x3) · x3 ⊕ (a1 ⊕ a2 ⊕ a1 ⊕ c2) · x1 · x2⊕(a1 ⊕ a3 ⊕ a1 ⊕ c3) · x1 · x3 ⊕ (a2 ⊕ a3 ⊕ a2 ⊕ c3) · x2 · x3⊕(a1 ⊕ a2 ⊕ a3 ⊕ a1 ⊕ a2 ⊕ a3) · x1 · x2 · x3 (A7)

Example 47. The Galois group: of every finite field extension of a finite field is finite and cyclic; conversely,given a finite field F and a finite cyclic group G, there is a finite field extension of F whose Galois group isG.

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